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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

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

EXAMPLE

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.

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

PROG

(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

CROSSREFS

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

Sequence in context: A029832 A320535 A174479 * A172444 A277146 A026611

Adjacent sequences:  A134266 A134267 A134268 * A134270 A134271 A134272

KEYWORD

nonn

AUTHOR

Anthony C Robin, Jan 15 2008

EXTENSIONS

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

More terms from Antti Karttunen, Nov 09 2018

STATUS

approved

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Last modified May 25 17:53 EDT 2020. Contains 334595 sequences. (Running on oeis4.)