A277173 Numbers m such that b^sigma(m) == b^phi(m) == b^numdiv(m) == b^m (mod m) for every integer b.

1, 2, 6, 12, 24, 60, 120, 126, 240, 420, 480, 504, 672, 780, 1248, 1260, 2340, 2520, 3360, 4680, 5040, 5460, 6240, 6552, 8160, 8736, 9360, 10080, 11424, 16380, 21216, 26208, 27360, 32760, 38304, 43680, 57120, 65520, 71136, 74592, 106080, 131040, 147168, 148512, 171360, 191520, 202464, 325920, 355680, 372960

Offset: 1

Author

David A. Corneth and Altug Alkan, Oct 02 2016

Keywords

• nonn

Comments

Are terms products products of primes of the form 2^i*3^j + 1, A058383, for some nonnegative i and j? This is true for all terms up to 7.6*10^6. 7600320 is divisible by 29, which isn't of the form 2^j*3^i+1. Up to 10^8, all of the terms are divisible by only 16 distinct prime factors. That is: omega(lcm(all terms up to 10^8)) = 16.

Subsequence of A124240.

Examples

			6 is a term because for the primes up to 6, (2, 3 and 5), b^sigma(6) == b^phi(6) == b^numdiv(6) == b^6 (mod 6). This is sufficient to prove for all values b up to 6.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..510

Crossrefs

Cf. A124240.

Programs

Mathematica

fQ[n_] := Block[{b = 2, s = DivisorSigma[1, n], e = EulerPhi[n], d = DivisorSigma[0, n]}, While[b < n && PowerMod[b, s, n] == PowerMod[b, e, n] == PowerMod[b, d, n] == PowerMod[b, n, n], b = NextPrime@ b]; b >= n]; lst = {1}; k = 2; While[k < 400000, If[ fQ@ k, AppendTo[lst, k]]; k ++]; lst (* Robert G. Wilson v, Nov 04 2016 *)

PARI

isk(n, k) = {Mod(k, n)^sigma(n)==Mod(k, n)^n && Mod(k, n)^eulerphi(n)==Mod(k, n)^n && Mod(k, n)^numdiv(n)==Mod(k, n)^n}
is(n) = my(i);forprime(i=2, n, if(isk(n, i)==0,return(0))) ; 1
upto(lim) = my(l=List());for(n=1, lim, if(is(n), listput(l,n))); l