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

1, 10, 18, 26, 31, 35, 39, 80, 49, 47, 57, 53, 63, 59, 65, 67, 248, 73, 71, 79, 85, 77, 93, 105, 332, 83, 89, 111, 97, 482, 95, 103, 101, 674, 135, 129, 115, 107, 800, 113, 1040, 121, 1010, 119, 127, 125, 153, 159, 133, 1136, 145, 131, 171, 1304, 137, 151, 1520

Offset: 0

Author

Reinhard Zumkeller, Nov 30 2006

Keywords

• nonn

Comments

A125688(a(n)) = n and A125688(m) <> n for m < a(n).

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A125688.

Programs

Mathematica

nmax = 300; kmax = 10000; c = ConstantArray[Null, nmax];
For[k = 1, k <= kmax, k++,
  l = Length@Select[IntegerPartitions[k, {3}, Prime@Range@kmax], #[[1]] > #[[2]] > #[[3]] &];
  If[l <= nmax && c[[l]] == Null, c[[l]] = k];
];
Prepend[c[[1 ;; First@FirstPosition[c, Null] - 1]], 1] (* Robert Price, Apr 25 2025 *)

PARI

\\ here b(n) is A125688.
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}
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]}
{ my(nn=100); nn++; sols(nn, 10^4, i->b(i, nn)+1) } \\ Andrew Howroyd, Jan 06 2020

Extensions

Terms a(40) and beyond from Andrew Howroyd, Jan 06 2020