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 1-6 of 6 results.

A366438 The number of divisors of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 8, 4, 4, 2, 8, 2, 6, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 2, 4, 2, 8, 4, 8, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 8, 2, 4, 4, 4, 4, 12, 2, 2, 8, 2, 8, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 16, 4
Offset: 1

Views

Author

Amiram Eldar, Oct 10 2023

Keywords

Comments

1 is the only odd term in this sequence.

Links

Crossrefs

Cf. A000005, A268335, A366439.
Similar sequences: A048691, A072048, A076400, A358040, A363194, A363195.

Programs

  • Mathematica
    f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, OddQ], Times @@ (e + 1), Nothing]]; f[1] = 1; Array[f, 150]
  • PARI
    lista(max) = for(k = 1, max, my(e = factor(k)[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(vecprod(apply(x -> x+1, e)), ", ")));
  • Python
    from math import prod
from itertools import count, islice
from sympy import factorint
def A366438_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e&1 for e in f):
            yield prod(e+1 for e in f)
A366438_list = list(islice(A366438_gen(),30)) # Chai Wah Wu, Oct 10 2023

Formula

a(n) = A000005(A268335(n)).

A343443 If n = Product (p_j^k_j) then a(n) = Product (k_j + 2), with a(1) = 1.

Original entry on oeis.org

1, 3, 3, 4, 3, 9, 3, 5, 4, 9, 3, 12, 3, 9, 9, 6, 3, 12, 3, 12, 9, 9, 3, 15, 4, 9, 5, 12, 3, 27, 3, 7, 9, 9, 9, 16, 3, 9, 9, 15, 3, 27, 3, 12, 12, 9, 3, 18, 4, 12, 9, 12, 3, 15, 9, 15, 9, 9, 3, 36, 3, 9, 12, 8, 9, 27, 3, 12, 9, 27, 3, 20, 3, 9, 12, 12, 9, 27, 3, 18
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 15 2021

Keywords

Comments

Inverse Moebius transform of A056671.
a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, May 03 2021

Links

Crossrefs

Cf. A000005, A001221, A005361, A007425, A025847, A034444, A056671, A064549, A074816, A107758, A107759, A348018, A363194.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((#[[2]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 80}]
a[n_] := Sum[If[GCD[d, n/d] == 1, DivisorSigma[0, d], 0], {d, Divisors[n]}]; Table[a[n], {n, 80}]
  • PARI
    a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, numdiv(d))) \\ Andrew Howroyd, Apr 15 2021
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2)/(1-X)^2)[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023
  • Python
    from math import prod
from sympy import factorint
def A343443(n): return prod(e+2 for e in factorint(n).values()) # Chai Wah Wu, Feb 21 2025

Formula

a(n) = 2^omega(n) * tau_3(n) / tau(n), where omega = A001221, tau = A000005 and tau_3 = A007425.
a(n) = Sum_{d|n, gcd(d, n/d) = 1} tau(d).
From Bernard Schott, May 03 2021: (Start)
a(p^k) = k+2 for p prime, or signature [k].
a(A006881(n)) = 9 for signature [1, 1].
a(A054753(n)) = 12 for signature [2, 1].
a(A065036(n)) = 15 for signature [3, 1].
a(A085986(n)) = 16 for signature [2, 2].
a(A178739(n)) = 18 for signature [4, 1].
a(A143610(n)) = 20 for signature [3, 2].
a(A007304(n)) = 27 for signature [1, 1, 1]. (End)
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 1/p^s - 1/p^(2*s)). - Vaclav Kotesovec, Feb 11 2023
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A064549(n)).
a(n) = A363194(A348018(n)). (End)

A363195 Number of divisors of the n-th cubefull number A036966(n).

Original entry on oeis.org

1, 4, 5, 4, 6, 7, 5, 4, 8, 16, 6, 9, 4, 20, 10, 5, 20, 7, 24, 16, 11, 25, 4, 28, 24, 20, 12, 8, 4, 5, 30, 16, 6, 16, 32, 30, 24, 13, 4, 20, 35, 20, 28, 9, 4, 36, 36, 28, 14, 16, 25, 20, 40, 16, 24, 35, 4, 40, 5, 42, 7, 32, 15, 6, 20, 32, 16, 20, 10, 30, 45, 20
Offset: 1

Views

Author

Amiram Eldar, May 21 2023

Keywords

Links

Crossrefs

Cf. A000005, A036966, A362986.
Similar sequences: A072048, A076400, A363194.

Programs

  • Mathematica
    DivisorSigma[0, Select[Range[25000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]
  • PARI
    lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 2, print1(numdiv(k), ", ")));
  • Python
    from itertools import count, islice
from math import prod
from sympy import factorint
def A363195_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e>2 for e in f):
            yield prod(e+1 for e in f)
A363195_list = list(islice(A363195_gen(),20)) # Chai Wah Wu, May 21 2023

Formula

a(n) = A000005(A036966(n)).
Sum_{A036966(k) < x} a(k) = c_1 * x^(1/3) * log(x)^3 + c_2 * x^(1/3) * log(x)^2 + c_3 * x^(1/3) * log(x) + c_4 * x^(1/3) + O(x^(7/24 + eps)), where c_1, c_2, c_3 and c_4 are constants. c_1 = Product_{p prime} ((1-1/p)^4 * (1 + 1/((p^(1/3) - 1)^2*p^(1/3)) + 3/(p-p^(2/3))))/162 = 0.1346652397135839416... . [corrected Sep 21 2024]

A366441 The number of divisors of the 5-rough numbers (A007310).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 3, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 4, 2, 4, 2, 2, 3, 2, 6, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 2, 4, 4, 4, 2, 2, 2, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4
Offset: 1

Views

Author

Amiram Eldar, Oct 10 2023

Keywords

Links

Crossrefs

Cf. A000005, A001620, A007310, A366442.
Similar sequences: A048691, A072048, A076400, A099774, A358040, A363194, A363195.

Programs

  • Mathematica
    a[n_] := DivisorSigma[0, 2*Floor[3*n/2] - 1]; Array[a, 100]
  • PARI
    a(n) = numdiv((3*n)\2 << 1 - 1)
  • Python
    from sympy import divisor_count
def A366441(n): return divisor_count((n+(n>>1)<<1)-1) # Chai Wah Wu, Oct 10 2023

Formula

a(n) = A000005(A007310(n)).
Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1 + 2*log(6)) * n / 3, where gamma is Euler's constant (A001620).

A370328 The number of powerful divisors of the powerful numbers.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 3, 5, 4, 2, 6, 6, 4, 4, 6, 2, 3, 7, 8, 2, 4, 6, 9, 4, 5, 8, 10, 2, 8, 3, 2, 6, 8, 12, 4, 4, 6, 9, 2, 12, 4, 12, 6, 4, 6, 8, 10, 2, 15, 8, 2, 6, 10, 9, 10, 4, 6, 14, 4, 4, 16, 6, 3, 6, 2, 6, 4, 4, 10, 12, 2, 18, 8, 12, 2, 8, 15, 12, 8, 11, 4, 7
Offset: 1

Views

Author

Amiram Eldar, Feb 15 2024

Keywords

Comments

The product of the exponents of the prime factorization of the powerful numbers.

Links

Crossrefs

Cf. A000005, A001694, A005361, A306458, A363194, A370329.

Programs

  • Mathematica
    f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[n == 1 || Min[e] > 1, Times @@ e, Nothing]]; Array[f, 2500]
  • PARI
    lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1 || vecmin(e) > 1, print1(vecprod(e), ", ")));}

Formula

a(n) = A005361(A001694(n)).
a(n) = A000005(A306458(n)).

A377138 Powerful numbers that have more divisors than any smaller powerful number.

Original entry on oeis.org

1, 4, 8, 16, 32, 36, 72, 144, 216, 288, 432, 576, 864, 900, 1728, 1800, 3600, 5400, 7200, 10800, 14400, 21600, 32400, 43200, 64800, 86400, 88200, 176400, 264600, 352800, 529200, 705600, 1058400, 1587600, 2116800, 3175200, 4233600, 6350400, 8467200, 10584000, 12700800
Offset: 1

Views

Author

Amiram Eldar, Oct 17 2024

Keywords

Comments

First differs from A283052 at n = 12.
Indices of records in A357669.
The corresponding record values are 1, 3, 4, 5, 6, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 36, ... (see the link for more values).

Links

Crossrefs

Subsequence of A001694 and A025487.
Cf. A000005, A002182, A283052, A357669, A363194.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1, 1, e + 1]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; With[{v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq]
Showing 1-6 of 6 results.

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

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