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A355717 Smallest number of generalized pentagonal numbers which sum to n.

%I A355717 #59 Mar 22 2025 23:34:27
%S A355717 0,1,1,2,2,1,2,1,2,2,2,3,1,2,2,1,2,2,3,2,2,3,1,2,2,3,1,2,2,2,2,2,3,2,
%T A355717 2,1,2,2,2,3,1,2,2,3,2,2,3,2,2,3,2,1,2,2,3,2,2,1,2,2,3,2,2,2,2,3,2,3,
%U A355717 3,2,1,2,2,2,3,2,3,1,2,2,2,3,2,2,2,2,2,3,3,2,3,2,1,2,2,3,2,2,3,2,1
%N A355717 Smallest number of generalized pentagonal numbers which sum to n.
%C A355717 From Euler's Pentagonal Number Theorem, every number is expressible as the sum of at most three generalized pentagonal numbers (A001318) (see Richard K. Guy reference).
%C A355717 Corresponding sums of only pentagonal numbers of positive rank are A100878(n). Those numbers are a subset of the generalized pentagonals so that a(n) <= A100878(n).
%C A355717 More specifically, by the definition given in the name, we understand the following: Given n >= 0 we seek a multiset S such that (1) S is a multiset of GPN = {0, 1, 2, 5, ...} = A001318; (2) Sum(S) = n; (3) if T is a multiset of GPN and Sum(T) = n then card(T) >= card(S). Additionally one might require that the set is not empty. If a multiset satisfies these three conditions, then a(n) = card(S). Note that no actual summation has to be performed to decide the value of a(n); only membership in GPN needs to be tested, as shown in the Maple and Python program. - _Peter Luschny_, Jul 18 2022
%D A355717 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.
%H A355717 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>.
%F A355717 a(n) <= 3.
%F A355717 a(A001318(n)) = 1.
%e A355717 Let GPN = {0, 1, 2, 5, ...} be the generalized pentagonal numbers.
%e A355717 a(0) = 0 since {} is a multiset of GPN, Sum {} = 0, and card({}) = 0.
%e A355717 a(1) = 1 since {1} is a multiset of GPN, Sum {1} = 1, and card({1}) = 1.
%e A355717 a(3) = 2 since {1, 2} is a multiset of GPN, Sum {1, 2} = 3, and card({1, 2}) = 2.
%e A355717 a(11) = 3 since {2, 2, 7} is a multiset of GPN, Sum {2, 2, 7} = 11, card({2, 2, 7}) = 3, and no other multiset S of GPN with Sum(S) = 11 has less members.
%p A355717 A355717_list := proc(upto) local P, Q, k, q, isgpn; P := []; Q := [0];
%p A355717 isgpn := k -> ormap(n -> 0 = 8*k - (n + irem(n,2)) * (3*n + 2 - irem(n,2)), [$0..k]);
%p A355717 for k from 1 to upto do
%p A355717     q := 3;
%p A355717     if isgpn(k) then
%p A355717         P := [op(P), k]; q := 1;
%p A355717         elif ormap(p -> member(k - p, P), P) then q := 2 fi:
%p A355717         Q := [op(Q), q];
%p A355717 od: Q end:
%p A355717 print(A355717_list(100));  # _Peter Luschny_, Jul 18 2022
%o A355717 (Python)
%o A355717 def A355717_list(ln: int) -> list[int]:
%o A355717     P: list[int] = []
%o A355717     Q: list[int] = [0]
%o A355717     def is_gpn(k: int) -> bool:
%o A355717         return any(8 * k == ((n + n % 2) * (3 * n + 2 - n % 2)) for n in range(k + 1))
%o A355717     for k in range(1, ln):
%o A355717         q = 3
%o A355717         if is_gpn(k):
%o A355717             P.append(k)
%o A355717             q = 1
%o A355717         elif any([(k - p) in P for p in P]):
%o A355717             q = 2
%o A355717         Q.append(q)
%o A355717     return Q
%o A355717 print(A355717_list(100))  # _Peter Luschny_, Jul 18 2022
%Y A355717 Cf. A001318, A093519 (indices of 3's).
%Y A355717 Cf. A100878.
%K A355717 nonn
%O A355717 0,4
%A A355717 _Bernard Schott_, Jul 15 2022