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A346551
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3-Sondow numbers: numbers k such that p^s divides k/p + 3 for every prime power divisor p^s of k.
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8
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OFFSET
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1,2
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COMMENTS
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Numbers k such that A235137(k) == 3 (mod k).
A positive integer k is a 3-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 3 for every prime power divisor p^s of k.
2) 3/k + Sum_{prime p|k} 1/p is an integer.
3) 3 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 3 (mod k).
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LINKS
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Table of n, a(n) for n=1..9.
Github, Jonathan Sondow (1943 - 2020)
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.
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MATHEMATICA
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Sondow[mu_][n_]:= Sondow[mu][n]= Module[{fa=FactorInteger[n]}, IntegerQ[mu/n+Sum[1/fa[[i, 1]], {i, Length[fa]}]]]
Select[Range[1000000], Sondow[3][#]&]
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CROSSREFS
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Cf. A054377, A007850, A235137, A348058, A348059.
(-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: A317714 A055260 A254059 * A180591 A330083 A322951
Adjacent sequences: A346548 A346549 A346550 * A346552 A346553 A346554
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KEYWORD
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nonn,more
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AUTHOR
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José María Grau Ribas, Dec 04 2021
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EXTENSIONS
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a(8)-a(9) from Martin Ehrenstein, Dec 31 2021
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STATUS
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approved
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