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A235061
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Number of ways to write n = k*(k+1)/2 + m with k > 0 and m > 0 such that prime(k*(k+1)/2) + phi(m) is prime, where phi(.) is Euler's totient function.
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1
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0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 3, 2, 2, 1, 1, 1, 2, 3, 3, 2, 1, 1, 3, 3, 3, 3, 2, 3, 2, 3, 2, 3, 4, 2, 4, 4, 2, 3, 1, 4, 4, 1, 4, 3, 3, 3, 4, 5, 4, 3, 1, 3, 3, 5, 4, 4, 5, 1, 5, 3, 5, 5, 4, 2, 2, 5, 4, 5, 1, 1, 6, 5, 6, 6, 4, 5, 5, 8, 5, 2, 1, 4, 6, 4, 6, 7, 3, 3, 6, 4, 7, 5, 2, 7, 6
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OFFSET
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1,13
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COMMENTS
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Conjecture: (i) a(n) > 0 for all n > 8.
(ii) Any integer n > 100 can be written as k^2 + m with k > 0 and m > 0 such that phi(k^2) + prime(m) is prime.
(iii) Any integer n > 187 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m) is a triangular number. Also, each integer n > 45 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a triangular number.
(iv) Any integer n > 293 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m) is a square. Also, each integer n > 83 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a square.
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LINKS
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EXAMPLE
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a(10) = 1 since 10 = 2*(2+1)/2 + 7 = 3 + 7 with prime(3) + phi(7) = 5 + 6 = 11 prime.
a(20) = 1 since 20 = 3*(3+1)/2 + 14 = 6 + 14 with prime(6) + phi(14) = 13 + 6 = 19 prime.
a(86) = 1 since 86 = 12*(12+1)/2 + 8 = 78 + 8 with prime(78) + phi(8) = 397 + 4 = 401 prime.
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MATHEMATICA
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a[n_]:=Sum[If[PrimeQ[Prime[k(k+1)/2]+EulerPhi[n-k(k+1)/2]], 1, 0], {k, 1, (Sqrt[8n-7]-1)/2}]; 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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