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A022894
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Number of solutions to c(1)*prime(1) +...+ c(2n+1)*prime(2n+1) = 0, where c(i) = +-1 for i > 1, c(1) = 1.
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37
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0, 1, 1, 2, 5, 13, 39, 122, 392, 1286, 4341, 14860, 51085, 178402, 634511, 2260918, 8067237, 29031202, 105250449, 383579285, 1404666447, 5171065198, 19141008044, 71124987313, 263548339462, 983424096451, 3684422350470, 13818161525284, 51938115653565
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OFFSET
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0,4
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COMMENTS
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c(1)*prime(1) + ... + c(2n)*prime(2n) = 0 has no solution, because the l.h.s. has an odd number of odd terms and the r.h.s. is even.
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LINKS
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FORMULA
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a(n) is the constant term in expansion of (1/2) * Product_{k=1..2*n+1} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 25 2024
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EXAMPLE
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a(1) = 1 because 2 + 3 - 5 = 0,
a(2) = 1 because 2 - 3 + 5 + 7 - 11 = 0,
a(3) = 2 because
2 + 3 - 5 - 7 + 11 + 13 - 17 =
2 + 3 - 5 + 7 - 11 - 13 + 17 = 0.
a(4) = 5 because
2 - 3 - 5 + 7 + 11 + 13 + 17 - 19 - 23 =
2 - 3 + 5 - 7 + 11 + 13 - 17 + 19 - 23 =
2 - 3 + 5 + 7 - 11 - 13 + 17 + 19 - 23 =
2 - 3 + 5 + 7 - 11 + 13 - 17 - 19 + 23 =
2 + 3 + 5 - 7 - 11 - 13 + 17 - 19 + 23 = 0
and there are no others up through the ninth prime.
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MAPLE
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sp:= proc(n) sp(n):= `if`(n=1, 0, ithprime(n)+sp(n-1)) end:
b := proc(n, i) option remember; `if`(n>sp(i), 0, `if`(i=1, 1,
b(n+ithprime(i), i-1)+ b(abs(n-ithprime(i)), i-1)))
end:
a:= n-> b(2, 2*n+1):
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MATHEMATICA
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Do[a = Table[ Prime[i], {i, 1, n} ]; c = 0; k = 2^(n - 1); While[k < 2^n, If[ Apply[ Plus, a*(-1)^(IntegerDigits[k, 2] + 1)] == 0, c++ ]; k++ ]; Print[c], {n, 1, 32, 2} ]
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PROG
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(PARI) A022894={a(n, s=0-prime(1), p=1)=if(n<=s, if(s==p, n==s, a(abs(n-p), s-p, precprime(p-1))+a(n+p, s-p, precprime(p-1))), if(s<=0, a(abs(s), max(sum(i=p+1, p+(p>1)+2*n, prime(i)), 1), prime(p+(p>1)+2*n))))} \\ M. F. Hasler, Aug 09 2015
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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