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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.

%I A346552 #16 Jan 21 2022 05:07:43
%S A346552 1,5,8,24,168,7224,188232,8858009688,209981586408
%N A346552 4-Sondow numbers: numbers k such that p^s divides k/p + 4 for every prime power divisor p^s of k.
%C A346552 Numbers k such that A235137(k) == 4 (mod k).
%C A346552 A positive integer k is a 4-Sondow number if satisfies any of the following equivalent properties:
%C A346552 1) p^s divides k/p + 4 for every prime power divisor p^s of k.
%C A346552 2) 4/k + Sum_{prime p|k} 1/p is an integer.
%C A346552 3) 4 + Sum_{prime p|k} k/p == 0 (mod k).
%C A346552 4) Sum_{i=1..k} i^phi(k) == 4 (mod k).
%C A346552 Other numbers in the sequence: 8858009688, 209981586408, 33961686334238753642827085044344
%H A346552 Github, <a href="https://jonathansondow.github.io/">Jonathan Sondow (1943 - 2020)</a>
%H A346552 J. M. Grau, A. M. Oller-Marcén and D. Sadornil, <a href="https://arxiv.org/abs/2111.14211">On µ-Sondow Numbers</a>, arXiv:2111.14211 [math.NT], 2021.
%H A346552 J. M. Grau, A. M. Oller-Marcen and J. Sondow, <a href="https://arxiv.org/abs/1309.7941">On the congruence 1^n + 2^n + ... + n^n = d (mod n), where d divides n</a>, arXiv:1309.7941 [math.NT], 2013-2014.
%t A346552 Sondow[mu_][n_]:=Sondow[mu][n]=Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]]
%t A346552 Select[Range[10000000],Sondow[4][#]&]
%o A346552 (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
%Y A346552 Cf. A054377, A007850, A235137, A348058, A348059.
%Y A346552 (-1) and (-2) -Sondow numbers: A326715, A330069.
%Y A346552 1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, A346555, A346556, A346557.
%K A346552 nonn,more
%O A346552 1,2
%A A346552 _José María Grau Ribas_, Jan 11 2022
%E A346552 a(8)-a(9) verified by _Martin Ehrenstein_, Jan 21 2022