Sequences
which I have written in the OEIS
The OEIS is
the best resource for integer sequences on the internet and is maintained and
presided over by Neil Sloane with the assistance of an editorial panel of
around 70 mathematicians and computer programmers. It currently contains around
200000 sequences, and those listed below are my personal contributions (given
in no particular order):
1. 
Sum of the proper
infinitary divisors of n. 




2. 
Smaller member of
an infinitary amicable pair. 




3. 
Larger member of an
infinitary amicable pair. 




4. 
The number of
infinitary amicable pairs (i,j) with i<j and i<=10^n. 




5. 
Odd, infinitary abundant
numbers. 




6. 
Lengths of the
infinitary aliquot sequences generated by n. 




7. 
Sum of the proper exponential
divisors of n. 




8. 
Larger member of
each exponential amicable pair. 




9. 
Larger member of a reduced
infinitary amicable pair. 




10. 
Smaller member of
an augmented infinitary amicable pair. 




11. 
Larger member of an
augmented infinitary amicable pair. 




12. 
Number of augmented
infinitary amicable pairs (i,j) with i<j and i<=10^n. 




13. 
Odd integers that
do not generate monotonically decreasing infinitary aliquot sequences. 




14. 
Unitary abundance
of the integers. 




15. 
Numbers which are
the cube roots of the products of their proper divisors. 




16. 
The smaller member
of each exponential amicable pair. 




17. 
Integers whose
aliquot sequences terminate by encountering a prime number. 




18. 
Composite numbers whose
aliquot sequences terminate by encountering a prime number. 




19. 
Integers whose
aliquot sequences terminate by encountering the prime number 3 (aka the prime
family 3). 




20. 
Integers whose
aliquot sequences terminate by encountering the prime number 7 (aka the prime
family 7). 




21. 
Lengths of the exponential
aliquot sequences. 




22. 
Integers whose
exponential aliquot sequences end in an eperfect number. 




23. 
Exponential aspiring
numbers. 




24. 
Exponential
amicable numbers. 




25. 
Integers whose
exponential aliquot sequences end in an exponential amicable pair. 




26. 
Integers whose
infinitary aliquot sequences end in an infinitary perfect number. 




27. 
Infinitary aspiring
numbers. 




28. 
Infinitary amicable
numbers. 




29. 
Numbers whose
infinitary aliquot sequences end in an infinitary amicable pair (aka the
infinitary 2cycle attractor set). 




30. 
The smaller member
of a reduced infinitary amicable pair. 




31. 
Integers whose
unitary aliquot sequences terminate in 0, including 1 but excluding the other
trivial cases in which n is itself either a prime or a prime power. 




32. 
Unitary aspiring
numbers. 




33. 
Numbers that are
members of A048945 but are not members
of A111398. 




34. 
Highly abundant
numbers with an odd divisor sum. 




35. 
Unitary deficient
numbers. 




36. 
Consider primitive
Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B)
= 




37. 
Consider primitive
Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B)
= 




38. 
Consider all
Pythagorean triangles A^2 + B^2 = C^2 with A<B<C; sequence gives values
of B, sorted to correspond to increasing A (A009004 (n)). 




39. 
Consider all
Pythagorean triangles A^2 + B^2 = C^2 with A<B<C; sequence gives values
of C, sorted to correspond to increasing A (A009004 (n)). 




40. 
Integers whose
unitary aliquot sequences are longer than their ordinary aliquot sequences. 




41. 
Highly abundant
numbers that are not superabundant. 




42. 
Highly abundant
numbers that are not products of consecutive primes with nonincreasing
exponents, i.e. that are not of the form n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2>=e_3>=...>=e_p. 




43. 
Highly abundant
numbers that are not Harshad numbers. 




44. 
Digital roots of
the Mersenne primes. 




45. 
Perfect squares
that can be expressed as a sum of three consecutive triangular numbers. 




46. 
Numbers n not
divisible by 6 such that sigma(n)>3n. 




47. 
Numbers whose
unitary aliquot sequences end in a unitary amicable pair, but which are not
unitary amicable numbers themselves. 




48. 
Odd unitary
abundant numbers. 




49. 
Odd unitary
abundant numbers that are not odd, squarefree, ordinary abundant numbers. 




50. 
Unitary abundancy of nth unitary
abundant number: usigma(k)2k if this is >0. 




51. 
Records for unitary
abundant numbers, i.e. those integers which set a record for having a greater
unitary abundance than any of their predecessors. 




52. 
Infinitary abundant
numbers. 




53. 
Infinitary
deficient numbers. 




54. 
a(n)=a(n1)^22 with a(1)=10. This is the
LucasLehmer sequence with starting value 10. 




55. 
Number of digits in
the nth Cullen prime. 




56. 
Consider primitive
Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B)
= 




57. 
Trapezoidal numbers. 




58. 
The complement of
the trapezoidal numbers. 




59. 
Indices of the least
triangular numbers for which three consecutive triangular numbers sum to a
perfect square. 




60. 
Number of reduced
infinitary amicable pairs (i,j) with i<j and i<=10^n. 




61. 
Number of betrothed
pairs (m,n) with m <=10^k (and k=1,2,3,...),
where a betrothed pair satisfies sigma(m)=sigma(n)=m+n+1 and m<n. 




62. 
Number of augmented
amicable pairs (m,n) with
m<n and for which m<=10^k, k=1,2,3,... 




63. 
Number of unitary
amicable pairs (i,j) with i<j
and i<=10^n. 




64. 
Number of super
unitary amicable pairs (i,j) with i<j and i<=10^n. 




65. 
Number of primitive
exponential amicable pairs (i,j) with i<j and i<=10^n. 




66. 
Odd, doubly
abundant numbers. 




67. 
Exponential
abundant numbers. 




68. 
Digital roots of
the Fermat numbers. 




69. 
Prime values of k
for which k 2^n+1 is composite for all positive integers n. 




70. 
a(n)=2^(2^n+n)1. Such integers are
simultaneously Mersenne and Woodall numbers. 




71. 
Number of digits in
the nth Woodall prime. 




72. 
Integers that can occur as either leg in more than one Pythagorean triple. 




73. 
The number of
primitive Pythagorean triples A^2+B^2=C^2 with 0 < A < B < C and gcd(A,B)=1, and both legs less than n. 




74. 
The number of
primitive Pythagorean triples A^2+B^2=C^2 with 0 < A < B < C and gcd(A,B)=1 that have an hypotenuse C that is less than or
equal to n. 




75. 
The ordered set of
a + b  c as (a,b,c) runs through all Pythagorean
triples with a<b<c. Also called the excess of a Pythagorean triangle,
and is equal to the diameter of its incircle. 




76. 
Perimeters of Pythagorean
triangles that can be constructed in exactly five different ways. 




77. 
The total number of
distinct Pythagorean triples with an area numerically equal to n times their perimeters.
The members of this sequence are also 1/2 the number of divisors of 8n^2. 




78. 
Inradii of
primitive Pythagorean triples a^2+b^2=c^2, 0<a<b<c with gcd(a,b)=1, and sorted to correspond to increasing a (given
in A020884
). 




79. 
Perfect squares
that can be expressed as a sum of four consecutive triangular numbers. 




80. 
Integers k for
which k(k+1)(k+2) is a triangular number. 




81. 
Integers with
precisely two partitions into sums of four squares of nonnegative numbers. 




82. 
The complete set of
integers that cannot be partitioned into a sum of six positive squares. 




83. 
The units’ digits
of the odd square numbers (or the centred octagonal numbers). 




84. 
Digital roots of
the odd square numbers (or the centred octagonal numbers). 




85. 
Units digits of the
positive, even squares 




86. 
Digital roots of
the positive, even squares 




87. 
Integers that cannot
be partitioned into a sum of an odd square, an even square and a triangular
number 




88. 
Integers that do
not have a partition into a sum of an odd square and two (not necessarily distinct)
triangular numbers 




89. 
Integers that are a
sum of two nonzero triangular numbers and also two nonzero square numbers 




90. 
Integers that are a
sum of two triangular numbers and two square numbers (including zeros) 




91. 
Digital roots of
the nonzero pentagonal numbers 




92. 
Digital roots of
the nonzero hexagonal numbers 




93. 
Units digits of the
nonzero hexagonal numbers 




94. 
Digital roots of
the nonzero heptagonal numbers 




95. 
Units digits of the
nonzero heptagonal numbers 




96. 
Digital roots of
the nonzero octagonal numbers 




97. 
Units digits of the
nonzero octagonal numbers 




98. 
Digital roots of
the nonzero nonagonal numbers 




99. 
Units digits of the
nonzero nonagonal numbers 




100. 
Units digits of the
nonzero decagonal numbers 




101. 
Integers that are
kgonal for precisely 3 distinct values of k, where k>=3 




102. 
Integers that are
kgonal for precisely 4 distinct values of k, where k>=3 




103. 
Integers that are
simultaneously triangular (A000217) and
decagonal (A001107). This sequence
began its life as A199086 (defined for natural numbers only), but was
subsequently merged with A133216. 




104. 
Indices of those
triangular numbers (A000217) that are
also decagonal (A001107). Similarly,
this sequence began its life as A199087 (defined for natural numbers only),
but was later merged with A133217. 




105. 
Indices of those decagonal
numbers (A001107) that are also triangular (A000217).
Similarly, this sequence began its life as A199088 (defined for natural
numbers only), but was later merged with A133218. 




106. 
Numbers which are
both decagonal and pentagonal. 




107. 
Indices of
pentagonal numbers which are also decagonal. 




108. 
Indices of
decagonal numbers which are also pentagonal. 




109. 
Numbers which are
both decagonal and hexagonal. 




110. 
Indices of
hexagonal numbers that are also decagonal. 




111. 
Indices of decagonal
numbers that are also hexagonal. 




112. 
Numbers which are
both heptagonal and decagonal. 




113. 
Indices of heptagonal
numbers that are also decagonal. 




114. 
Indices of
decagonal numbers that are also heptagonal. 




115. 
Numbers which are
both decagonal and octagonal. 




116. 
Indices of
octagonal numbers which are also decagonal. 




117. 
Indices of decagonal
numbers which are also octagonal. 




118. 
Numbers which are
both decagonal and nonagonal. 




119. 
Indices of nonagonal
numbers which are also decagonal. 




120. 
Indices of
decagonal numbers which are also nonagonal. 




121. 
Period 6: repeat (1,
6, 5, 6, 1, 0), or equivalently the units’ digits of the nonzero square
triangular numbers. 




122. 
Period 12; repeat
(1, 8, 4, 9, 7, 8, 7, 9, 4, 8, 1, 9) –
or equivalently the digital roots of the indices of those nonzero
triangular numbers that are also perfect squares. 




123. 
Period 6; repeat
(1, 8, 9, 8, 1, 0) – or equivalently the units digits of the
indices of nonzero triangular numbers that are also perfect squares. 




124. 
Period 12; repeat
(1, 6, 8, 6, 1, 9, 8, 3, 1, 3, 8, 9) –
or equivalently the digital roots of the indices of those nonzero
square numbers that are also triangular. 




125. 
Period 6; repeat
(1, 6, 5, 4, 9, 0) – or equivalently the units’ digits of the
indices of those nonzero square numbers that are also triangular. 




126. 
Period 20, repeat
1, 4, 0, 9, 1, 6, 4, 5, 9, 6, 6, 9, 5, 4, 6, 1, 9, 0, 4, 1 (or equivalently
the units’ digits of the centered triangular numbers A005448(n). 




127. 
Odd heptagonal
pyramidal numbers A002413(n). 




128. 
Even heptagonal
pyramidal numbers A002413(n). 




129. 
Odd octagonal pyramidal
numbers A002414(n). 




130. 
Even octagonal
pyramidal numbers A002414(n). 




131. 
Odd nonagonal
pyramidal numbers A007584(n). 




132. 
Even nonagonal
pyramidal numbers A007584(n). 




133. 
Odd decagonal
pyramidal numbers A007585(n). 




134. 
Even, nonzero
decagonal pyramidal numbers A007585(n). 




135. 
Decimal expansion of
Sierpinski's second constant, 




136. 
Decimal expansion
of Sierpinski's third constant, K3 = lim n>infinity, (1 / n * sum(i = 1 to n (A004018(i))^2 – 4*
log(n)). 




137. 
Suppose that (m,
m+1) is a pair of consecutive powerful numbers as defined by A001694. This
sequence gives the values of m for which neither m nor m+1
are perfect squares. 




138. 
Odd primes p>3
for which 14*p+1 is also prime. 