

A274397


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


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
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OFFSET

1,1


COMMENTS

See the subsequence A274685 of odd terms for a remark on frequent pairs of the form (30k3, 30k1).
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]&] (* JeanFrançois 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



