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A125689 a(n) is the smallest number having exactly n partitions into three distinct primes.

%I A125689 #13 Apr 26 2025 08:28:20
%S A125689 1,10,18,26,31,35,39,80,49,47,57,53,63,59,65,67,248,73,71,79,85,77,93,
%T A125689 105,332,83,89,111,97,482,95,103,101,674,135,129,115,107,800,113,1040,
%U A125689 121,1010,119,127,125,153,159,133,1136,145,131,171,1304,137,151,1520
%N A125689 a(n) is the smallest number having exactly n partitions into three distinct primes.
%C A125689 A125688(a(n)) = n and A125688(m) <> n for m < a(n).
%H A125689 Andrew Howroyd, <a href="/A125689/b125689.txt">Table of n, a(n) for n = 0..1000</a>
%t A125689 nmax = 300; kmax = 10000; c = ConstantArray[Null, nmax];
%t A125689 For[k = 1, k <= kmax, k++,
%t A125689   l = Length@Select[IntegerPartitions[k, {3}, Prime@Range@kmax], #[[1]] > #[[2]] > #[[3]] &];
%t A125689   If[l <= nmax && c[[l]] == Null, c[[l]] = k];
%t A125689 ];
%t A125689 Prepend[c[[1 ;; First@FirstPosition[c, Null] - 1]], 1] (* _Robert Price_, Apr 25 2025 *)
%o A125689 (PARI) \\ here b(n) is A125688.
%o A125689 b(n, brk=oo)={my(s=0); forprime(p=2, n\3, if((n-p)%2==0, forprime(q=p+1, (n-p)/2-1, if(isprime(n-p-q), s++; if(s>=brk, return(-1))) ))); s}
%o A125689 sols(n, lim, f)={my(u=vector(n), r=n); for(i=1, lim, my(t=f(i)); if(t>0 && t<=#u && !u[t], u[t]=i; r--; if(r==0, return(u)))); my(m=1); while(m<=#u && u[m], m++); u[1..m-1]}
%o A125689 { my(nn=100); nn++; sols(nn, 10^4, i->b(i, nn)+1) } \\ _Andrew Howroyd_, Jan 06 2020
%Y A125689 Cf. A125688.
%K A125689 nonn
%O A125689 0,2
%A A125689 _Reinhard Zumkeller_, Nov 30 2006
%E A125689 Terms a(40) and beyond from _Andrew Howroyd_, Jan 06 2020