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 A379957 #20 Jan 23 2025 17:43:50
%S A379957 0,1,1,2,1,4,1,4,2,6,1,9,1,8,4,9,1,15,1,15,5,16,1,23,2,22,5,25,1,37,1,
%T A379957 31,7,38,4,49,1,48,9,55,1,73,1,66,12,76,1,93,2,99,11,101,1,129,5,124,
%U A379957 14,142,1,167,1,168,17,174,5,223,1,211,17,247,1,269,1,286,24,293,4,355,1,347,21,392,1,432,6,452,25
%N A379957 Number of partitions of n where the smallest part is a divisor d > 1 of n, and the other parts are positive powers of that divisor.
%H A379957 Antti Karttunen, <a href="/A379957/b379957.txt">Table of n, a(n) for n = 1..20000</a>
%H A379957 <a href="/index/Par#part">Index entries for sequences related to partitions</a>.
%F A379957 For all n >= 1, a(n) <= A072721(n).
%F A379957 G.f.: Sum_{k>=2} x^k/Product_{j>=1} (1 - x^(k^j)). - _Andrew Howroyd_, Jan 23 2025
%e A379957 The a(2) = 1 through a(12) = 9 integer partitions (A = 10, B = 11, C = 12):
%e A379957 (2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (C)
%e A379957 (22) (33) (44) (333) (55) (66)
%e A379957 (42) (422) (82) (93)
%e A379957 (222) (2222) (442) (444)
%e A379957 (4222) (822)
%e A379957 (22222) (3333)
%e A379957 (4422)
%e A379957 (42222)
%e A379957 (222222)
%e A379957 Note how this differs from A072721 first at n=12 (that has value A072721(12)=10 instead of 9) because this doesn't count the partition (84) of 12, as although both 8 and 4 are powers of 2 (which is a divisor of 12), the 2 itself is not included in that partition as its smallest term and 8 is not a power of 4.
%o A379957 (PARI)
%o A379957 powers_of_d_reversed(n, d) = vecsort(vector(logint(n, d), i, d^i),,4);
%o A379957 partitions_into_parts(n, parts, from=1) = if(0==n, 1 , my(s=0); for(i=from, #parts, if(parts[i]<=n, s += partitions_into_parts(n-parts[i], parts, i))); (s));
%o A379957 A379957(n) = if(!n,1,sumdiv(n, d, if(1==d, 0, partitions_into_parts(n-d, powers_of_d_reversed(n, d)))));
%o A379957 (PARI) A379957(n) = sumdiv(n, d, if(d>1, polcoef(1/prod(j=1, logint(n,d), 1 - 'x^(d^j), Ser(1, 'x, n-d+1)), n-d)));
%o A379957 (PARI) seq(n)={Vec(sum(d=2, n, x^d/prod(j=1, logint(n,d), 1 - x^(d^j), Ser(1,x,1+n-d))), -n)} \\ _Andrew Howroyd_, Jan 23 2025
%Y A379957 First differs from A322968 at n=16, where a(16) = 9, while A322968(16) = 10.
%Y A379957 Cf. also A072721, A322900.
%K A379957 nonn
%O A379957 1,4
%A A379957 _Antti Karttunen_, Jan 22 2025