A356457 a(n) is the least number that can be written in exactly n ways as p*q + q*r + p*r where (p,q,r) is an unordered triple of distinct primes.

1, 31, 71, 151, 191, 491, 671, 887, 311, 1151, 1391, 1751, 1031, 2711, 2831, 3911, 1991, 3191, 5351, 9551, 7031, 20951, 8951, 8711, 10631, 5591, 15431, 10391, 15791, 28031, 20471, 17111, 48191, 27191, 31391, 39191, 52631, 35591, 42311, 61871, 50951, 92231, 70391, 108071, 99431, 103991, 96071

Offset: 0

Author

J. M. Bergot and Robert Israel, Aug 07 2022

Keywords

• nonn

Comments

Empirical observation: It appears that the majority of the members of this sequence end in 1, and nearly all the rest end in 9.

Conjecture: a(n) = A003415(A007304(k)) and a(n) is the least number where at least n solutions for k exist. - Thomas Scheuerle, Aug 08 2022

Examples

			a(3) = 151 because 151 is the first number that can be written in exactly 3 ways: 151 = 3*7 + 3*13 + 7*13 = 3*5 + 3*17 + 5*17 = 2*3 + 2*29 + 3*29.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..319

Crossrefs

Cf. A003415, A007304, A087053.

Programs

Maple

M:= 10^5: # to get terms before the first term > M
V:= Vector(M): p:= 1:
do
p:= nextprime(p);
if 5*p+6 > M then break fi;
q:= 1;
do
    q:= nextprime(q);
  if q = p or p*q + 2*(p+q) > M then break fi;
  r:= 1;
  do
    r:= nextprime(r);
    if r = q then break fi;
    v:= p*q + p*r + q*r;
    if v > M then break fi;
    V[v]:= V[v]+1;
od od od:
m:= max(V):
W:= Array(0..m):
for i from 1 to M do
if W[V[i]] = 0 then W[V[i]]:= i fi
od:
if member(0,W,'k') then m:= k-1 fi:
convert(W[0..m],list);