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A233542
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Number of ways to write n = k^2 + m with k > 0 and m > 0 such that phi(k^2)*phi(m) - 1 is prime, where phi(.) is Euler's totient function (A000010).
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11
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0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 2, 3, 3, 3, 2, 4, 2, 2, 2, 4, 3, 1, 2, 4, 4, 4, 3, 2, 4, 2, 3, 3, 2, 3, 4, 4, 5, 4, 4, 2, 1, 3, 4, 5, 4, 4, 3, 1, 6, 5, 5, 5, 2, 4, 4, 3, 2, 3, 4, 5, 4, 5, 4, 2, 3, 6, 4, 3, 5, 6, 3, 4, 6, 3, 4, 6, 6, 4, 4, 3, 8, 1, 3, 6, 5, 5, 4, 2, 2, 4, 5, 4, 5, 2, 5, 6, 3, 4, 6
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OFFSET
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1,8
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COMMENTS
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Conjecture: (i) a(n) > 0 for all n > 5.
(ii) Any integer n > 7 can be written as k^2 + m with k > 0 and m > 0 such that phi(k)^2*phi(m) - 1 is prime.
(iii) If n > 1 is not equal to 36, then n can be written as k^2 + m with k > 0 and m > 0 such that sigma(k)^2*phi(m) + 1 is prime, where sigma(k) is the sum of all (positive) divisors of k.
We have verified part (i) of the conjecture for n up to 2*10^7.
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LINKS
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EXAMPLE
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a(6) = 1 since 6 = 1^2 + 5 with phi(1^2)*phi(5) - 1 = 1*4 - 1 = 3 prime.
a(7) = 1 since 7 = 2^2 + 3 with phi(2^2)*phi(3) - 1 = 2*2 - 1 = 3 prime.
a(23) = 1 since 23 = 4^2 + 7 with phi(4^2)*phi(7) - 1 = 8*6 - 1 = 47 prime.
a(42) = 1 since 42 = 6^2 + 6 with phi(6^2)*phi(6) - 1 = 12*2 - 1 = 23 prime.
a(49) = 1 since 49 = 2^2 + 45 with phi(2^2)*phi(45) - 1 = 2*24 - 1 = 47 prime.
a(83) = 1 since 83 = 9^2 + 2 with phi(9^2)*phi(2) - 1 = 54*1 - 1 = 53 prime.
a(188) = 1 since 188 = 6^2 + 152 with phi(6^2)*phi(152) - 1 = 12*72 - 1 = 863 prime.
a(327) = 1 since 327 = 5^2 + 302 with phi(5^2)*phi(302) - 1 = 20*150 - 1 = 2999 prime.
a(557) = 1 since 557 = 12^2 + 413 with phi(12^2)*phi(413) - 1 = 48*348 - 1 = 16703 prime.
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MATHEMATICA
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a[n_]:=Sum[If[PrimeQ[EulerPhi[k^2]*EulerPhi[n-k^2]-1], 1, 0], {k, 1, Sqrt[n-1]}]
Table[a[n], {n, 1, 100}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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