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

1,1

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

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.

Formula

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

Example

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.

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 *)

Prog

(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))

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).

Sequence in context: A227881 A294581 A290185 * A178130 A227029 A260004

Adjacent sequences: A274394 A274395 A274396 * A274398 A274399 A274400

Keyword

nonn

Author

M. F. Hasler, Jul 02 2016

Extensions

Edited by M. F. Hasler, Jul 10 2016

Status

approved