A015706 -id:A015706 - cp's OEIS frontend

A015707 Values of n where (phi(n) * sigma(n))/n is an integer and increases.

1, 6, 18, 24, 28, 40, 54, 72, 96, 117, 135, 196, 200, 216, 224, 288, 360, 384, 468, 496, 600, 640, 756, 775, 864, 891, 1000, 1080, 1152, 1372, 1488, 1521, 1568, 1701, 1792, 1944, 2016, 2160, 2176, 2480, 2592, 3100, 3159, 3200, 3375, 3456, 3744, 4320, 4455, 4608

Offset: 1

Robert G. Wilson v

nonn

This sequence is infinite since there are infinitely many numbers m with an arbitrarily large integer value of (phi(m) * sigma(m))/m. For example, if m = 2 * 3^k, then this value is (3^(k+1)-1)/2. - Amiram Eldar, Dec 05 2018

Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Cf. A015706.

seq = {}; rm = 0; Do[r = EulerPhi[n]*DivisorSigma[1, n]/n; If[IntegerQ[r] && r > rm, rm = r; AppendTo[seq, n]], {n, 1, 5000}]; seq (* Amiram Eldar, Dec 05 2018 *)

Title improved by Sean A. Irvine, Dec 05 2018

a(1)=1 inserted and more terms added by Amiram Eldar, Dec 05 2018

A386871 Least k such that A056100(k) = n or -1 if no such k exists.

Original entry on oeis.org

6, 124357252657, 4, 33, 8, 145, 9, 37063859, 16, 51, 26, 1441, 15, 2353, 34, 69, 20, 1011377103546119, 27, 7201, 25, 87, 115, 9911837, 56, 385, 58, 45, 62, 86125529, 57, 30721, 35, 123, 74, 295, 90, 15686608811, 82, 141, 86, 70561, 49, 77857739, 94, 159, 329, 34884199

Offset: 1

Michel Marcus, Aug 06 2025

Wanted terms: a(18), a(38), a(72), a(80).
a(38) = 15686608811 (found by Tomas Rokicki), a(80) = 4222433393407, and a(18), a(72) > 6*10^12. - Giovanni Resta, Aug 09 2025
a(72) > a(18). - Martin Ehrenstein, Aug 10 2025

Michel Marcus, Extension to A056100, Seqfan, Aug 05 2025.
Neil Sloane, Smallest inverse to f(n) = phi(n)*sigma(n) + 1 mod n ?, Math Fun, Aug 05 2025.

Cf. A056100, A386856 (odd bisection), A015706.

a[n_]:=Module[{k=0},Until[Mod[DivisorSigma[1, k]*EulerPhi[k] + 1, k]==n,k++];k] (* James C. McMahon, Aug 06 2025 *)

a(2) from Tomas Rokicki, Aug 05 2025
a(18) from Martin Ehrenstein, Aug 10 2025
a(19)-a(37) from Hugo Pfoertner, Aug 11 2025
a(38) from Tomas Rokicki, Aug 09 2025
More terms from Hugo Pfoertner, Aug 11 2025

cp's OEIS Frontend

A015707 Values of n where (phi(n) * sigma(n))/n is an integer and increases.

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