A240712 -id:A240712 - cp's OEIS frontend

A276034 a(n) is the number of decompositions of 2n into an unordered sum of two primes in A274987.

0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 3, 2, 1, 2, 2, 2, 1, 2, 1, 0, 2, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 1, 2, 4, 3, 1, 5, 3, 2, 5, 1, 2, 2, 2, 5, 2, 3, 4, 5, 3, 2, 5, 2, 1, 4, 0, 1, 5, 3, 1, 3, 5, 4, 4, 3, 2, 4, 3, 3, 4, 2, 3, 7, 2, 2, 3, 2, 2, 2

Offset: 1

Lei Zhou, Nov 15 2016

The two primes are allowed to be the same.
It is conjectured that the primes in A274987 (a subset of all primes) are sufficient to decomposite even numbers into two primes in A274987 when n > 958.
This sequence provides a very tight alternative of the Goldbach conjecture for all positive integers, in which indices of zero terms form a complete sequence {1, 2, 16, 26, 64, 97, 107, 122, 146, 167, 194, 391, 451, 496, 707, 856, 958}.
There is no more zero terms of a(n) tested up to n = 100000.

			A274987 = {3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 59, 61, 73, 79, 83, 89, 101, 103, 109, ...}.
For n=3, 2n=6 = 3+3, one case of decomposition, so a(3)=1;
for n=4, 2n=8 = 3+5, one case of decomposition, so a(4)=1;
...
for n=17, 2n=34 = 3+31 = 11+23 = 17+17, three cases of decompositions, so a(17)=3.

Lei Zhou, Table of n, a(n) for n = 1..10000
Lei Zhou, List plot of the first 10000 terms of a(n).

Cf. A002375, A045917, A001031, A274987, A171611, A240708, A240712, A230443.

p = 3; sp = {p}; a = Table[m = 2*n; l = Length[sp]; While[sp[[l]] < m, While[p = NextPrime[p]; cp = 2*3^(Floor[Log[3, 2*p - 1]]) - p; ! PrimeQ[cp]]; AppendTo[sp, p]; l++]; ct = 0; Do[If[(2*sp[[i]] <= m) && (MemberQ[sp, m - sp[[i]]]), ct++], {i, 1, l}]; ct, {n, 1, 87}]

A276520 a(n) is the number of decompositions of n into unordered form p + c*q, where p, q are terms of A274987, c=1 for even n-s and c=2 for odd n-s.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 1, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 0, 3, 3, 1, 2, 4, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 0, 2, 2, 0, 1, 3, 1, 3, 2, 0, 2, 3, 3, 3, 3, 3, 2, 3, 2, 2, 2, 2, 2, 3, 3, 2, 2, 4, 1, 2, 2, 3, 4, 4, 3, 4

Offset: 1

Lei Zhou, Nov 11 2016

p=q is allowed.
It is conjectured that the primes p, q in A274987 (a subset of all primes) are sufficient to decomposite all numbers into p and c*q (c=1 when n is even, 2 when c is odd) when n > 2551.
This sequence provides a very tight alternative of the Goldbach conjecture for all positive integers, in which indices of zero terms form a complete sequence {1, 2, 3, 4, 5, 7, 32, 52, 55, 61, 128, 194, 214, 244, 292, 334, 388, 782, 902, 992, 1414, 1571, 1712, 1916, 2551}.
There are no more zero terms of a(n) up to n = 100000.

			A274987 = {3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 59, 61, 73, 79, 83, 89, 101, 103, 109, ...}
For n=6, 6 = 3+3, one case of decomposition, so a(6)=1;
For n=7, 7 < 3+2*3=9, no eligible case could be found, so a(7)=0;
...
For n=17, 17 = 3+2*7 = 7+2*5 = 11+2*3, three cases of decompositions, so a(17)=3.

Lei Zhou, Table of n, a(n) for n = 1..10000
Lei Zhou, List plot of the first 10000 terms of a(n).

Cf. A002375, A045917, A001031, A274987, A171611, A240708, A240712, A230443, A276034, A103151, A001031.

p = 3; sp = {p}; Table[l = Length[sp]; While[sp[[l]] < n, While[p = NextPrime[p]; cp = 2*3^(Floor[Log[3, 2*p - 1]]) - p; ! PrimeQ[cp]]; AppendTo[sp, p]; l++]; c = 2 - Mod[n + 1, 2]; ct = 0; Do[If[MemberQ[sp, n - c*sp[[i]]], If[c == 1, If[(2*sp[[i]]) <= n, ct++], ct++]], {i, 1, l}]; ct, {n, 1, 87}]

A240713 Number of decompositions of 2n=p1+p2 (prime p1 <= p2), where at least one other such pair 2n=p3+p4 (prime p3 <= p4) exists such that |p1-p3|= 6 or 12.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 3, 4, 2, 2, 4, 2, 3, 5, 3, 3, 5, 2, 4, 6, 2, 4, 5, 0, 4, 5, 4, 4, 7, 3, 4, 6, 3, 4, 9, 3, 5, 7, 3, 5, 7, 3, 5, 7, 5, 6, 10, 5, 6, 12, 2, 5, 10, 2, 6, 7, 4, 4, 4, 5, 7, 9, 6, 5, 11, 0, 6, 10, 3, 7, 8, 4, 4, 13, 8

Offset: 1

Lei Zhou, Apr 10 2014

It is conjectured that a(n)=0 only when n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 34, 76, 229.
The 2n decompositions counted in this sequence are a subset of 2n decompositions as of in Goldbach conjecture (A002375).
Per definition, all nonzero terms are greater than 1.

			For n = 11, 2n=22. 22 = 3 + 19 = 5 + 17 = 11 + 11. |5-11|=6 so pair 5+17 and 11+11 are counted. So a(11)=2.
...
For n = 17, 2n=34. 34 = 3 + 31 = 5 + 29 = 11 + 23 = 17 + 17. |5-11|=6, so 5+29 and 11+23 are counted. Also since |11-17|=6, 17+17 is also counted (where 11+23 is already counted). In case |5-17|=12, both instances are already counted. So overall three instances are found. a(17)=3.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A000040, A002375, A240712.

Table[s = 2*n; ct = 0; p = 1; While[p = NextPrime[p]; p <= n, If[PrimeQ[s - p], ok = 0; a1 = p - 12; b1 = s - a1; a2 = p - 6; b2 = s - a2; a3 = p + 6; b3 = s - a3; a4 = p + 12; b4 = s - a4; If[a1 > 0, If[PrimeQ[a1] && PrimeQ[b1], ok = 1]]; If[a2 > 0, If[PrimeQ[a2] && PrimeQ[b2], ok = 1]]; If[a3 <= n, If[PrimeQ[a3] && PrimeQ[b3], ok = 1]]; If[a4 <= n, If[PrimeQ[a4] && PrimeQ[b4], ok = 1]]; If[ok == 1, ct++]]]; ct, {n, 1, 85}]

A254041 Number of decompositions of 2n into an unordered sum of two sexy primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 1, 3, 4, 2, 2, 4, 2, 3, 5, 3, 3, 5, 2, 4, 6, 2, 4, 6, 2, 4, 6, 4, 3, 6, 4, 3, 7, 4, 3, 8, 3, 4, 7, 3, 4, 7, 4, 5, 7, 5, 5, 9, 5, 5, 12, 4, 4, 10, 3, 5, 7, 4, 5, 6, 5, 6, 8, 4, 5, 9, 2, 5, 8, 3, 5, 8, 4, 4, 9, 6, 4, 9

Offset: 1

Lei Zhou, Jan 23 2015

"Sexy primes" are listed in A136207.

It is conjectured that a(n) > 0 for n > 4.

			When n = 79, 2n = 158 = 7 + 151 = 19 + 139 = 31 + 127 = 61 + 97 = 79 + 79 has five "two prime decompositions". Among the involved prime numbers 7, 19, 31, 61, 79, 97, 127, 139, 151, prime 127 and 139 are not sexy primes. So only three decompositions, 158 = 7 + 151 = 61 + 97 = 79 + 79 satisfy the definition of this sequence. Thus a(79) = 3.

Lei Zhou, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Math, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021]
Wikipedia, Sexy Primes
Lei Zhou, Plot of a(n) up to n=1000000.

Cf. A136207, A023201, A240712, A002375.

Table[e = 2 n; ct = 0; p = 2; While[p = NextPrime[p]; p <= n, q = e - p; If[PrimeQ[q], If[(((p > 6) && PrimeQ[p - 6]) || PrimeQ[p + 6]) && (((q > 6) && PrimeQ[q - 6]) || PrimeQ[q + 6]), ct++]]]; ct, {n, 87}]

A376510 a(n) is the number of pairs of primes p+q=2*(n+4) with 5 <= p <= n such that either p+6 or q+6 is also prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 1, 3, 4, 1, 2, 4, 2, 3, 4, 3, 3, 5, 2, 3, 6, 1, 4, 6, 2, 4, 5, 4, 4, 6, 4, 4, 8, 3, 3, 8, 3, 5, 7, 2, 4, 7, 4, 5, 6, 5, 6, 9, 5, 4, 12, 3, 5, 10, 2, 5, 7, 5, 5, 6, 6, 5, 11, 5, 4, 11, 2, 7, 8, 3, 6, 10, 5, 4, 9, 7, 5, 11, 6

Offset: 1

Lei Zhou, Sep 25 2024

It is hypothesized that all terms of this sequence are positive integers.

If the above hypothesis is true, the Goldbach Hypothesis is true, since for every even number 2n, if there is a Goldbach decomposition p+q=2n meets the condition of this sequence, p+q+6=2n+6 forms at least one Goldbach decomposition of 2n+6.

			For n=1, 2*(n+4)=10, 10=5+5, and 5+6=11 is a prime. Thus a(1)=1;
For n=2, 2*(n+4)=12, 12=5+7, and 5+6=11 is a prime. Thus a(2)=1;
...
For n=14, 2*(n+4)=36, 36=5+31 (5+6=11); 7+29 (7+6=13); 13+23 (13+6=19); 17+19 (17+6=23), four cases found.  Thus a(14)=4.

Cf. A240712, A171611, A254041.

res = {}; Do[n[2] = i*6; n[1] = n[2] - 2; n[3] = n[2] + 2;
 Do[c[j] = 0; p[j] = NextPrime[n[j]/2 - 1];
  While[q[j] = n[j] - p[j];
   If[PrimeQ[q[j]] && q[j] > 3,
    If[PrimeQ[p[j] + 6] || PrimeQ[q[j] + 6], c[j]++]];
   p[j] < n[j] - 5, p[j] = NextPrime[p[j]]], {j, 1, 3}];
 AppendTo[res, c[1]]; AppendTo[res, c[2]];
 AppendTo[res, c[3]], {i, 2, 29}]; Print[res]

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A276034 a(n) is the number of decompositions of 2n into an unordered sum of two primes in A274987.

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A276520 a(n) is the number of decompositions of n into unordered form p + c*q, where p, q are terms of A274987, c=1 for even n-s and c=2 for odd n-s.

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A240713 Number of decompositions of 2n=p1+p2 (prime p1 <= p2), where at least one other such pair 2n=p3+p4 (prime p3 <= p4) exists such that |p1-p3|= 6 or 12.

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A254041 Number of decompositions of 2n into an unordered sum of two sexy primes.

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A376510 a(n) is the number of pairs of primes p+q=2*(n+4) with 5 <= p <= n such that either p+6 or q+6 is also prime.

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