A053033 Numbers which are the average of two primes in more ways than any smaller number.

1, 2, 5, 11, 17, 24, 30, 39, 42, 45, 57, 60, 84, 90, 105, 150, 165, 195, 210, 255, 315, 390, 420, 495, 525, 570, 630, 735, 825, 840, 945, 1050, 1155, 1365, 1575, 1785, 1995, 2100, 2205, 2310, 2625, 2730, 3045, 3255, 3465, 3990, 4095

Offset: 1

Author

Len Smiley, Feb 23 2000

Keywords

• nonn

Comments

From Ahmad J. Masad, Dec 09 2019: (Start)

Conjecture 1: This sequence is infinite.

Conjecture 2: If this sequence is infinite, then for each prime number p > 2, there exists a minimum sufficiently large number k such that all terms >= k are multiples of p. (End)

Apparently, all terms >= 90 are multiples of 15. - Hugo Pfoertner, Dec 09 2019

Positions of records in A045917. - Sean A. Irvine, Dec 04 2021

Examples

			a(1) = 1: average of 0 pairs of primes;
a(2) = 2: average of 1 pair of primes (2,2);
a(3) = 5: average of 2 pairs of primes (3,7), (5,5);
a(4) = 11: average of 3 pairs of primes (3,19), (5,17), (11,11);
a(5) = 17: average of 4 pairs of primes (3,31), (5,29), (11,23), (17,17).
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..266 (terms < 10^7)

Crossrefs

Cf. A002375, A045917.

Programs

Maple

(for n>0): printlevel := -1:maxx := 0:for j from 2 to 1000 do count := 0; for k from 0 to j-2 do if (isprime(j-k) and isprime(j+k)) then count := count+1 fi od; if count>maxx then print(j,count); maxx := count fi od;

Extensions

More terms from James Sellers, Feb 25 2000