%I #61 Jan 02 2023 12:30:52

%S 8,19,24,27,29,38,40,54,56,57,58,59,72,76,79,87,88,89,95,104,108,109,

%T 114,116,118,120,128,133,135,136,139,145,149,152,158,168,171,174,177,

%U 178,179,184,189,190,199,200,203,209,216,218,228,229,232,236,237,239,247,248,261,264,266,267,269,270,278,280,285,290,295,296,297

%N Positive integers m such that sigma(m) is divisible by 5.

%C See the subsequence A274685 of odd terms for a remark on frequent pairs of the form (30k-3, 30k-1).

%C If m is in the sequence and gcd(k,m)=1, then k*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 primitive terms are the primes and powers of primes within the sequence, cf. below.

%C Integers m > 0 where an integer k exists such that A000203(m) = A008587(k). - _Felix Fröhlich_, Jul 02 2016

%C For any prime p <> 5 there is an exponent k in {1, 3, 4} (depending on whether p is in A030433, A003631 or A030430) such that p^k is in this sequence. Given these p^k, the sequence consists of all numbers of the form n*p^(q*(k+1)-1) where n is coprime to p and q >= 1. Otherwise said, all numbers m which have some prime factor p with multiplicity q*(k+1)-1, where k = k(p) in {1, 3, 4} as introduced before. - _M. F. Hasler_, Jul 10 2016

%H Robert Israel, <a href="/A274397/b274397.txt">Table of n, a(n) for n = 1..10000</a>

%H Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, <a href="https://arxiv.org/abs/2007.03088">Arithmetic properties of the sum of divisors</a>, arXiv:2007.03088 [math.NT], 2020. See p. 20.

%H N. J. A. Sloane, <a href="http://list.seqfan.eu/oldermail/seqfan/2016-July/016528.html">Needed: smallest number k with sigma(sigma(k)) = 5k</a>, SeqFan list, Jul 02 2016.

%F lim_{n->oo} a(k)/k = 2 (conjectured; cf. Examples).

%e Some values for a(2^k): We have a(2) = 19, a(4) = 27, a(8) = 54, a(16) = 87, a(32) = 145, a(64) = 270, a(128) = 488, a(256) = 919, a(512) = 1736, a(1024) = 3267, a(2048) = 6258, a(4096) = 12035, a(8192) = 23160, a(16384) = 44878, a(32768) = 87207, a(65536) = 169911, a(131072) = 332009, a(262144) = 650031, a(524288) = 1274569, a(1048576) = 2503510, a(2097152) = 4924370, a(4194304) = 9697475, a(8388608) = 19116191.

%p select(t -> numtheory:-sigma(t) mod 5 = 0, [$1..1000]); # _Robert Israel_, Jul 12 2016

%t Select[Range[300], Divisible[DivisorSigma[1, #], 5]&] (* _Jean-François Alcover_, Apr 09 2019 *)

%o (PARI) is(n)=sigma(n)%5==0

%o (PARI) is(n)=for(i=1,#n=factor(n)~,n[1,i] != 5 && (n[2,i]+1) % [5,4,4,2][n[1,i]%5] == 0 && return(1))

%Y Cf. A000203, 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

%E Edited by _M. F. Hasler_, Jul 10 2016