A006511 Largest inverse of totient function (A000010): a(n) is the largest x such that phi(x) = m, where m = A002202(n) is the n-th number in the range of phi.

2, 6, 12, 18, 30, 22, 42, 60, 54, 66, 46, 90, 58, 62, 120, 126, 150, 98, 138, 94, 210, 106, 162, 174, 118, 198, 240, 134, 142, 270, 158, 330, 166, 294, 276, 282, 420, 250, 206, 318, 214, 378, 242, 348, 354, 462, 254, 510, 262, 414, 274, 278, 426, 630, 298, 302

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Always even, as phi(2n) = phi(n) when n is odd. - Alain Jacques (thegentleway(AT)bigpond.com), Jun 15 2006

References

• J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• Richard K. Guy, Letter to N. J. A. Sloane, Jun 1991.

Crossrefs

Cf. A000010, A002202, A002181, A057826.

For records see A036913, A132154, A036912.

Programs

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]]; Last/@Select[phiinv/@Range[1, 200], #!={}&] (* 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 *)

PARI

g(n) = if(n%2, 2*(n==1), forstep(k = floor(exp(Euler)*n*log(log(n^2))+2.5*n/log(log(n^2))), n, -1, if(eulerphi(k)==n, return(k)); if(k==n, return(0)))); \\ A057635
lista(nn) = for(m = 1, nn, if(istotient(m), print1(g(m), ", "))); \\ Jinyuan Wang, Aug 29 2019

PARI

lista(nmax) = my(s); for(n = 1, nmax, s = invphiMax(n); if(s > 0, print1(s, ", "))); \\ Amiram Eldar, Nov 14 2024, using Max Alekseyev's invphi.gp

Perl

use ntheory ":all"; my $k=1; for my $i (1..100) { my @v; do{@v=inverse_totient($k++)} until @v; print "$i $v[-1]\n"; } # Dana Jacobsen, Mar 04 2019

Formula

a(n) = A057635(A002202(n)). - T. D. Noe