A002374 -id:A002374 - cp's OEIS frontend

A277581 Goldbach's problem extended to squares of nonnegative differences of primes: smallest integer >= ((A112823(n) - A234345(n))^2)/n for n >= 2.

0, 0, 1, 0, 1, 0, 5, 2, 4, 0, 1, 0, 3, 2, 3, 0, 1, 0, 2, 1, 15, 0, 5, 6, 2, 3, 12, 0, 1, 0, 11, 2, 2, 5, 3, 0, 9, 1, 1, 0, 1, 0, 1, 1, 20, 0, 3, 12, 1, 6, 7, 0, 4, 11, 1, 2, 16, 0, 1, 0, 6, 2, 1, 3, 2, 0, 14, 1, 1, 0, 1, 0, 13, 1, 1, 2, 2, 0, 5, 1, 11, 0, 2, 7, 1, 10, 4, 0

Offset: 2

Juri-Stepan Gerasimov, Oct 21 2016

nonn

Where A112823(n) + A234345(n) = 2n and A112823(n) <= A234345(n) (or nonnegative differences of primes). If n is prime, then a(n) = 0.

Conjecture: 1 <= a(n) <= m for all n, where m is largest value of a(n), i.e., the sequence of records in a(n) {1, 5, 15, 20, ..., m} is finite.

			a(8) = 5 because ((A112823(8) - A234345(8))^2)/8 = ((5 - 11)^2)/8 < 5, where 5(prime) + 11(prime) = 2*8;
a(9) = 2 because ((A112823(9) - A234345(9))^2)/9 = ((7 - 11)^2)/9 < 2, where 7(prime) + 11(prime) = 2*9;
a(10) = 4 because ((A112823(10) - A234345(10))^2)/10 = ((7 - 13)^2)/10 < 4, where 7(prime) + 13(prime) = 2*10.

Cf. A112823 (2 together with A002374), A234345, A277583 (Goldbach's problem extended to squares of prime gaps >= 2).

A328298 The smaller prime in the decomposition of 2n (>=6) into a sum of two odd primes obtained from increasing the smaller prime of such a decomposition of 2n-2.

Original entry on oeis.org

3, 3, 5, 5, 7, 5, 7, 7, 11, 11, 13, 11, 13, 13, 17, 17, 19, 17, 19, 13, 17, 19, 19, 23, 23, 19, 29, 29, 31, 23, 29, 31, 29, 31, 37, 29, 31, 37, 41, 41, 43, 41, 43, 31, 41, 43, 37, 41, 43, 43, 47, 47, 43, 53, 53, 43, 47, 53, 61, 53, 59, 61, 59, 61, 67, 53, 59

Offset: 3

Lei Zhou, Oct 11 2019

This sequence is different from A002374 from the 23rd term on.

			For the 3rd even number 6, 6=3+3;
For the 4th number 8, increasing the first prime in 6=3+3 by 2, we get 8=5+3, 5 and 3 are both primes, choose the smaller one, the second term of this sequence is 3, which makes 8=3+5;
...
For the 23rd even number 46, increasing the first prime in 44=13+31 by 2, we get 46=15+31.  15 is not prime, keep increasing: 46=17+29.  Both 17 and 29 are primes, so the 23rd term of this sequence is 17, as of 46=17+29;
...
For 28th even number 56, increasing the first prime in 54=23+31 by 2, we get 56=25+31.  25 is not prime, keep increasing, 56 = 27+29 = 29+27 = 31+25 = 33+23 = 35+21 = 37+19.  Both 37 and 19 are primes, and 19 is smaller.  So the 28th term of this sequence is 19, as of 56=19+37.

Cf. A002374, A112823.

e = 4; p1 = 1; p2 = 3; a = Table[e = e + 2; If[p1 < p2, p1 = p1 + 2, p2 = p2 + 2];
  While[! (PrimeQ[p1] && PrimeQ[p2]), p1 = p1 + 2; p2 = p2 - 2];
  If[p1 > p2, p1 = p2; p2 = e - p1]; p1, {i, 1, 67}]

cp's OEIS Frontend

A277581 Goldbach's problem extended to squares of nonnegative differences of primes: smallest integer >= ((A112823(n) - A234345(n))^2)/n for n >= 2.

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