A274685 Odd numbers n such that sigma(n) is divisible by 5.

19, 27, 29, 57, 59, 79, 87, 89, 95, 109, 133, 135, 139, 145, 149, 171, 177, 179, 189, 199, 203, 209, 229, 237, 239, 247, 261, 267, 269, 285, 295, 297, 319, 323, 327, 343, 349, 351, 359, 377, 379, 389, 395, 399, 409, 413, 417, 419, 435, 437, 439, 445, 447, 449, 459, 475, 479, 493, 499

Offset

1,1

Comments

The subsequence of odd terms in A274397.

If n is in the sequence and gcd(n,m)=1 for some odd m, then n*m is also in the sequence. One might call "primitive" those terms which are not of this form, i.e., not a "coprime" multiple of an earlier term. The list of these primitive terms is (19, 27, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 343, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, ...). The primitive terms are the primes and powers of primes within the sequence. If a prime power p^k (k >= 1) is in the sequence, then p^(m(k+1)-1) is in the sequence for any m >= 1, since 1+p+...+p^(m(k+1)-1) = (1+p+...+p^k)(1+p^(k+1)+...+p^((m-1)*(k+1))). For example, with the prime p=19 we also have all odd powers 19^3, 19^5, ..., and with 27 = 3^3, we also have 27^5, 27^9, ... in the sequence.

On the other hand, for any prime p <> 5 there is an exponent k in {1, 3, 4} such that p^k is in this sequence (and therewith all higher powers of the form given above).

One may notice that there are many pairs of the form (30k-3, 30k-1), e.g., 27,29; 57,59; 87,89; 177,179; 237,239; 295,299; ... Indeed, it is likely that 30k-1 is prime and in this case, if 10k-1 is also prime, then sigma(30k-3) = 40k is divisible by 5 and sigma(30k-1) = 30k is also divisible by 5.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) ~ 2n. - Charles R Greathouse IV, Jul 16 2016

Example

Some values of a(2^k): a(2) = 27, a(4) = 57, a(8) = 89, a(16) = 171, a(32) = 297, a(64) = 545, a(128) = 1029, a(256) = 1937, a(512) = 3625, a(1024) = 6939, a(2048) = 13257, a(4096) = 25483, a(8192) = 49319, a(16384) = 95695, a(32768) = 185991, a(65536) = 362725, a(131072) = 708887, a(262144) = 1388367, a(524288) = 2722639, a(1048576) = 5346681, a(2097152) = 10514679, a(4194304) = 20698531, a(8388608) = 40790203.

Mathematica

Select[Range[1, 500, 2], Divisible[DivisorSigma[1, #], 5] &] (* Michael De Vlieger, Jul 16 2016 *)

Prog

(PARI) is_A274685(n)=sigma(n)%5==0&&bittest(n, 0)
(PARI) forstep(n=1, 999, 2, sigma(n)%5||print1(n", "))

Crossrefs

Cf. A000203 (sigma), A028983 (sigma even), A087943 (sigma = 3k), A248150 (sigma = 4k); A028982 (sigma is odd), A248151 (sigma is not divisible by 4); A272930(sigma(sigma(k)) = nk).

Sequence in context: A039342 A043165 A043945 * A152013 A160036 A032701

Adjacent sequences: A274682 A274683 A274684 * A274686 A274687 A274688

Keyword

nonn

Author

M. F. Hasler, Jul 02 2016

Status

approved