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
KEYWORD
nonn,changed
AUTHOR
M. F. Hasler, Jul 02 2016
EXTENSIONS
Edited by M. F. Hasler, Jul 10 2016
STATUS
approved