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.

Previous Showing 11-17 of 17 results.

A338506 a(n) is the number of subsets of divisors of n.

Original entry on oeis.org

2, 4, 4, 8, 4, 16, 4, 16, 8, 16, 4, 64, 4, 16, 16, 32, 4, 64, 4, 64, 16, 16, 4, 256, 8, 16, 16, 64, 4, 256, 4, 64, 16, 16, 16, 512, 4, 16, 16, 256, 4, 256, 4, 64, 64, 16, 4, 1024, 8, 64, 16, 64, 4, 256, 16, 256, 16, 16, 4, 4096, 4, 16, 64, 128, 16, 256, 4, 64
Offset: 1

Views

Author

Rémy Sigrist, Oct 31 2020

Keywords

Comments

In contrast to A100587, we take into account the empty set.

Examples

			For n = 6:
- 6 has 4 divisors,
- so a(n) = 2^4 = 16.

Links

Crossrefs

Cf. A000005, A100587.

Programs

  • Mathematica
    Table[2^DivisorSigma[0, n], {n, 68}]
  • PARI
    a(n) = 2^numdiv(n)

Formula

a(n) = 2^A000005(n) = A100587(n) + 1.

A229253 Total number of elements of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 4, 4, 12, 4, 32, 4, 32, 12, 32, 4, 192, 4, 32, 32, 80, 4, 192, 4, 192, 32, 32, 4, 1024, 12, 32, 32, 192, 4, 1024, 4, 192, 32, 32, 32, 2304, 4, 32, 32, 1024, 4, 1024, 4, 192, 192, 32, 4, 5120, 12, 192, 32, 192, 4, 1024, 32, 1024, 32, 32, 4, 24576, 4, 32, 192
Offset: 1

Views

Author

Jaroslav Krizek, Sep 29 2013

Keywords

Comments

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

Examples

			For n = 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}; total number  of elements of subsets = 1 + 1 + 1 + 2 + 2 + 2 + 3 = 12.

Links

Crossrefs

Cf. A000005, A000337, A001787, A003506, A100587.

Programs

Formula

a(n) = A001787(A000005(n)) = A000005(n) * 2^(A000005(n)-1) = A100587(n) + A000337(n-1) = tau(n) * 2^(tau(n)-1).

A338507 Irregular table T(n, k) read by rows, n > 0 and k = 1..A000005(n); T(n, k) is the number of subsets of divisors of n with least common multiple of elements equal to the k-th divisor of n.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 10, 2, 2, 2, 2, 4, 8, 2, 2, 4, 2, 2, 2, 10, 2, 2, 2, 2, 2, 4, 10, 44, 2, 2, 2, 2, 2, 10, 2, 2, 2, 10, 2, 2, 4, 8, 16, 2, 2, 2, 2, 2, 10, 4, 44, 2, 2, 2, 2, 4, 2, 10, 44, 2, 2, 2, 10, 2, 2, 2, 10, 2, 2, 2, 2, 2, 4, 10, 8, 44, 184
Offset: 1

Views

Author

Rémy Sigrist, Oct 31 2020

Keywords

Comments

All terms are even (as the presence of 1 in a set does not change the least common multiple of its elements).

Examples

			Triangle begins:
     1: [2]
     2: [2, 2]
     3: [2, 2]
     4: [2, 2, 4]
     5: [2, 2]
     6: [2, 2, 2, 10]
     7: [2, 2]
     8: [2, 2, 4, 8]
     9: [2, 2, 4]
    10: [2, 2, 2, 10]
    11: [2, 2]
    12: [2, 2, 2, 4, 10, 44]
    13: [2, 2]
    14: [2, 2, 2, 10]
    15: [2, 2, 2, 10]

Crossrefs

Cf. A000005, A027750, A076078, A100587, A338508 (GCD variant).

Programs

  • PARI
    row(n) = { my (d=divisors(n), r=vector(#d)); for (m=0, 2^#d-1, r[setsearch(d, lcm(vecextract(d, m)))]++); r }

Formula

Sum_{k = 1..A000005(n)} T(n, k) = 1 + A100587(n).
T(n, A000005(n)) = A076078(n) for any n > 1.
T(n, 1) = 2.
T(n, k) = A338508(n, A000005(n)+1-k) for k = 2..A000005(n).

A338508 Irregular table T(n, k) read by rows, n > 0 and k = 1..A000005(n); T(n, k) is the number of nonempty subsets of divisors of n with greatest common divisor of elements equal to the k-th divisor of n.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 2, 1, 2, 1, 10, 2, 2, 1, 2, 1, 8, 4, 2, 1, 4, 2, 1, 10, 2, 2, 1, 2, 1, 44, 10, 4, 2, 2, 1, 2, 1, 10, 2, 2, 1, 10, 2, 2, 1, 16, 8, 4, 2, 1, 2, 1, 44, 4, 10, 2, 2, 1, 2, 1, 44, 10, 2, 4, 2, 1, 10, 2, 2, 1, 10, 2, 2, 1, 2, 1, 184, 44, 8, 10, 4, 2, 2, 1
Offset: 1

Views

Author

Rémy Sigrist, Oct 31 2020

Keywords

Examples

			Triangle begins:
     1: [1]
     2: [2, 1]
     3: [2, 1]
     4: [4, 2, 1]
     5: [2, 1]
     6: [10, 2, 2, 1]
     7: [2, 1]
     8: [8, 4, 2, 1]
     9: [4, 2, 1]
    10: [10, 2, 2, 1]
    11: [2, 1]
    12: [44, 10, 4, 2, 2, 1]
    13: [2, 1]
    14: [10, 2, 2, 1]
    15: [10, 2, 2, 1]

Crossrefs

Cf. A000005, A076078, A100587, A338507 (LCM variant).

Programs

  • Mathematica
    Array[Tally[Map[GCD @@ # &, Rest[Subsets@ Divisors[#]]]][[All, -1]] &, 24] // Flatten (* Michael De Vlieger, Nov 04 2020 *)
  • PARI
    row(n) = { my (d=divisors(n), r=vector(#d)); for (m=1, 2^#d-1, r[setsearch(d, gcd(vecextract(d, m)))]++); r }

Formula

Sum_{k = 1..A000005(n)} T(n, k) = A100587(n).
T(n, 1) = A076078(n).
T(n, k) = A338507(n, A000005(n)+1-k) for k = 1..A000005(n)-1.
T(n, A000005(n)) = 1.

A339638 Number of nonempty sets of distinct positive integers that have a least common multiple <= n.

Original entry on oeis.org

1, 3, 5, 9, 11, 21, 23, 31, 35, 45, 47, 91, 93, 103, 113, 129, 131, 175, 177, 221, 231, 241, 243, 427, 431, 441, 449, 493, 495, 713, 715, 747, 757, 767, 777, 1177, 1179, 1189, 1199, 1383, 1385, 1603, 1605, 1649, 1693, 1703, 1705, 2457, 2461, 2505, 2515, 2559, 2561, 2745, 2755
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 11 2020

Keywords

Comments

Partial sums of A076078.

Examples

			a(5) = 11 sets: {1}, {2}, {3}, {4}, {5}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 4} and {1, 2, 4}.

Links

Crossrefs

Cf. A000005, A076078, A100587.

Programs

  • Mathematica
    Table[Sum[Sum[MoebiusMu[k/d] (2^DivisorSigma[0, d] - 1), {d, Divisors[k]}], {k, n}], {n, 55}]
Accumulate[Table[Sum[MoebiusMu[k/d] (2^DivisorSigma[0, d] - 1), {d, Divisors[k]}], {k, 1, 60}]] (* Vaclav Kotesovec, Dec 25 2020 *)
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, moebius(k/d) * (2^numdiv(d) - 1))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} mu(k/d) * (2^tau(d) - 1), where tau = A000005.

A339667 Number of nonempty subsets of divisors of n having a common factor > 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 3, 5, 1, 19, 1, 5, 5, 15, 1, 19, 1, 19, 5, 5, 1, 71, 3, 5, 7, 19, 1, 37, 1, 31, 5, 5, 5, 111, 1, 5, 5, 71, 1, 37, 1, 19, 19, 5, 1, 271, 3, 19, 5, 19, 1, 71, 5, 71, 5, 5, 1, 347, 1, 5, 19, 63, 5, 37, 1, 19, 5, 37, 1, 703, 1, 5, 19, 19, 5, 37, 1, 271
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2020

Keywords

Examples

			a(12) = 19 subsets: {2}, {3}, {4}, {6}, {12}, {2, 4}, {2, 6}, {2, 12}, {3, 6}, {3, 12}, {4, 6}, {4, 12}, {6, 12}, {2, 4, 6}, {2, 4, 12}, {2, 6, 12}, {3, 6, 12}, {4, 6, 12} and {2, 4, 6, 12}.

Links

Crossrefs

Cf. A000005, A008683, A027750, A076078, A100587, A109511.

Programs

  • Mathematica
    Table[-DivisorSum[n, MoebiusMu[n/#] (2^DivisorSigma[0, #] - 1) &, # < n &], {n, 80}]
  • PARI
    A339667(n) = -sumdiv(n, d, if(d==n,0, moebius(n/d)*((2^numdiv(d))-1))); \\ Antti Karttunen, Dec 15 2021

Formula

a(n) = -Sum_{d|n, d < n} mu(n/d) * (2^tau(d) - 1), where tau = A000005, and mu = A008683.
a(n) = A100587(n) - A076078(n).
a(p) = 1 for p prime.

A229334 Product of numbers of elements of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 2, 2, 24, 2, 20736, 2, 20736, 24, 20736, 2, 11501279977342425366528000000, 2, 20736, 20736, 309586821120, 2, 11501279977342425366528000000, 2, 11501279977342425366528000000, 20736, 20736, 2
Offset: 1

Views

Author

Jaroslav Krizek, Sep 30 2013

Keywords

Comments

Number of nonempty subsets of divisors of n = A100587(n).
Also product of sizes of all the subsets of set of divisors of n.

Examples

			For n = 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 numbers of elements of subsets = 1*1*1*2*2*2*3 = 24.
For n = 4; tau(4) = 3; a(4) = [1^(3!/((3-1)!*1!))] * [2^(3!/((3-2)!*2!))] * [3^(3!/((3-3)!*3!))] = 1^3 * 2^3 * 3^1 = 24.

Crossrefs

Cf. A000005, A100587, A229333.

Programs

  • Mathematica
    Table[Times @@ Rest[Length /@ Subsets[Divisors[n]]], {n, 23}] (* T. D. Noe, Oct 01 2013 *)

Formula

a(n) = product[k=1..tau(n)] k^C(tau(n),k) = product[k=1..tau(n)] k^(tau(n)!/((tau(n)-k)!*k!)).
Previous Showing 11-17 of 17 results.

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