A357898 a(n) is the least k such that phi(k) + d(k) = 2^n, or -1 if there is no such k, where phi(k) = A000010(k) is Euler's totient function and d(k) = A000005(k) is the number of divisors of k.

1, 3, 7, 21, 31, 77, 127, 301, 783, 1133, 3399, 4781, 8191, 16637, 37367, 101601, 131071, 305837, 524287, 1073581, 3220743, 4201133, 8544103, 18404669, 34012327, 67139117, 135255431, 300528877, 824583699, 1073862029, 2147483647, 4295564381, 8603449703, 25807607829

Offset: 1

Author

J. M. Bergot and Robert Israel, Oct 19 2022

Keywords

• nonn

Comments

All primes in this sequence are primes of the form 2^n - 1. This is true because phi(p) = 2^n - 2 if p = 2^n - 1 is a Mersenne prime. - Thomas Scheuerle, Oct 19 2022

274878976349 = a(38) < a(37) = 274881227398. - Martin Ehrenstein, Oct 24 2022

d(k) <= A070319(2^n). - David A. Corneth, Oct 25 2022

Examples

			a(3) = 7 because phi(7)+d(7) = 6+2 = 2^3, and 7 is the least number that works.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..160 (a(35)..a(38) from Martin Ehrenstein; a(39)..a(49) from David A. Corneth)

Crossrefs

Cf. A000005, A000010, A061468, A070319, A073757.

Programs

Maple

V:= Array(0..23): count:= 0:
for n from 1 while count < 23 do
  s:= phi(n)+tau(n);
  t:= padic:-ordp(s,2);
  if V[t] = 0 and s = 2^t then
     V[t]:= n; count:= count+1;
  fi
od:
convert(V,list)[2..-1];

Extensions

a(27)-a(33) from Giorgos Kalogeropoulos, Oct 22 2022

a(34) from Martin Ehrenstein, Oct 24 2022