A078539 Least non-balanced x (i.e., not in A020492) such that sigma(2n-1,x)/phi(x) is an integer.
38, 46, 295, 38, 235, 749, 38, 3687, 6128, 38, 1415, 46, 38, 4254, 10451, 38, 46, 8351, 38, 334, 4511, 38, 3398, 295, 38, 1286, 46, 38, 148870, 11015, 38, 46, 35519, 38, 10239, 14072, 38, 235, 76088, 38, 5991, 46, 38, 718, 295, 38, 46, 11654, 38, 30761
Offset: 2
Keywords
Examples
n=7: 2n-1 = 13, cases of sigma(13,x)/phi(x) is an integer listed in A015771: 1, 2, 3,6, 12, etc,; the first term which is non-balanced, i.e., not in A020492 is a(7) = 749 = A020492(31); increasing n, the trend of a(n) is roughly the same. If 2n-1 = 3s, i.e., is divisible by 3, then a(3s) = 38. Similar relationships hold for 2n - 1 = 5s, 7s, 11s, etc.
Links
- Amiram Eldar, Table of n, a(n) for n = 2..1200
Programs
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Mathematica
Table[fl=1; Do[s1=DivisorSigma[1, n]/EulerPhi[n]; sk=DivisorSigma[2*k-1, n]/EulerPhi[n]; If[ !IntegerQ[s1]&&IntegerQ[sk]&&Equal[fl, 1], Print[{n, 2*k-1}]; fl=0], {n, 1, 1000000}], {k, 2, 100}]
Formula
a(n) = min{x; sigma(1,x) mod phi(x) = 0 but sigma(2n-1, x) mod phi(x) is not 0}.
Extensions
a(31) corrected by Amiram Eldar, Jul 21 2019