A081378 Numbers k for which the sums of prime factors (ignoring multiplicity) of sigma(k) and phi(k) are equal but the sets of prime factors of sigma and phi are different.
412, 1142, 1236, 1328, 1339, 1703, 2855, 2875, 2884, 3406, 3426, 3668, 3708, 3984, 4017, 5109, 5356, 5710, 5750, 5924, 6003, 6281, 6399, 6413, 6640, 6812, 7994, 8054, 8318, 8515, 8565, 8611, 8625, 8652, 8843, 8858, 9373, 9707, 9991
Offset: 1
Keywords
Examples
k = 412 = 2*2*103: sigma(412) = 728 = 2*2*2*7*13, phi(412) = 204 = 2*2*3*17, the sums of prime factors are identical (2 + 7 + 13 = 22 = 2 + 3 + 17) but the prime sets are different: {2,7,13} vs. {2,7,17}.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; spf[x_] := Apply[Plus, ba[x]]; k=0; Do[s=ba[DivisorSigma[1, n]]; s1=ba[EulerPhi[n]]; s3=spf[DivisorSigma[1, n]]; s4=spf[EulerPhi[n]]; If[ !Equal[s, s1]&&Equal[s3, s4], k=k+1; Print[{n, s, s1, ba[n], s3}]], {n, 1, 10000}] -
PARI
is(n) = {my(f = factor(n), p1 = factor(sigma(f))[, 1], p2 = factor(eulerphi(f))[, 1]); p1 != p2 && vecsum(p1) == vecsum(p2) ;} \\ Amiram Eldar, Mar 25 2024