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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.

%I A356457 #14 Aug 14 2022 10:20:18
%S A356457 1,31,71,151,191,491,671,887,311,1151,1391,1751,1031,2711,2831,3911,
%T A356457 1991,3191,5351,9551,7031,20951,8951,8711,10631,5591,15431,10391,
%U A356457 15791,28031,20471,17111,48191,27191,31391,39191,52631,35591,42311,61871,50951,92231,70391,108071,99431,103991,96071
%N 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.
%C A356457 Empirical observation: It appears that the majority of the members of this sequence end in 1, and nearly all the rest end in 9.
%C A356457 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
%H A356457 Robert Israel, <a href="/A356457/b356457.txt">Table of n, a(n) for n = 0..319</a>
%e A356457 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.
%p A356457 M:= 10^5: # to get terms before the first term > M
%p A356457 V:= Vector(M): p:= 1:
%p A356457 do
%p A356457 p:= nextprime(p);
%p A356457 if 5*p+6 > M then break fi;
%p A356457 q:= 1;
%p A356457 do
%p A356457     q:= nextprime(q);
%p A356457   if q = p or p*q + 2*(p+q) > M then break fi;
%p A356457   r:= 1;
%p A356457   do
%p A356457     r:= nextprime(r);
%p A356457     if r = q then break fi;
%p A356457     v:= p*q + p*r + q*r;
%p A356457     if v > M then break fi;
%p A356457     V[v]:= V[v]+1;
%p A356457 od od od:
%p A356457 m:= max(V):
%p A356457 W:= Array(0..m):
%p A356457 for i from 1 to M do
%p A356457 if W[V[i]] = 0 then W[V[i]]:= i fi
%p A356457 od:
%p A356457 if member(0,W,'k') then m:= k-1 fi:
%p A356457 convert(W[0..m],list);
%Y A356457 Cf. A003415, A007304, A087053.
%K A356457 nonn
%O A356457 0,2
%A A356457 _J. M. Bergot_ and _Robert Israel_, Aug 07 2022