A330433 - OEIS

A330433

Numbers k such that if there is a prime partition of k with least part p, then there exists at least one other prime partition of k with least part p.

63, 161, 195, 235, 253, 425, 513, 581, 611, 615, 635, 667, 767, 779, 791, 803, 959, 1001, 1015, 1079, 1095, 1121, 1127, 1251, 1253, 1265, 1267, 1547, 1557, 1595, 1617, 1625, 1647, 1649, 1681, 1683, 1687, 1771, 1817, 1829, 1915, 1921, 2071, 2125, 2159, 2185

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OFFSET

1,1

COMMENTS

If k is prime then [k] is the only prime partition of k with least part k, and therefore k cannot be in this sequence. If k > 2 is even, then (assuming the validity of Goldbach's conjecture) there is a prime partition [p,q] of k (p <= q) in which p is the greatest possible least part and therefore no other partition of k is possible with least part p, so k is not a term. Therefore all terms of this sequence are odd composites.

LINKS

Table of n, a(n) for n=1..46.

EXAMPLE

9 is not a term because [3,3,3] is the only prime partition of 9 having 3 as least part.

63 is a term because every possible prime partition is accounted for as follows, where (m,p) means m partitions of 63 with least part p: (2198,2), (323,3), (60,5), (15,7), (5,11), (2,13), (2,17), (sum of m values = 2605 = A000607(63)). 63 must be in the sequence because (1,p) does not appear in this list, and is the smallest such number because every odd composite < 63 has at least one prime partition with unique least part (as for 9 above).

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 k; for k from a(n-1)+1

while 1 in {coeffs(b(k, 2, x))} do od; k

end: a(0):=1:

seq(a(n), n=1..40); # Alois P. Heinz, Mar 21 2020

MATHEMATICA

b[n_, p_, t_] := b[n, p, t] = If[n == 0, 1, If[p > n, 0, Function[q, Sum[b[n - p j, q, 1], {j, 1, n/p}] t^p + b[n, q, t]][NextPrime[p]]]];

a[0] = 1;

a[n_] := a[n] = Module[{k}, For[k = a[n-1]+1, MemberQ[CoefficientList[b[k, 2, x], x], 1], k++]; k];