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A346552
4-Sondow numbers: numbers k such that p^s divides k/p + 4 for every prime power divisor p^s of k.
8
1, 5, 8, 24, 168, 7224, 188232, 8858009688, 209981586408
OFFSET
1,2
COMMENTS
Numbers k such that A235137(k) == 4 (mod k).
A positive integer k is a 4-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 4 for every prime power divisor p^s of k.
2) 4/k + Sum_{prime p|k} 1/p is an integer.
3) 4 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 4 (mod k).
Other numbers in the sequence: 8858009688, 209981586408, 33961686334238753642827085044344
LINKS
J. M. Grau, A. M. Oller-Marcén and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
J. M. Grau, A. M. Oller-Marcen and J. Sondow, On the congruence 1^n + 2^n + ... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013-2014.
MATHEMATICA
Sondow[mu_][n_]:=Sondow[mu][n]=Module[{fa=FactorInteger[n]}, IntegerQ[mu/n+Sum[1/fa[[i, 1]], {i, Length[fa]}]]]
Select[Range[10000000], Sondow[4][#]&]
PROG
(PARI) isok(k) = {my(f=factor(k)); for (i=1, #f~, my(p=f[i, 1]); for (j=1, f[i, 2], if ((k/p + 4) % p^j, return(0))); ); return(1); } \\ Michel Marcus, Jan 17 2022
CROSSREFS
(-1) and (-2) -Sondow numbers: A326715, A330069.
1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, A346555, A346556, A346557.
Sequence in context: A140113 A357117 A368484 * A375350 A166652 A286571
KEYWORD
nonn,more
AUTHOR
EXTENSIONS
a(8)-a(9) verified by Martin Ehrenstein, Jan 21 2022
STATUS
approved