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%I #29 Mar 16 2019 19:41:00
%S 19,27,29,57,59,79,87,89,95,109,133,135,139,145,149,171,177,179,189,
%T 199,203,209,229,237,239,247,261,267,269,285,295,297,319,323,327,343,
%U 349,351,359,377,379,389,395,399,409,413,417,419,435,437,439,445,447,449,459,475,479,493,499
%N Odd numbers n such that sigma(n) is divisible by 5.
%C The subsequence of odd terms in A274397.
%C 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.
%C 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).
%C 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.
%H Seiichi Manyama, <a href="/A274685/b274685.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) ~ 2n. - _Charles R Greathouse IV_, Jul 16 2016
%e 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.
%t Select[Range[1, 500, 2], Divisible[DivisorSigma[1, #], 5] &] (* _Michael De Vlieger_, Jul 16 2016 *)
%o (PARI) is_A274685(n)=sigma(n)%5==0&&bittest(n,0)
%o (PARI) forstep(n=1,999,2,sigma(n)%5||print1(n","))
%Y 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).
%K nonn
%O 1,1
%A _M. F. Hasler_, Jul 02 2016
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