This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A322968 #20 Jan 23 2025 20:05:42
%S A322968 1,0,1,1,2,1,4,1,4,2,6,1,9,1,8,4,10,1,14,1,16,5,16,1,24,2,22,5,28,1,
%T A322968 37,1,36,7,38,4,55,1,48,9,63,1,73,1,76,12,76,1,105,2,98,11,116,1,128,
%U A322968 5,143,14,142,1,186,1,168,18,202,5,223,1,240,17,247,1,305,1,286,23
%N A322968 Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.
%C A322968 First differs from A072721 at a(12) = 9, A072721(12) = 10.
%C A322968 First differs from A379957 at a(16) = 10, A379957(16) = 9.
%H A322968 Andrew Howroyd, <a href="/A322968/b322968.txt">Table of n, a(n) for n = 0..10000</a>
%H A322968 <a href="/index/Par#part">Index entries for sequences related to partitions</a>.
%F A322968 From _Andrew Howroyd_, Jan 23 2025: (Start)
%F A322968 G.f.: 1 + Sum_{k>=2} -1 + 1/Product_{j>=1} (1 - x^(A005117(k)^j)).
%F A322968 a(p) = 1 for prime p. (End)
%e A322968 The a(2) = 1 through a(12) = 9 integer partitions (A = 10, B = 11):
%e A322968 (2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (66)
%e A322968 (22) (33) (44) (333) (55) (84)
%e A322968 (42) (422) (82) (93)
%e A322968 (222) (2222) (442) (444)
%e A322968 (4222) (822)
%e A322968 (22222) (3333)
%e A322968 (4422)
%e A322968 (42222)
%e A322968 (222222)
%e A322968 The a(20) = 16 integer partitions:
%e A322968 (10,10), (16,4),
%e A322968 (8,8,4), (16,2,2),
%e A322968 (5,5,5,5), (8,4,4,4), (8,8,2,2),
%e A322968 (4,4,4,4,4), (8,4,4,2,2),
%e A322968 (4,4,4,4,2,2), (8,4,2,2,2,2),
%e A322968 (4,4,4,2,2,2,2), (8,2,2,2,2,2,2),
%e A322968 (4,4,2,2,2,2,2,2),
%e A322968 (4,2,2,2,2,2,2,2,2),
%e A322968 (2,2,2,2,2,2,2,2,2,2).
%t A322968 radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
%t A322968 powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
%t A322968 Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],And@@powsqfQ/@#,SameQ@@radbase/@#]&]],{n,30}]
%o A322968 (PARI) a(n)={if(n==0, 1, sumdiv(n, d, if(d>1&&issquarefree(d), polcoef(1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)), n))))} \\ _Andrew Howroyd_, Jan 23 2025
%o A322968 (PARI) seq(n)={Vec(1 + sum(d=2, n, if(issquarefree(d), -1 + 1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)))))} \\ _Andrew Howroyd_, Jan 23 2025
%Y A322968 Cf. A001597, A005117, A018819, A023893, A052410, A072720, A072721, A072774, A102430, A322900, A322903, A322911, A322912, A379957.
%K A322968 nonn
%O A322968 0,5
%A A322968 _Gus Wiseman_, Jan 01 2019
%E A322968 a(66) onwards from _Andrew Howroyd_, Jan 23 2025