cp's OEIS Frontend

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.

%I A357898 #43 Jan 20 2023 21:44:00
%S A357898 1,3,7,21,31,77,127,301,783,1133,3399,4781,8191,16637,37367,101601,
%T A357898 131071,305837,524287,1073581,3220743,4201133,8544103,18404669,
%U A357898 34012327,67139117,135255431,300528877,824583699,1073862029,2147483647,4295564381,8603449703,25807607829
%N 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.
%C A357898 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
%C A357898 274878976349 = a(38) < a(37) = 274881227398. - _Martin Ehrenstein_, Oct 24 2022
%C A357898 d(k) <= A070319(2^n). - _David A. Corneth_, Oct 25 2022
%H A357898 Max Alekseyev, <a href="/A357898/b357898.txt">Table of n, a(n) for n = 1..160</a> (a(35)..a(38) from Martin Ehrenstein; a(39)..a(49) from David A. Corneth)
%e A357898 a(3) = 7 because phi(7)+d(7) = 6+2 = 2^3, and 7 is the least number that works.
%p A357898 V:= Array(0..23): count:= 0:
%p A357898 for n from 1 while count < 23 do
%p A357898   s:= phi(n)+tau(n);
%p A357898   t:= padic:-ordp(s,2);
%p A357898   if V[t] = 0 and s = 2^t then
%p A357898      V[t]:= n; count:= count+1;
%p A357898   fi
%p A357898 od:
%p A357898 convert(V,list)[2..-1];
%Y A357898 Cf. A000005, A000010, A061468, A070319, A073757.
%K A357898 nonn
%O A357898 1,2
%A A357898 _J. M. Bergot_ and _Robert Israel_, Oct 19 2022
%E A357898 a(27)-a(33) from _Giorgos Kalogeropoulos_, Oct 22 2022
%E A357898 a(34) from _Martin Ehrenstein_, Oct 24 2022