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

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, 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, 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

Offset: 1

Author

Anthony C Robin, Jan 15 2008

Keywords

• nonn

Comments

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.

Examples

			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.
a(2) = 2 refers to 2^2 - 2^1 = 2 and 3^1 - 3^0 = 2.
a(6) = 2 as 6 = 3^2 - 3^1 = 7^1 - 7^0.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000010, A014197, A114871, A114873, A114874.

Programs

Maple

A134269 := proc(n)
    local a,p,r ;
    a := 0 ;
    p :=2 ;
    while p <= n+1 do
        r := n/(p-1) ;
        if type(r,'integer') then
            if r = 1 then
                a := a+1 ;
            else
                r := ifactors(r)[2] ;
                if nops(r) = 1 then
                    if op(1,op(1,r)) = p then
                        a := a+1 ;
                    end if;
                end if;
            end if;
        end if;
        p := nextprime(p) ;
    end do:
    return a;
end proc: # R. J. Mathar, Aug 06 2013

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

PARI

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); };
v134269 = A134269list(up_to);
A134269(n) = v134269[n]; \\ Antti Karttunen, Nov 09 2018

Extensions

a(2) corrected by Michel Marcus, Aug 06 2013

More terms from Antti Karttunen, Nov 09 2018