A240710 -id:A240710 - cp's OEIS frontend

A240712 Number of decompositions of 2n into an unordered sum of two terms of A240710.

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

Offset: 1

Lei Zhou, Apr 10 2014

a(n) differs from A171611 beginning at term a(264).  To show the difference, the first 270 terms are listed.
Conjecture: a(n) > 0 for all n > 4.
This is a much stronger version of the Goldbach Conjecture.

			For n < 264, please refer to examples at A171611.
For n = 264, 2n=528. A240710 has terms {5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521} up to 528, where prime number 523 < 528 is not in the set, such that 528 = 5 + 523 is not counted in this sequence but is counted in A171611. So a(264) = A171611(264)-1 = 25-1 = 24.

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

Cf. A000040, A002375, A171611, A240710.

a240710 = {5}; Table[s = 2*n; While[a240710[[-1]] < s, p = a240710[[-1]]; While[p = NextPrime[p]; ok = 0; a1 = p - 12; a2 = p - 6; a3 = p + 6; a4 = p + 12; If[a1 > 0, If[PrimeQ[a1], ok = 1]]; If[a2 > 0, If[PrimeQ[a2], ok = 1]]; If[PrimeQ[a3], ok = 1]; If[PrimeQ[a4], ok = 1]; ok == 0]; AppendTo[a240710, p]]; pos = 0; ct = 0; While[pos++; pos <= Length[a240710], p = a240710[[pos]]; If[p <= n, If[MemberQ[a240710, s - p], ct++]]]; ct, {n, 1, 270}]

A240709 Primes p such that no number among p+-6 and p+-12 is also a prime.

Original entry on oeis.org

2, 3, 523, 617, 691, 701, 709, 719, 743, 787, 911, 937, 967, 1153, 1171, 1259, 1381, 1399, 1409, 1637, 1667, 1723, 1787, 1831, 1847, 1931, 1933, 1949, 1951, 2053, 2113, 2161, 2179, 2203, 2221, 2311, 2437, 2477, 2503, 2521, 2593, 2617, 2749, 2767, 2819, 2833

Offset: 1

Lei Zhou, Apr 10 2014

The union of A240709 and A240710 is the set of all prime numbers, i.e., A000040.

			For 2, 2+-6 and 2+-12 are all even numbers and composite. So 2 is included.
For 3, 3+-6 and 3+-12 are all multiples of 3. So 3 is included.
For each prime number p between 5 and 521, at least one number among p+-6 and p+-12 is a prime number, thus p is excluded.
For 523, 523 - 12 = 511 = 7*73, 523 - 6 = 517 = 11*47, 523 + 6 = 529 = 23^2, 523 + 12 = 535 = 5*107. They are all composites. So 523 is included.

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

Cf. A000040, A240710.

p = 1; Table[While[p = NextPrime[p]; ok = 0; a1 = p - 12; a2 = p - 6; a3 = p + 6; a4 = p + 12; If[a1 > 0, If[PrimeQ[a1], ok = 1]]; If[a2 > 0, If[PrimeQ[a2], ok = 1]]; If[PrimeQ[a3], ok = 1]; If[PrimeQ[a4], ok = 1]; ok != 0]; p, {n, 1, 46}]

cp's OEIS Frontend

A240712 Number of decompositions of 2n into an unordered sum of two terms of A240710.

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