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

1,2

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

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)

Example

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

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];

Crossrefs

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

Sequence in context: A323585 A003585 A108102 * A065523 A046528 A018572

Adjacent sequences: A357895 A357896 A357897 * A357899 A357900 A357901

Keyword

nonn

Author

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

Extensions

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

a(34) from Martin Ehrenstein, Oct 24 2022

Status

approved