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.

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A359421 a(n) = number of abelian groups of order p^2 - 1, where p = prime(n).

Original entry on oeis.org

1, 3, 3, 5, 3, 3, 14, 6, 5, 3, 11, 6, 5, 3, 7, 9, 3, 3, 3, 10, 10, 7, 3, 10, 22, 6, 5, 9, 9, 7, 44, 3, 5, 3, 6, 10, 3, 15, 5, 3, 6, 6, 15, 15, 12, 20, 3, 11, 3, 3, 10, 7, 14, 18, 30, 5, 9, 21, 3, 5, 3, 6, 6, 5, 5, 3, 3, 14, 3, 6, 11, 10, 7, 3, 9, 22, 3, 6, 14
Offset: 1

Views

Author

Ali Ramsey, Dec 31 2022

Keywords

Examples

			For p = 5, p^2 - 1 = 24 = 2^3 * 3^1. The number of abelian groups of order 24 = (the number of partitions of 3)*(the number of partitions of 1) = 3*1 = 3.

Crossrefs

Subsequence of A000688.
Cf. A000040, A000041, A084920.

Programs

  • Maple
    a:= n-> mul(combinat[numbpart](i[2]), i=ifactors(ithprime(n)^2-1)[2]):
seq(a(n), n=1..79);  # Alois P. Heinz, Dec 31 2022
  • Mathematica
    A000688[n_] := Times @@ (PartitionsP /@ FactorInteger[n][[All, 2]]);
a[n_] := A000688[Prime[n]^2 - 1];
Table[a[n], {n, 1, 79}] (* Jean-François Alcover, Feb 03 2025 *)
  • Python
    from sympy import factorint, npartitions
from math import prod
def A359421(n): return prod(npartitions(d) for d in factorint(prime(n)**2-1).values()) # Chai Wah Wu, Jan 12 2023

Formula

a(n) = A000688(prime(n)^2-1) = A000688(A000040(n)^2-1) = A000688(A084920(n)).

A380472 a(n) = gcd_{primes P >= prime(n+1)} Product_{i=1..n} (P^2-i^2).

Original entry on oeis.org

1, 24, 360, 40320, 1814400, 479001600, 43589145600, 20922789888000, 3201186852864000, 2432902008176640000, 562000363888803840000, 620448401733239439360000, 201645730563302817792000000, 304888344611713860501504000000, 132626429906095529318154240000000, 263130836933693530167218012160000000
Offset: 0

Views

Author

Lily Bétaz, Martin Chevalier, and Basile Fusil, Jun 23 2025

Keywords

Comments

a(n) is the GCD of all numbers of the form Product_{i=1..n} (P^2-i^2) where P is a prime larger than or equal to the (n+1)-th prime.

Examples

			a(1) = 24 because 24 = GCD{P^2-1^2} GCD is taken on all numbers of the form P^2-1^2 with P a prime and P>3. This implies that for all primes P>3, P^2-1 is divisible by 24.
a(2) = 360 because 360 = GCD{(P^2-1^2)(P^2-2^2)} GCD is taken on all numbers of the form (P^2-1^2)(P^2-2^2) with P a prime and P>5. This implies that for all primes P>5, (P^2-1^2)(P^2-2^2) is divisible by 360.
a(3) = 40320 because 40320 = GCD{(P^2-1^2)(P^2-2^2)(P^2-3^2)}.
b(5) = 231 = a(7)/a(6).
c(2) = 112 = a(3)/a(2).

Crossrefs

Cf. A014634 (odd ratio), A014635 (even ratio, multiplied by 4), A084920.

Programs

  • Maple
    seq((2*n + 2)!*(3/4 - (-1)^n/4), n = 0..20)
  • Mathematica
    Table[(2*n + 2)!*(3/4 - (-1)^n/4), {n, 0, 20}]

Formula

E.g.f.: Sum_{n >= 0} a(n)/(2*n)!*z^(2*n) = (1 + 12*z^2 + 12*z^4 + 20*z^6 + 3*z^8)/(1 - z^4)^3.
a(n) = (2*n+2)!*(3/4-(-1)^n/4).
b(n) = (2*n+1)*(4*n+1) = a(2n)/a(2n-1) for n>=1 gives the odd ratios of a(n) (A014634).
c(n) = 4*2*n*(4*n-1) = a(2n-1)/a(2n-2) for n>=1 gives the even ratios of a(n) (4 times A014635).
Sum_{n>=0} 1/a(n) = 3*cosh(1)/2 - cos(1)/2 - 1. - Amiram Eldar, Jul 03 2025

A140392 Triples of height (a prime p), base length x and side length y=z of isosceles triangles.

Original entry on oeis.org

3, 8, 5, 5, 24, 13, 7, 48, 25, 11, 120, 61, 13, 168, 85, 17, 288, 145, 19, 360, 181, 23, 528, 265, 29, 840, 421, 31, 960, 481, 37, 1368, 685, 41, 1680, 841, 43, 1848, 925, 47, 2208, 1105, 53, 2808, 1405, 59, 3480, 1741, 61, 3720, 1861, 67, 4488, 2245, 71, 5040, 2521
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 13 2008

Keywords

Comments

Or: consecutive triples of p=A000040(j), x=2*A084920(j), y=z= A066885(j), j>=2.
The area of the triangles is half the product of height and base length, p*x/2=A127918(j).

Examples

			Contains (p,x,y) = (3,8,5), (5,24,13), (7,48,25), (11,120,61), ...

Extensions

Edited and extended by R. J. Mathar, Jun 17 2008

A301812 Numbers of the form p^2 - 1 where p is a prime of the form 3*k-1 (A003627).

Original entry on oeis.org

3, 24, 120, 288, 528, 840, 1680, 2208, 2808, 3480, 5040, 6888, 7920, 10200, 11448, 12768, 17160, 18768, 22200, 27888, 29928, 32040, 36480, 38808, 51528, 54288, 57120, 63000, 66048, 69168, 72360, 78960, 85848, 96720, 100488, 120408, 124608, 128880, 146688
Offset: 1

Views

Author

Peter Luschny, Mar 27 2018

Keywords

Crossrefs

Cf. A003627, A084920.

Programs

  • Maple
    A301812List := proc(len) local p, n, L; L := 3; p := 5;
for n from 2 to len do if isprime(p) then L := L,(p^2 - 1) fi;
p := p + 6; od: L end:
A301812List(65);
  • Mathematica
    Flatten[Table[n^2 - 1, {n, {2, Select[Range[5, 385, 6], PrimeQ]}}]]

Formula

a(n) = A003627(n)^2 - 1. - Altug Alkan, Mar 28 2018
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