cp's OEIS Frontend

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.

%I A330433 #28 Jul 18 2021 02:24:47
%S A330433 63,161,195,235,253,425,513,581,611,615,635,667,767,779,791,803,959,
%T A330433 1001,1015,1079,1095,1121,1127,1251,1253,1265,1267,1547,1557,1595,
%U A330433 1617,1625,1647,1649,1681,1683,1687,1771,1817,1829,1915,1921,2071,2125,2159,2185
%N 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.
%C A330433 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.
%e A330433 9 is not a term because [3,3,3] is the only prime partition of 9 having 3 as least part.
%e A330433 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).
%p A330433 b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
%p A330433       add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
%p A330433     end:
%p A330433 a:= proc(n) option remember; local k; for k from a(n-1)+1
%p A330433       while 1 in {coeffs(b(k, 2, x))} do od; k
%p A330433     end: a(0):=1:
%p A330433 seq(a(n), n=1..40);  # _Alois P. Heinz_, Mar 21 2020
%t A330433 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]]]];
%t A330433 a[0] = 1;
%t A330433 a[n_] := a[n] = Module[{k}, For[k = a[n-1]+1, MemberQ[CoefficientList[b[k, 2, x], x], 1], k++]; k];
%t A330433 Table[Print[n, " ", a[n]]; a[n], {n, 1, 40}] (* _Jean-François Alcover_, Nov 26 2020, after _Alois P. Heinz_ *)
%Y A330433 Cf. A000040, A000607, A051034, A331634, A332861, A333365.
%K A330433 nonn
%O A330433 1,1
%A A330433 _David James Sycamore_, Mar 01 2020