A332861 Primes p with the property that if q<p is the least part of a partition of p into primes, then p has at least one other prime partition with the same least part.

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

1,1

Links

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

Example

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].

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 *)

Crossrefs

Cf. A000607, A331634.

Sequence in context: A238423 A133370 A237283 * A182047 A291544 A208149

Adjacent sequences: A332858 A332859 A332860 * A332862 A332863 A332864

Keyword

nonn

Author

David James Sycamore, Feb 27 2020

Extensions

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

Status

approved