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 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 (list; graph; refs; listen; history; text; internal format)
 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]; Table[Print[n, " ", a[n]]; a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *) CROSSREFS Cf. A000040, A000607, A051034, A331634, A332861, A333365. Sequence in context: A044314 A044695 A211850 * A044395 A044776 A077263 Adjacent sequences: A330430 A330431 A330432 * A330434 A330435 A330436 KEYWORD nonn AUTHOR David James Sycamore, Mar 01 2020 STATUS approved

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Last modified June 18 14:18 EDT 2024. Contains 373481 sequences. (Running on oeis4.)