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.

Showing 1-10 of 18 results. Next

A301854 Number of positive special sums of integer partitions of n.

Original entry on oeis.org

1, 3, 7, 13, 25, 40, 67, 100, 158, 220, 336, 452, 649, 862, 1228, 1553, 2155, 2738, 3674, 4612, 6124, 7497, 9857, 12118, 15524, 18821, 24152, 28863, 36549, 44002, 54576, 65125, 80943, 95470, 117991, 139382, 169389, 199144, 242925, 283353, 342139, 400701, 479001
Offset: 1

Views

Author

Gus Wiseman, Mar 27 2018

Keywords

Comments

A positive special sum of an integer partition y is a number n > 0 such that exactly one submultiset of y sums to n.

Examples

			The a(4) = 13 special positive subset-sums:
1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111),
1<=(211),  3<=(211),  4<=(211),
1<=(31),   3<=(31),   4<=(31),
2<=(22),   4<=(22),
4<=(4).

Crossrefs

Cf. A000712, A108917, A122768, A275972, A276024, A284640, A299701, A299702, A299729, A301829, A301830, A301854, A301855, A301856.

Programs

  • Mathematica
    uqsubs[y_]:=Join@@Select[GatherBy[Union[Rest[Subsets[y]]],Total],Length[#]===1&];
Table[Total[Length/@uqsubs/@IntegerPartitions[n]],{n,25}]
  • Python
    from collections import Counter
from sympy.utilities.iterables import partitions, multiset_combinations
def A301854(n): return sum(sum(1 for r in Counter(sum(q) for l in range(1,len(p)+1) for q in multiset_combinations(p,l)).values() if r==1) for p in (tuple(Counter(x).elements()) for x in partitions(n))) # Chai Wah Wu, Sep 26 2023

Extensions

a(21)-a(35) from Alois P. Heinz, Apr 08 2018
a(36)-a(43) from Chai Wah Wu, Sep 26 2023

A301855 Number of divisors d|n such that no other divisor of n has the same Heinz weight A056239(d).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 4, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 4, 3, 4, 4, 6, 2, 6, 2, 6, 4, 4, 4, 5, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 4, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 4, 2, 4, 4, 7, 4, 8, 2, 6, 4, 6, 2, 4, 2, 4, 6, 6, 4, 8, 2, 6, 5, 4, 2, 6, 4, 4, 4, 8, 2, 6, 4, 6, 4, 4, 4, 4, 2, 6, 6, 9, 2, 8, 2, 8, 8
Offset: 1

Views

Author

Gus Wiseman, Mar 27 2018

Keywords

Examples

			The a(24) = 4 special divisors are 1, 2, 12, 24.

Links

Crossrefs

Cf. A000712, A056239, A108917, A112798, A122768, A275972, A276024, A284640, A296150, A299701, A299702, A299729, A301830, A301854, A301855, A301856.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uqsubs[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Total],Length[#]===1&];
Table[Length[uqsubs[primeMS[n]]],{n,100}]
  • PARI
    A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A301855(n) = if(1==n,n,my(m=Map(),w,s); fordiv(n,d,w = A056239(d); if(!mapisdefined(m, w, &s), mapput(m,w,Set([d])), mapput(m,w,setunion(Set([d]),s)))); sumdiv(n,d,(1==length(mapget(m,A056239(d)))))); \\ Antti Karttunen, Jul 01 2018

Extensions

More terms from Antti Karttunen, Jul 01 2018

A328706 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^2.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 6, 3, 6, 2, 12, 2, 6, 6, 9, 2, 12, 2, 12, 6, 6, 2, 24, 3, 6, 6, 12, 2, 22, 2, 14, 6, 6, 6, 28, 2, 6, 6, 24, 2, 22, 2, 12, 12, 6, 2, 42, 3, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2, 52, 2, 6, 12, 22, 6, 22, 2, 12, 6, 22, 2, 60, 2, 6, 12, 12, 6, 22, 2, 42
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

Comments

Dirichlet convolution of A045778 with itself.

Links

Crossrefs

Cf. A045778, A301830.

Programs

  • Mathematica
    Block[{f}, f[m_, 1] := 1; f[1, n_] := 0; f[1, 1] := 1; f[0, n_] := 0; f[m_, n_] := f[m, n] = Total[f[# - 1, n/#] & /@ Select[Divisors[n], # <= m &]]; Table[DivisorSum[n, f[n/#, n/#]*f[#, #] &], {n, 80}]] (* Michael De Vlieger, Nov 04 2020 *)

Formula

a(n) = Sum_{d|n} A045778(n/d) * A045778(d).

A339318 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^3.

Original entry on oeis.org

1, 3, 3, 9, 3, 12, 3, 22, 9, 12, 3, 39, 3, 12, 12, 51, 3, 39, 3, 39, 12, 12, 3, 105, 9, 12, 22, 39, 3, 57, 3, 108, 12, 12, 12, 135, 3, 12, 12, 105, 3, 57, 3, 39, 39, 12, 3, 258, 9, 39, 12, 39, 3, 105, 12, 105, 12, 12, 3, 201, 3, 12, 39, 221, 12, 57, 3, 39, 12, 57
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 3 kinds.

Examples

			From _Antti Karttunen_, Dec 15 2021: (Start)
For n = 8, A001055(8) = 3, as it has three ordinary factorizations: (8), (4*2), (2*2*2). When allowing each of the factors appear in three different guises (here indicated with a subscript), and where neither the order of factors nor their subscripts matter, we get the following 22 different factorizations:
  (8_3), (8_2), (8_1),
  (4_3 * 2_3), (4_3 * 2_2), (4_3 * 2_1),
  (4_2 * 2_3), (4_2 * 2_2), (4_2 * 2_1),
  (4_1 * 2_3), (4_1 * 2_2), (4_1 * 2_1),
  (2_3 * 2_3 * 2_3), (2_3 * 2_3 * 2_2), (2_3 * 2_3 * 2_1),
  (2_3 * 2_2 * 2_2), (2_3 * 2_2 * 2_1), (2_3 * 2_1 * 2_1),
  (2_2 * 2_2 * 2_2), (2_2 * 2_2 * 2_1), (2_2 * 2_1 * 2_1),
  (2_1 * 2_1 * 2_1),
therefore a(8) = 22. (End)

Links

Crossrefs

Cf. A000716, A001055, A301830, A339319, A339320, A339321, A339322, A339323, A339324.

Programs

Formula

a(p^k) = A000716(k) for prime p.

A301856 Number of subset-products (greater than 1) of factorizations of n into factors greater than 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 3, 4, 1, 12, 1, 4, 4, 14, 1, 12, 1, 12, 4, 4, 1, 29, 3, 4, 7, 12, 1, 17, 1, 27, 4, 4, 4, 36, 1, 4, 4, 29, 1, 17, 1, 12, 12, 4, 1, 62, 3, 12, 4, 12, 1, 29, 4, 29, 4, 4, 1, 53, 1, 4, 12, 47, 4, 17, 1, 12, 4, 17, 1, 90, 1, 4, 12, 12, 4, 17
Offset: 1

Views

Author

Gus Wiseman, Mar 27 2018

Keywords

Comments

For a finite multiset p of positive integers greater than 1 with product n, a pair (t > 1, p) is defined to be a subset-product if there exists a nonempty submultiset of p with product t.

Examples

			The a(12) = 12 subset-products:
12<=(2*2*3), 6<=(2*2*3), 4<=(2*2*3), 3<=(2*2*3), 2<=(2*2*3),
12<=(2*6),   6<=(2*6),   4<=(3*4),   3<=(3*4),   2<=(2*6),
12<=(3*4),
12<=(12).
The a(16) = 14 subset-products:
16<=(16),
16<=(4*4),
16<=(2*8),     8<=(2*8),     4<=(4*4),     2<=(2*8),
16<=(2*2*4),   8<=(2*2*4),   4<=(2*2*4),   2<=(2*2*4),
16<=(2*2*2*2), 8<=(2*2*2*2), 4<=(2*2*2*2), 2<=(2*2*2*2).

Crossrefs

Cf. A001055, A000712, A045778, A108917, A162247, A276024, A281116, A284640, A292886, A301829, A301830.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[Union[Times@@@Rest[Subsets[f]]]],{f,facs[n]}],{n,100}]

A339319 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^4.

Original entry on oeis.org

1, 4, 4, 14, 4, 20, 4, 40, 14, 20, 4, 76, 4, 20, 20, 105, 4, 76, 4, 76, 20, 20, 4, 236, 14, 20, 40, 76, 4, 116, 4, 252, 20, 20, 20, 306, 4, 20, 20, 236, 4, 116, 4, 76, 76, 20, 4, 656, 14, 76, 20, 76, 4, 236, 20, 236, 20, 20, 4, 476, 4, 20, 76, 574, 20, 116, 4, 76, 20, 116
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 4 kinds.

Crossrefs

Cf. A001055, A023003, A301830, A339318, A339320, A339321, A339322, A339323, A339324.

Formula

a(p^k) = A023003(k) for prime p.

A339320 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^5.

Original entry on oeis.org

1, 5, 5, 20, 5, 30, 5, 65, 20, 30, 5, 130, 5, 30, 30, 190, 5, 130, 5, 130, 30, 30, 5, 455, 20, 30, 65, 130, 5, 205, 5, 506, 30, 30, 30, 595, 5, 30, 30, 455, 5, 205, 5, 130, 130, 30, 5, 1405, 20, 130, 30, 130, 5, 455, 30, 455, 30, 30, 5, 955, 5, 30, 130, 1265, 30, 205, 5, 130
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 5 kinds.

Crossrefs

Cf. A001055, A023004, A301830, A339318, A339319, A339321, A339322, A339323, A339324.

Formula

a(p^k) = A023004(k) for prime p.

A339321 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^6.

Original entry on oeis.org

1, 6, 6, 27, 6, 42, 6, 98, 27, 42, 6, 204, 6, 42, 42, 315, 6, 204, 6, 204, 42, 42, 6, 792, 27, 42, 98, 204, 6, 330, 6, 918, 42, 42, 42, 1044, 6, 42, 42, 792, 6, 330, 6, 204, 204, 42, 6, 2682, 27, 204, 42, 204, 6, 792, 42, 792, 42, 42, 6, 1716, 6, 42, 204, 2492, 42, 330
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 6 kinds.

Crossrefs

Cf. A001055, A023005, A301830, A339318, A339319, A339320, A339322, A339323, A339324.

Formula

a(p^k) = A023005(k) for prime p.

A339322 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^7.

Original entry on oeis.org

1, 7, 7, 35, 7, 56, 7, 140, 35, 56, 7, 301, 7, 56, 56, 490, 7, 301, 7, 301, 56, 56, 7, 1281, 35, 56, 140, 301, 7, 497, 7, 1547, 56, 56, 56, 1701, 7, 56, 56, 1281, 7, 497, 7, 301, 301, 56, 7, 4711, 35, 301, 56, 301, 7, 1281, 56, 1281, 56, 56, 7, 2849, 7, 56, 301, 4522
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 7 kinds.

Crossrefs

Cf. A001055, A023006, A301830, A339318, A339319, A339320, A339321, A339323, A339324.

Formula

a(p^k) = A023006(k) for prime p.

A339323 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^8.

Original entry on oeis.org

1, 8, 8, 44, 8, 72, 8, 192, 44, 72, 8, 424, 8, 72, 72, 726, 8, 424, 8, 424, 72, 72, 8, 1960, 44, 72, 192, 424, 8, 712, 8, 2464, 72, 72, 72, 2620, 8, 72, 72, 1960, 8, 712, 8, 424, 424, 72, 8, 7768, 44, 424, 72, 424, 8, 1960, 72, 1960, 72, 72, 8, 4456, 8, 72, 424
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 8 kinds.

Crossrefs

Cf. A001055, A023007, A301830, A339318, A339319, A339320, A339321, A339322, A339324.

Formula

a(p^k) = A023007(k) for prime p.
Showing 1-10 of 18 results. Next

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