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A134269
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Number of solutions to the equation p^k - p^(k-1) = n, where k is a positive integer and p is prime.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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;
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PROG
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(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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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