cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A152677 Subsequence of odd terms in A000203 (sum-of-divisors function sigma), in the order in which they occur and with repetitions.

Original entry on oeis.org

1, 3, 7, 15, 13, 31, 39, 31, 63, 91, 57, 93, 127, 195, 121, 171, 217, 133, 255, 403, 363, 183, 399, 465, 403, 399, 511, 819, 307, 847, 549, 381, 855, 961, 741, 1209, 931, 1023, 553, 1651, 921, 781, 1815, 1281, 1143, 1093, 1767, 1953, 871, 2223, 2821, 993, 1995
Offset: 1

Views

Author

Omar E. Pol, Dec 10 2008

Keywords

Comments

Equivalently: subsequence of A000203 (sigma) with indices equal to a square or twice a square (A028982).
See A060657 for the set of odd values in the range of the sigma function, i.e., the list of odd values in ordered by increasing size and without repetitions.

Links

Crossrefs

Cf. A000203 (sigma = sum-of-divisors function), A152678 (even terms in A000203), A028982 (squares and twice the squares).
See A062700 and A023195 for the subsequence resp. subset of primes; A023194 for the indices of A000203 which yield these primes.
Cf. A002117.

Programs

  • Magma
    [d:k in [1..1000]|IsOdd(d) where d is DivisorSigma(1,k)]; // Marius A. Burtea, Jan 09 2020
  • Mathematica
    Select[DivisorSigma[1, Range[1000]], OddQ[#] &] (* Giovanni Resta, Jan 08 2020 *)
With[{max = 1000}, DivisorSigma[1, Union[Range[Sqrt[max]]^2, 2*Range[Sqrt[max/2]]^2]]] (* Amiram Eldar, Nov 28 2023 *)
  • PARI
    A152677_upto(lim)=apply(sigma,vecsort(concat(vector(sqrtint(lim\1), i, i^2), vector(sqrtint(lim\2), i, 2*i^2)))) \\ Gives [a(n) = sigma(k) with k = A028982(n) <= lim]. - Charles R Greathouse IV, Feb 15 2013, corrected by M. F. Hasler, Jan 08 2020

Formula

a(n) = A000203(A028982(n)). - R. J. Mathar, Dec 12 2008
Sum_{k=1..n} a(k) ~ c * n^3, where c = (16-10*sqrt(2))*zeta(3)/Pi^2 = 0.226276... . - Amiram Eldar, Nov 28 2023

Extensions

Extended by R. J. Mathar, Dec 12 2008
Edited and definition reworded by M. F. Hasler, Jan 08 2020

A331036 Odd values of the sum-of-divisors function sigma (A000203), listed by increasing size and with multiplicity.

Original entry on oeis.org

1, 3, 7, 13, 15, 31, 31, 39, 57, 63, 91, 93, 121, 127, 133, 171, 183, 195, 217, 255, 307, 363, 381, 399, 399, 403, 403, 465, 511, 549, 553, 741, 781, 819, 847, 855, 871, 921, 931, 961, 993, 1023, 1093, 1143, 1209, 1281, 1407, 1651, 1659, 1723, 1729, 1767, 1767, 1815, 1893, 1953
Offset: 1

Views

Author

M. F. Hasler, Jan 08 2020

Keywords

Comments

See A060657 for the range (without repeated terms) and A152677 for the subsequence of odd values in A000203.

Links

Crossrefs

Cf. A060657, A023195 (subset of primes), A152677 (subsequence of odd values in A000203), A300869 (repeated terms).

Programs

  • Maple
    N:= 2000: # for terms <= N
Res:= NULL:
for m from 1 to floor(sqrt(N)) by 2 do
  sm:= numtheory:-sigma(m^2);
  for k from 1 to floor(log[2](N/sm+1)) do
    v:= sm*(2^k-1);
    if v <= N then Res:= Res, v; count:= count+1 fi;
  od
od:
sort([Res]); # Robert Israel, Jan 14 2020
  • Mathematica
    Sort@ Select[DivisorSigma[1, Range@ 2000], OddQ[#] && # < 2000 &] (* Giovanni Resta, Jan 08 2020 *)
  • PARI
    list(lim)=select(k->k<=lim, vecsort(apply(sigma, concat(vector(sqrtint(lim\1), i, i^2), vector(sqrtint(lim\2), i, 2*i^2))))) \\ Charles R Greathouse IV, Feb 15 2013 [originally added in A152677]

A275671 Even values produced by the sigma function A000203, in increasing order.

Original entry on oeis.org

4, 6, 8, 12, 14, 18, 20, 24, 28, 30, 32, 36, 38, 40, 42, 44, 48, 54, 56, 60, 62, 68, 72, 74, 78, 80, 84, 90, 96, 98, 102, 104, 108, 110, 112, 114, 120, 124, 126, 128, 132, 138, 140, 144, 150, 152, 156, 158, 160, 162, 164, 168, 174, 176, 180, 182, 186, 192, 194
Offset: 1

Views

Author

Jaroslav Krizek, Aug 04 2016

Keywords

Comments

Even terms of A002191.
Complement of A060657 with respect to A002191.

Examples

			8 is in the sequence because sigma(7) = 8 and it is an even number.

Links

Crossrefs

Cf. A000203, A002191, A060657, A152678.

Programs

  • Magma
    Set(Sort([SumOfDivisors(n): n in[1..1000] | not IsOdd(SumOfDivisors(n)) and SumOfDivisors(n) le 1000]));
  • PARI
    is(k) = !(k % 2) && invsigmaNum(k) > 0; \\ Amiram Eldar, Dec 26 2024, using Max Alekseyev's invphi.gp
Showing 1-3 of 3 results.

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