cp's OEIS Frontend

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.

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A072721 Number of partitions of n into parts which are each positive powers of a single number >1 (which may vary between partitions).

Original entry on oeis.org

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, 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, 5, 145, 14, 142, 1, 189, 1, 168, 19, 202, 5, 223, 1, 241, 17, 247, 1, 309, 1, 286, 24, 333, 4
Offset: 0

Views

Author

Henry Bottomley, Jul 05 2002

Keywords

Comments

First differs from A322968 at a(12) = 10, A322968(12) = 9.

Examples

			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).
From _Gus Wiseman_, Jan 01 2019: (Start)
The a(2) = 1 through a(12) = 10 integer partitions (A = 10, B = 11, C = 12):
  (2)  (3)  (4)   (5)  (6)    (7)  (8)     (9)    (A)      (B)  (C)
            (22)       (33)        (44)    (333)  (55)          (66)
                       (42)        (422)          (82)          (84)
                       (222)       (2222)         (442)         (93)
                                                  (4222)        (444)
                                                  (22222)       (822)
                                                                (3333)
                                                                (4422)
                                                                (42222)
                                                                (222222)
(End)
Compare above to the example section of A379957. - _Antti Karttunen_, Jan 23 2025

Links

Crossrefs

Cf. A018819, A023894, A052410, A072720, A072721, A102430, A175082, A322900, A322902, A322903, A322968.
Cf. also A379957.

Programs

  • Mathematica
    radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],SameQ@@radbase/@#]&]],{n,30}] (* Gus Wiseman, Jan 01 2019 *)
  • 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
  • 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

Formula

a(n) = A072721(n)-A072721(n-1). a(p)=1 for p prime.
a(n) = A322900(n) - 1. - Gus Wiseman, Jan 01 2019
G.f.: 1 + Sum_{k>=2} -1 + 1/Product_{j>=1} (1 - x^(A175082(k)^j)). - Andrew Howroyd, Jan 23 2025
For n >= 1, a(n) >= A379957(n). - Antti Karttunen, Jan 23 2025

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.

Original entry on oeis.org

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, 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, 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
Offset: 1

Views

Author

Antti Karttunen, Jan 22 2025

Keywords

Examples

			The a(2) = 1 through a(12) = 9 integer partitions (A = 10, B = 11, C = 12):
  (2)  (3)  (4)   (5)  (6)    (7)  (8)     (9)    (A)      (B)  (C)
            (22)       (33)        (44)    (333)  (55)          (66)
                       (42)        (422)          (82)          (93)
                       (222)       (2222)         (442)         (444)
                                                  (4222)        (822)
                                                  (22222)       (3333)
                                                                (4422)
                                                                (42222)
                                                                (222222)
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.

Links

Crossrefs

First differs from A322968 at n=16, where a(16) = 9, while A322968(16) = 10.
Cf. also A072721, A322900.

Programs

  • PARI
    powers_of_d_reversed(n, d) = vecsort(vector(logint(n, d), i, d^i),,4);
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));
A379957(n) = if(!n,1,sumdiv(n, d, if(1==d, 0, partitions_into_parts(n-d, powers_of_d_reversed(n, d)))));
  • 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)));
  • 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

Formula

For all n >= 1, a(n) <= A072721(n).
G.f.: Sum_{k>=2} x^k/Product_{j>=1} (1 - x^(k^j)). - Andrew Howroyd, Jan 23 2025
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