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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.)