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A349193
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1-Sondow numbers: numbers j such that p divides j/p + 1 for every prime divisor p of j.
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7
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OFFSET
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1,2
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COMMENTS
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These are the weak primary pseudoperfect numbers mentioned in Grau-Oller-Sondow (2013).
Includes the primary pseudoperfect numbers (A054377). Any weak primary pseudoperfect number which is not a primary pseudoperfect number must have more than 58 distinct prime factors, and therefore must be greater than 10^110; none are known.
A positive integer j is a k-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides j/p + k for every prime power divisor p^s of j.
2) k/j + Sum_{prime p|j} 1/p is an integer.
3) k + Sum_{prime p|j} j/p == 0 (mod j).
4) Sum_{i=1..j} i^A000010(j) == k (mod j).
Numbers m such that A235137(m) == 1 (mod m).
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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[100000], Sondow[1][#]&]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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