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 1-3 of 3 results.

A229335 Sum of sums of elements of subsets of divisors of n.

Original entry on oeis.org

1, 6, 8, 28, 12, 96, 16, 120, 52, 144, 24, 896, 28, 192, 192, 496, 36, 1248, 40, 1344, 256, 288, 48, 7680, 124, 336, 320, 1792, 60, 9216, 64, 2016, 384, 432, 384, 23296, 76, 480, 448, 11520, 84, 12288, 88, 2688, 2496, 576, 96, 63488, 228, 2976, 576, 3136, 108
Offset: 1

Views

Author

Jaroslav Krizek, Sep 20 2013

Keywords

Comments

Number of nonempty subsets of divisors of n = A100587(n).

Examples

			For n = 2^2 = 4; divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; sum of sums of elements of subsets = 1 + 2 + 4 + 3 + 5 + 6 + 7 = 28 = (2^3-1) * 2^2 = 7 * 4.

Links

Crossrefs

Cf. A229336 (product of sums of elements of subsets of divisors of n).
Cf. A229337 (sum of products of elements of subsets of divisors of n).
Cf. A229338 (product of products of elements of subsets of divisors of n).

Programs

  • Maple
    A229335 := proc(n)
    numtheory[sigma](n)*A100577(n) ;
end proc:
seq(A229335(n),n=1..100) ; # R. J. Mathar, Nov 10 2017
  • Mathematica
    Table[Total[Flatten[Subsets[Divisors[n]]]], {n, 100}] (* T. D. Noe, Sep 21 2013 *)

Formula

a(n) = A000203(n) * A100577(n) = A000203(n) * (A100587(n) + 1) / 2 = A000203(n) * 2^(A000005(n) - 1) = sigma(n) * 2^(tau(n) - 1).
a(2^n) = (2^(n+1) - 1) * 2^n.

A229338 Product of products of elements of subsets of divisors of n.

Original entry on oeis.org

1, 4, 9, 4096, 25, 2821109907456, 49, 281474976710656, 531441, 10000000000000000, 121, 39939223824273992215667642551956428337968885602521915290518994217942463316460321327052965050967304175616, 169, 2177953337809371136, 6568408355712890625, 1461501637330902918203684832716283019655932542976
Offset: 1

Views

Author

Jaroslav Krizek, Sep 20 2013

Keywords

Examples

			For n = 4; divisors of 4: {1, 2, 4}; subsets of divisors of n: {}, {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; product of products of elements of subsets = 1*1*2*4*2*4*8*8 = 4096.

Crossrefs

Cf. A229335 (sum of sums of the elements), A229336 (product of sums of the elements), A229337 (sum of products of the elements).

Programs

  • Mathematica
    Table[Times@@Times@@@Subsets[Divisors[n]],{n,20}] (* Harvey P. Dale, Mar 05 2015 *)

Formula

Conjecture: a(n) = n^s(n); where s(n) = A057711(tau(n)) = A057711(A000005(n)) = tau(n)*2^(tau(n)-2).

A229336 Product of sums of elements of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 6, 12, 5040, 30, 77598259200, 56, 1307674368000, 168480, 12703122432000, 132, 52875224823823084892891318660312910903645116196873830400000000000000, 182, 440505199411200, 493242753024000, 8222838654177922817725562880000000, 306
Offset: 1

Views

Author

Jaroslav Krizek, Sep 20 2013

Keywords

Comments

Number of nonempty subsets of divisors of n = A100587(n).

Examples

			For n = 2^2 = 4; divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; product of sums of elements of subsets = 1*2*4*3*5*6*7 = 5040 = (2^3 - 1)! = 7!.

Crossrefs

Cf. A229335 (sum of sums of elements of nonempty subsets of divisors of n),
A229337 (sum of products of elements of nonempty subsets of divisors of n),
A229338 (product of products of elements of nonempty subsets of divisors of n).

Programs

  • Mathematica
    Table[Times@@(Total/@Rest[Subsets[Divisors[n]]]),{n,20}] (* Harvey P. Dale, Jan 22 2023 *)

Formula

a(2^n) = (2^(n+1) - 1)!.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.