A332861 Primes p with the property that if q

2, 3, 7, 13, 23, 31, 41, 79, 101, 107, 149, 163, 173, 191, 197, 269, 271, 293, 347, 419, 443, 523, 557, 647, 761, 769, 787, 1013, 1153, 1373, 1613, 1619, 1669, 1693, 1777, 1783, 1873, 2153, 2161, 2207, 2399, 2447, 2801, 2939, 2999, 3011, 3049, 3253, 3319, 3413

Offset: 1

Author

David James Sycamore, Feb 27 2020

Keywords

• nonn

Examples

			Prime 13 is a member, because the minimal primes in partitions of 13 into prime parts smaller than 13 occur at least twice: [2,2,2,2,2,3], [2,2,3,3,3], [2,2,2,2,5], [2,3,3,5], [2,2,2,7], [2,11], [3,3,7], [3,5,5]; 3 occurs twice, 2 occurs 6 times.
Prime 11 is not a member, because 3 occurs only once as a minimal prime in partitions of 11 into smaller primes: [2,2,2,2,3], [2,3,3,3], [2,2,2,5], [2,2,7], [3,3,5].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000607, A331634.

Programs

Maple

b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
a:= proc(n) option remember; local p; p:= a(n-1); do
      p:= nextprime(p); if (f-> andmap(i-> coeff(f, x, i)
          <>1, [$2..p-1]))(b(p, 2, x)) then return p fi od
    end: a(1):=2:
seq(a(n), n=1..33);  # Alois P. Heinz, Mar 13 2020

Mathematica

b[n_, p_, t_] := b[n, p, t] = If[n == 0, 1, If[p > n, 0, With[{q = NextPrime[p]}, Sum[b[n - p j, q, 1], {j, 1, n/p}] t^p + b[n, q, t]]]];
a[n_] := a[n] = Module[{p = a[n - 1]}, While[True, p = NextPrime[p]; If[AllTrue[Range[2, p-1], SeriesCoefficient[b[p, 2, x], {x, 0, #}] != 1&], Return [p]]]];
a[1] = 2;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 33}] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Extensions

a(13)-a(50) from Alois P. Heinz, Feb 28 2020