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

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

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}.

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}]

Prog

(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

Crossrefs

Cf. A000010, A000203, A065642, A076533, A081377.

Sequence in context: A055018 A201844 A105211 * A234190 A234185 A282127

Adjacent sequences: A081375 A081376 A081377 * A081379 A081380 A081381

Keyword

nonn

Author

Labos Elemer, Mar 26 2003

Status

approved