

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.


1



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



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(np*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(n1)+1
while 1 in {coeffs(b(k, 2, x))} do od; k
end: a(0):=1:


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[n1]+1, MemberQ[CoefficientList[b[k, 2, x], x], 1], k++]; k];


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



