A274397 Positive integers m such that sigma(m) is divisible by 5.

8, 19, 24, 27, 29, 38, 40, 54, 56, 57, 58, 59, 72, 76, 79, 87, 88, 89, 95, 104, 108, 109, 114, 116, 118, 120, 128, 133, 135, 136, 139, 145, 149, 152, 158, 168, 171, 174, 177, 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

Offset: 1

Author

M. F. Hasler, Jul 02 2016

Keywords

• nonn

Comments

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

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.

Integers m > 0 where an integer k exists such that A000203(m) = A008587(k). - Felix Fröhlich, Jul 02 2016

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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, Arithmetic properties of the sum of divisors, arXiv:2007.03088 [math.NT], 2020. See p. 20.
• N. J. A. Sloane, Needed: smallest number k with sigma(sigma(k)) = 5k, SeqFan list, Jul 02 2016.

Crossrefs

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).

Programs

Maple

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

Mathematica

Select[Range[300], Divisible[DivisorSigma[1, #], 5]&] (* Jean-François Alcover, Apr 09 2019 *)

PARI

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

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))

Formula

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

Extensions

Edited by M. F. Hasler, Jul 10 2016