A058214 Sum of solutions of phi(x) = 2^n.

3, 13, 35, 105, 231, 581, 1315, 3225, 6711, 15221, 32755, 74505, 154407, 339397, 718115, 1589145, 3243831, 6946421, 14482675, 31259145, 63894567, 135588037, 281203235, 601400985, 1219907127, 2557715317, 5267017715, 11123540745, 22600784679, 47205887429

Offset

0,1

Links

Amiram Eldar, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe)

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Formula

If there are only five Fermat primes, then a(n) = 2^(n-30) * 99852066765 for n > 31. - T. D. Noe, Jun 21 2012

Example

For n = 6, 2^n = 64; the solutions of phi(x) = 64 are {85,128,136,160,170,192,204,240}, whose sum is a(6) = 1315.

Mathematica

phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl=={}, Return[If[n==1, {1}, {}]]]; val={}; p=Last[pl]; For[e=0; pe=1, e==0||Mod[n, (p-1)pe/p]==0, e++; pe*=p, val=Join[val, pe*phiinv[If[e==0, n, n*p/pe/(p-1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1+Divisors[n], PrimeQ]]; Table[Plus@@phiinv[2^n], {n, 0, 30}] (* phiinv[n, pl] = list of x with phi(x)=n and all prime divisors of x in list pl. phiinv[n] = list of x with phi(x)=n *)

Prog

(PARI) a(n) = vecsum(invphi(2^n)); \\ Amiram Eldar, Nov 11 2024, using Max Alekseyev's invphi.gp

Crossrefs

Cf. A000010, A001317, A003401, A004729, A019434, A045544, A047999, A053576, A054432, A058213, A058215.

Sequence in context: A281868 A137976 A095661 * A108480 A322187 A146741

Adjacent sequences: A058211 A058212 A058213 * A058215 A058216 A058217

Keyword

nonn

Author

Labos Elemer, Nov 30 2000

Extensions

Edited by Dean Hickerson, Jan 25 2002

a(28)-a(29) from Donovan Johnson, Oct 22 2011

Status

approved