cp's OEIS Frontend

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

%I A053033 #34 Jul 02 2025 16:01:59
%S A053033 1,2,5,11,17,24,30,39,42,45,57,60,84,90,105,150,165,195,210,255,315,
%T A053033 390,420,495,525,570,630,735,825,840,945,1050,1155,1365,1575,1785,
%U A053033 1995,2100,2205,2310,2625,2730,3045,3255,3465,3990,4095
%N A053033 Numbers which are the average of two primes in more ways than any smaller number.
%C A053033 From _Ahmad J. Masad_, Dec 09 2019: (Start)
%C A053033 Conjecture 1: This sequence is infinite.
%C A053033 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)
%C A053033 Apparently, all terms >= 90 are multiples of 15. - _Hugo Pfoertner_, Dec 09 2019
%C A053033 Positions of records in A045917. - _Sean A. Irvine_, Dec 04 2021
%H A053033 Giovanni Resta, <a href="/A053033/b053033.txt">Table of n, a(n) for n = 1..266</a> (terms < 10^7)
%e A053033 a(1) = 1: average of 0 pairs of primes;
%e A053033 a(2) = 2: average of 1 pair of primes (2,2);
%e A053033 a(3) = 5: average of 2 pairs of primes (3,7), (5,5);
%e A053033 a(4) = 11: average of 3 pairs of primes (3,19), (5,17), (11,11);
%e A053033 a(5) = 17: average of 4 pairs of primes (3,31), (5,29), (11,23), (17,17).
%p A053033 (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;
%Y A053033 Cf. A002375, A045917.
%K A053033 nonn
%O A053033 1,2
%A A053033 _Len Smiley_, Feb 23 2000
%E A053033 More terms from _James Sellers_, Feb 25 2000