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 A072721 #29 Jan 23 2025 20:04:27
%S A072721 1,0,1,1,2,1,4,1,4,2,6,1,10,1,8,4,10,1,15,1,17,5,16,1,26,2,22,5,29,1,
%T A072721 37,1,36,7,38,4,57,1,48,9,65,1,73,1,77,13,76,1,108,2,99,11,117,1,130,
%U A072721 5,145,14,142,1,189,1,168,19,202,5,223,1,241,17,247,1,309,1,286,24,333,4
%N A072721 Number of partitions of n into parts which are each positive powers of a single number >1 (which may vary between partitions).
%C A072721 First differs from A322968 at a(12) = 10, A322968(12) = 9.
%H A072721 Andrew Howroyd, <a href="/A072721/b072721.txt">Table of n, a(n) for n = 0..10000</a>
%H A072721 <a href="/index/Par#part">Index entries for sequences related to partitions</a>.
%F A072721 a(n) = A072721(n)-A072721(n-1). a(p)=1 for p prime.
%F A072721 a(n) = A322900(n) - 1. - _Gus Wiseman_, Jan 01 2019
%F A072721 G.f.: 1 + Sum_{k>=2} -1 + 1/Product_{j>=1} (1 - x^(A175082(k)^j)). - _Andrew Howroyd_, Jan 23 2025
%F A072721 For n >= 1, a(n) >= A379957(n). - _Antti Karttunen_, Jan 23 2025
%e A072721 a(5)=1 since the only partition without 1 as a part is 5 (a power of 5). a(6)=4 since 6 can be written as 6 (powers of 6), 3+3 (powers of 3) and 4+2 and 2+2+2 (both powers of 2).
%e A072721 From _Gus Wiseman_, Jan 01 2019: (Start)
%e A072721 The a(2) = 1 through a(12) = 10 integer partitions (A = 10, B = 11, C = 12):
%e A072721 (2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (C)
%e A072721 (22) (33) (44) (333) (55) (66)
%e A072721 (42) (422) (82) (84)
%e A072721 (222) (2222) (442) (93)
%e A072721 (4222) (444)
%e A072721 (22222) (822)
%e A072721 (3333)
%e A072721 (4422)
%e A072721 (42222)
%e A072721 (222222)
%e A072721 (End)
%e A072721 Compare above to the example section of A379957. - _Antti Karttunen_, Jan 23 2025
%t A072721 radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
%t A072721 Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],SameQ@@radbase/@#]&]],{n,30}] (* _Gus Wiseman_, Jan 01 2019 *)
%o A072721 (PARI) a(n)={if(n==0, 1, sumdiv(n, d, if(d>1&&!ispower(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 A072721 (PARI) seq(n)={Vec(1 + sum(d=2, n, if(!ispower(d), -1 + 1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)))))} \\ _Andrew Howroyd_, Jan 23 2025
%Y A072721 Cf. A018819, A023894, A052410, A072720, A072721, A102430, A175082, A322900, A322902, A322903, A322968.
%Y A072721 Cf. also A379957.
%K A072721 nonn
%O A072721 0,5
%A A072721 _Henry Bottomley_, Jul 05 2002