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 1-7 of 7 results.

A289132 Indices of records in A063974.

Original entry on oeis.org

1, 12, 24, 60, 72, 216, 240, 720, 1440, 2160, 2880, 4320, 8640, 10080, 12960, 17280, 20160, 25920, 30240, 40320, 43200, 51840, 60480, 90720, 103680, 120960, 181440, 241920, 302400, 362880, 483840, 604800, 725760, 1088640, 1209600, 1451520, 1814400, 2419200
Offset: 1

Views

Author

Amiram Eldar, Jun 25 2017

Keywords

Comments

Numbers n such that usigma(x) = n has more solutions x than any smaller n, where usigma(x) is the sum of unitary divisors of x (A034448).
The unitary version of A145899.
The corresponding number of solutions for each term is: 1, 2, 3, 4, 6, 7, 11, 18, 27, 30, 32, 48, 63, 65, 71, 88, 89, 102, 121, 122, 131, 144, 188, 190, 203, 262, 313, 364, 377, 472, 483, 584, 668, 725, 810, 928, 1076, 1138.
Is this a subsequence of A025487? - David A. Corneth, Jun 25 2017

Examples

			There are 3 solutions to usigma(x) = 24: usigma(14) = usigma(15) = usigma(23) = 24. For all m < 24, there are 2 or fewer solutions to usigma(x) = m, thus 24 is in the sequence.

Links

Crossrefs

Cf. A025487, A034448, A063974, A145899.

Programs

  • Mathematica
    usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[#, n/#] == 1 &]]; t = Map[usigma, Range[10^7]]; t2 = Sort[Tally[t]]; mn = 0; t3 = {}; Do[If[t2[[n]][[2]] > mn, mn = t2[[n]][[2]]; AppendTo[t3, t2[[n]][[1]]]], {n, Length[t2]}]; t3 (* after T. D. Noe at A145899 *)

A332036 Number of integers whose bi-unitary divisors sum to n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 1, 1, 0, 0, 3, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 5, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 2, 0, 0, 0
Offset: 1

Views

Author

Amiram Eldar, Feb 05 2020

Keywords

Examples

			a(12) = 2 since there are 2 solutions to bsigma(x) = 12 (bsigma is A188999): 6 and 11.

Links

Crossrefs

Cf. A054973, A063974, A188999, A332038, A332040.

Programs

  • Mathematica
    fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); m = 100; v = Table[0, {m}]; Do[b = bsigma[k]; If[b <= m, v[[b]]++], {k, 1, m}]; v

A332038 Number of integers whose infinitary divisors sum to n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 2, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 1, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 5, 0, 1, 0, 0, 0, 1, 0, 3, 0, 1, 0, 2, 1, 0, 0
Offset: 1

Views

Author

Amiram Eldar, Feb 05 2020

Keywords

Examples

			a(12) = 2 since there are 2 solutions to isigma(x) = 12 (isigma is A049417): 6 and 11.

Links

Crossrefs

Cf. A054973, A049417, A063974, A332036, A332040.

Programs

  • Mathematica
    fun[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ (fun @@@ FactorInteger[n]); m = 100; v = Table[0, {m}]; Do[i = isigma[k]; If[i <= m, v[[i]]++], {k, 1, m}]; v

A332040 Number of integers whose exponential divisors sum to n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 0, 0, 1, 5, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 3, 1, 1, 0, 0, 1, 3, 1, 0, 1, 2, 1, 1, 1, 1, 0, 0, 1, 3, 1, 0, 0, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Feb 05 2020

Keywords

Examples

			a(6) = 2 since there are 2 solutions to esigma(x) = 6 (esigma is A051377): 4 and 6.

Links

Crossrefs

Cf. A054973, A051377, A063974, A332036, A332038.

Programs

  • Mathematica
    f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; m = 100; v = Table[0, {m}]; Do[e = esigma[k]; If[e <= m, v[[e]]++], {k, 1, m}]; v

A308041 Decimal expansion of lim_{m->oo} (1/log(m))*Sum_{k=1..m} 1/usigma(k), where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

7, 6, 8, 7, 1, 8, 3, 6, 2, 4, 4, 6, 4, 8, 5, 1, 9, 8, 6, 7, 2, 7, 3, 4, 3, 3, 2, 4, 5, 5, 3, 5, 0, 5, 2, 5, 2, 3, 4, 2, 5, 5, 7, 4, 0, 4, 1, 1, 9, 0, 4, 1, 1, 0, 7, 0, 1, 5, 4, 1, 3, 5, 2, 9, 3, 4, 8, 6, 0, 7, 7, 6, 8, 3, 3, 7, 9, 0, 8, 0, 3, 9, 3, 3, 2, 8, 8, 0, 7, 6, 4, 8, 9, 6, 9, 1, 4, 7, 5, 9, 5, 3, 3, 7, 2, 4
Offset: 0

Views

Author

Amiram Eldar, May 10 2019

Keywords

Examples

			0.76871836244648519867273433245535052523425574041190...

Links

Crossrefs

Cf. A034448, A063974, A308039 (corresponding limit with sigma).

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; m = 1000; f[p_] := 1 - (p - 1)/p*Sum[1/p^k/(p^k + 1), {k, 1, m}]; c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]*Range[0, m]];RealDigits[f[2]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Formula

From Amiram Eldar, Dec 23 2024: (Start)
Equals Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/(p^k+1))).
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A063974(k). (End)

Extensions

More digits from Vaclav Kotesovec, Jun 13 2021

A332042 Number of integers whose Dedekind psi function (A001615) values are n.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 9, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0
Offset: 1

Views

Author

Amiram Eldar, Feb 05 2020

Keywords

Examples

			a(6) = 2 since there are 2 solutions to psi(x) = 6: 4 and 5.

Links

Crossrefs

Cf. A001615, A054973, A063974, A065463, A332036, A332038, A332040.

Programs

  • Mathematica
    psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); m = 100; v = Table[0, {m}]; Do[i = psi[k]; If[i <= m, v[[i]]++], {k, 1, m}]; v

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.7044422... (A065463). - Amiram Eldar, Dec 25 2024

A141059 Number of numbers m such that n = 0 (mod usigma(m)), where usigma(m) is the sum of unitary divisors of m (A034448).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 1, 3, 3, 3, 1, 6, 1, 2, 3, 3, 2, 6, 1, 6, 2, 1, 1, 10, 2, 2, 3, 4, 1, 8, 1, 5, 3, 2, 2, 11, 1, 2, 2, 8, 1, 6, 1, 3, 4, 1, 1, 13, 1, 5, 3, 3, 1, 9, 2, 6, 2, 1, 1, 17, 1, 2, 3, 5, 3, 4, 1, 5, 2, 5, 1, 21, 1, 2, 3, 3, 1, 5, 1, 11, 3, 2, 1, 13, 3, 1, 2, 4, 1, 15, 1, 2, 2, 1, 2, 19, 1, 3, 4, 9, 1, 6
Offset: 1

Views

Author

Yasutoshi Kohmoto, Aug 01 2008

Keywords

Comments

If p is prime but not a Fermat prime then a(p)=1.
Least k such that a(k) = n: 1, 3, 6, 28, 32, 12, 112, 30, 54, 24, 36, 126, 48, 200, 90, 160, 60, 264, 96, 400, ..., . - Robert G. Wilson v, Aug 07 2008

Links

Crossrefs

Cf. A034448, A063974.

Programs

  • Mathematica
    usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; f[n_] := Block[{c = 0, m = 1}, While[m <= n, If[ Mod[n, usigma@ m] == 0, c++ ]; m++ ]; c]; Array[f, 102] (* Robert G. Wilson v, Aug 07 2008 *)

Extensions

More terms from Robert G. Wilson v, Aug 07 2008
Showing 1-7 of 7 results.

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