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A354516
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Smallest k such that m - gpf(m) = k has exactly n solutions m >= 2, gpf = A006530; or -1 if no such k exists.
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1
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OFFSET
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0,2
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COMMENTS
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Smallest k such that there are exactly n primes p such that gpf(k+p) = p (such p must be prime factors of k).
Smallest k having exactly n distinct prime factors p such that k+p is p-smooth.
Conjectures (if no term equals -1): (Start)
(1) Sequence is strictly increasing.
(2) All terms are squarefree.
(3) All terms are in A354525. (End)
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LINKS
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EXAMPLE
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a(4) = 1660577: 1660577 = 17*23*31*127, and we have 1660577+17 = 2*13^2*17^3 is 17-smooth, 1660577+23 = 2^3*5^2*19^2*23 is 23-smooth, 1660577+31 = 2^6*3^3*31^2 is 31-smooth, 1660577+137 = 2*11*19*29*137, so m - gpf(m) = 1660577 has 4 solutions m = 1660577+17 = 1660594, 1660577+23 = 1660600, 1660577+31 = 1660608, and 1660577+137 = 1660714.
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PROG
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(PARI) gpf(n) = vecmax(factor(n)[, 1]);
A354512(n) = my(f=factor(n)[, 1]); sum(i=1, #f, gpf(n+f[i])==f[i]);
a(n) = my(k=1); while(omega(k)<n || A354512(k) != n, k++); return(k)
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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