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A134269 Number of solutions to the equation p^k - p^(k-1) = n, where k is a positive integer and p is prime.

%I A134269 #22 Nov 09 2018 21:58:15
%S A134269 1,2,0,2,0,2,0,1,0,1,0,1,0,0,0,2,0,2,0,1,0,1,0,0,0,0,0,1,0,1,0,1,0,0,
%T A134269 0,1,0,0,0,1,0,2,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,
%U A134269 0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0
%N A134269 Number of solutions to the equation p^k - p^(k-1) = n, where k is a positive integer and p is prime.
%C A134269 The Euler phi function A000010 (number of integers less than n which are coprime with n) involves calculating the expression p^(k-1)*(p-1), where p is prime. For example phi(120) = phi(2^3*3*5) = (2^3-2^2)*(3-1)*(5-1) = 4*2*4 = 32.
%H A134269 Antti Karttunen, <a href="/A134269/b134269.txt">Table of n, a(n) for n = 1..65537</a>
%e A134269 Notice that it is not possible to have more than 2 solutions, but say when n=4 there are two solutions, namely 5^1 - 5^0 and 2^3 - 2^2.
%e A134269 a(2) = 2 refers to 2^2 - 2^1 = 2 and 3^1 - 3^0 = 2.
%e A134269 a(6) = 2 as 6 = 3^2 - 3^1 = 7^1 - 7^0.
%p A134269 A134269 := proc(n)
%p A134269     local a,p,r ;
%p A134269     a := 0 ;
%p A134269     p :=2 ;
%p A134269     while p <= n+1 do
%p A134269         r := n/(p-1) ;
%p A134269         if type(r,'integer') then
%p A134269             if r = 1 then
%p A134269                 a := a+1 ;
%p A134269             else
%p A134269                 r := ifactors(r)[2] ;
%p A134269                 if nops(r) = 1 then
%p A134269                     if op(1,op(1,r)) = p then
%p A134269                         a := a+1 ;
%p A134269                     end if;
%p A134269                 end if;
%p A134269             end if;
%p A134269         end if;
%p A134269         p := nextprime(p) ;
%p A134269     end do:
%p A134269     return a;
%p A134269 end proc: # _R. J. Mathar_, Aug 06 2013
%o A134269 (PARI) lista(N=100) = {tab = vector(N); for (i=1, N, p = prime(i); for (j=1, N, v = p^j-p^(j-1); if (v <= #tab, tab[v]++););); for (i=1, #tab, print1(tab[i], ", "));} \\ _Michel Marcus_, Aug 06 2013
%o A134269 (PARI)
%o A134269 A134269list(up_to) = { my(v=vector(up_to)); forprime(p=2,1+up_to, for(j=1,oo,my(d = (p^j)-(p^(j-1))); if(d>up_to,break,v[d]++))); (v); };
%o A134269 v134269 = A134269list(up_to);
%o A134269 A134269(n) = v134269[n]; \\ _Antti Karttunen_, Nov 09 2018
%Y A134269 Cf. A000010, A014197, A114871, A114873, A114874.
%K A134269 nonn
%O A134269 1,2
%A A134269 _Anthony C Robin_, Jan 15 2008
%E A134269 a(2) corrected by _Michel Marcus_, Aug 06 2013
%E A134269 More terms from _Antti Karttunen_, Nov 09 2018