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.

A280581 a(n) = the product of divisors of sum of divisors of n.

Original entry on oeis.org

1, 3, 8, 7, 36, 1728, 64, 225, 13, 5832, 1728, 21952, 196, 331776, 331776, 31, 5832, 1521, 8000, 3111696, 32768, 10077696, 331776, 46656000000, 31, 3111696, 2560000, 9834496, 810000, 139314069504, 32768, 250047, 254803968, 8503056, 254803968, 8281, 1444
Offset: 1

Views

Author

Jaroslav Krizek, Jan 05 2017

Keywords

Comments

a(n) < A007955(n) for numbers n in A219364.
a(n) | A007955(n) for numbers n in A219363.
A007955(n) | a(n) for numbers n in A219362.
n | a(n) for numbers n in A175200.

Examples

			For n = 5; a(n) = product of divisors of sigma(5) = 1*2*3*6 = 36.

Links

Crossrefs

Cf. A000203, A007955, A219362, A219363, A219364, A280582.

Programs

  • Magma
    [&*[d: d in Divisors(SumOfDivisors(n))]: n in [1..100]]
  • Mathematica
    Table[Times @@ Divisors@ DivisorSigma[1, n], {n, 37}] (* Michael De Vlieger, Jan 06 2017 *)
a[n_] := (s = DivisorSigma[1, n])^(DivisorSigma[0, s]/2); Array[a, 40] (* Amiram Eldar, Jun 26 2022 *)
  • PARI
    a(n) = my(k = 1); fordiv(sigma(n), d, k*=d); k; \\ Michel Marcus, Jan 06 2017
  • Python
    from math import isqrt
from sympy import divisor_count, divisor_sigma
def A280581(n): return (lambda m:isqrt(m)**c if (c:=divisor_count(m)) & 1 else m**(c//2))(divisor_sigma(n)) # Chai Wah Wu, Jun 25 2022

Formula

a(n) = A007955(A000203(n)).

A280684 a(n) = number of divisors of the product of the divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 9, 2, 7, 4, 9, 2, 28, 2, 9, 9, 11, 2, 28, 2, 28, 9, 9, 2, 65, 4, 9, 7, 28, 2, 125, 2, 16, 9, 9, 9, 100, 2, 9, 9, 65, 2, 125, 2, 28, 28, 9, 2, 126, 4, 28, 9, 28, 2, 65, 9, 65, 9, 9, 2, 637, 2, 9, 28, 22, 9, 125, 2, 28, 9, 125, 2, 247, 2, 9, 28
Offset: 1

Views

Author

Jaroslav Krizek, Jan 07 2017

Keywords

Examples

			For n = 4; a(n) = tau (1*2*4) = tau(8) = 4.

Links

Crossrefs

Cf. A000005, A000040, A001248, A007955, A008578, A052213, A280685, A280582.

Programs

Formula

a(n) = A000005(A007955(n)).
a(p) = 2 for p = primes (A000040).
a(n) = 4 for squares of primes (A001248).
a(n) = n for numbers 1, 2, 4, 28, 100, ...
a(n) = tau(n) for noncomposites (A008578).
a(n) = a(n+1) for numbers in A052213.

A280685 a(n) = sum of the divisors of the product of the divisors of n.

Original entry on oeis.org

1, 3, 4, 15, 6, 91, 8, 127, 40, 217, 12, 5080, 14, 399, 403, 2047, 18, 16395, 20, 19812, 741, 931, 24, 991111, 156, 1281, 1093, 50800, 30, 2929531, 32, 65535, 1729, 2149, 1767, 30203052, 38, 2667, 2379, 6397171, 42, 10506551, 44, 185928, 170508, 3871, 48
Offset: 1

Views

Author

Jaroslav Krizek, Jan 07 2017

Keywords

Examples

			For n = 4; a(n) = sigma (1*2*4) = sigma(8) = 15.

Crossrefs

Cf. A000040, A000203, A007955, A280582, A280684.

Programs

  • Magma
    [&+[d: d in Divisors(&*[d: d in Divisors(n)])]: n in [1..100]];
  • Python
    from math import isqrt
from sympy import divisor_sigma
def A280685(n): return divisor_sigma((isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2)) # Chai Wah Wu, Jun 25 2022

Formula

a(n) = A000203(A007955(n)).
a(p) = p + 1 for p = primes (A000040).
a(2^n) = 2*A007955(2^n) - 1. [corrected by Jason Yuen, Mar 08 2025]
Showing 1-3 of 3 results.

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