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A346553
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5-Sondow numbers: numbers k such that p^s divides k/p + 5 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) == 5 (mod k).
A positive integer k is a 5-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 5 for every prime power divisor p^s of k.
2) 5/k + Sum_{prime p|k} 1/p is an integer.
3) 5 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 5 (mod k).
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LINKS
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J. M. Grau, A. M. Oller-Marcén and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
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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[10^7], Sondow[5][#]&]
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PROG
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(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 + 5) % p^j, return(0))); ); return(1); } \\ Michel Marcus, Jan 17 2022
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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