cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A022894 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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A113040, A215036, A083309 (sums of odd primes).
Cf. A022895, A022896 (r.h.s. = 1 & 2, using all primes), A083309 and A022897 - A022899 (using primes >= 3), A022900 - A022902 (using primes >=5), A022903, A022904, A022920 (using primes >= 7); A261061 - A261063 & A261045 (r.h.s. = -1); A261057, A261059, A261060 & A261044 (r.h.s. = -2).
Bisection (odd part) of A306443.

Programs

  • Maple
    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):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 05 2012
  • Mathematica
    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} ]
  • 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

Formula

Conjecture: limit_{n->oo} a(n)^(1/n) = 4. - Vaclav Kotesovec, Jun 05 2019
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

Extensions

Edited by Robert G. Wilson v, Jan 29 2002
More terms from T. D. Noe, Jan 16 2007
Edited by M. F. Hasler, Aug 09 2015

A022920 Number of solutions to c(1)*prime(4) + ... + c(n)*prime(n+3) = 2, where c(i) = +-1 for i > 1, c(1) = 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 7, 0, 12, 0, 61, 0, 131, 0, 472, 0, 2039, 0, 5924, 0, 21095, 0, 76058, 0, 274023, 0, 1032989, 0, 3694643, 0, 12987172, 0, 48417270, 0, 174274092, 0, 642785629, 0, 2402825962, 0, 8918414212, 0, 32868915523, 0, 123145191037, 0
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Each second entry is 0 because the primes that are involved are all odd and the right hand side is even. - R. J. Mathar, Aug 06 2015

Links

Crossrefs

Cf. A022894, A022895, A022896 (r.h.s. = 0, 1 & 2, using all primes), A083309 and A022897 - A022899 (using primes >= 3), A022900 - A022902 (using primes >=5), A022903, A022904 (r.h.s. = 0 & 1, using primes >= 7); A261061 - A261063 & A261045 (r.h.s. = -1); A261057, A261059, A261060 & A261044 (r.h.s. = -2).

Programs

  • Mathematica
    b[n_, s_, p_] := b[n, s, p] = If[n <= s, If[s == p, Boole[n == s], b[Abs[n - p], s - p, NextPrime[p - 1, -1]] + b[n + p, s - p, NextPrime[p - 1, -1] ]], If[s <= 0, b[Abs[s], Sum[Prime[i], {i, p + 1, p + n - 1}], Prime[p + n - 1]]]] /. Null -> 0; a[n_] := b[n, 2 - Prime[4], 4]; Array[a, 50] (* Jean-François Alcover, Feb 14 2018, after M. F. Hasler *)
  • PARI
    A022920(n)={my(p=vector(n-1,i,prime(i+4)));sum(i=1,2^(n-1),sum(j=1,#p,(1-bittest(i,j-1)<<1)*p[j],7)==2)} \\ For illustrative purpose; too slow for n >> 20. - M. F. Hasler, Aug 08 2015
  • PARI
    a(n, s=2-prime(4), p=4)=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), sum(i=p+1, p+n-1, prime(i)), prime(p+n-1)))) \\ M. F. Hasler, Aug 09 2015

Formula

a(2n-1) = 0 for all n >= 1.

Extensions

Corrected by R. J. Mathar, Aug 06 2015
a(22)-a(49) from Alois P. Heinz, Aug 06 2015
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