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 1-4 of 4 results.

A000082 a(n) = n^2*Product_{p|n} (1 + 1/p).

Original entry on oeis.org

1, 6, 12, 24, 30, 72, 56, 96, 108, 180, 132, 288, 182, 336, 360, 384, 306, 648, 380, 720, 672, 792, 552, 1152, 750, 1092, 972, 1344, 870, 2160, 992, 1536, 1584, 1836, 1680, 2592, 1406, 2280, 2184, 2880, 1722, 4032, 1892, 3168, 3240, 3312, 2256
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

For n > 1: A006530(a(n)) = A076566(n-1). - Reinhard Zumkeller, Oct 03 2012
A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Jan 01 2017

References

  • B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 79.

Links

Crossrefs

Cf. A001615, A027748, A033196, A124010, A335762.

Programs

  • Haskell
    a000082 n = product $ zipWith (\p e -> p ^ (2*e - 1) * (p + 1))
                              (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Oct 03 2012
  • Maple
    proc(n) local b,d: b := n^2: for d from 1 to n do if irem(n,d) = 0 and isprime(d) then b := b*(1+d^(-1)): fi: od: RETURN(b): end:
  • Mathematica
    Table[ Fold[ If[ Mod[ n, #2 ]==0 && PrimeQ[ #2 ], #1*(1+1/#2), #1 ]&, n^2, Range[ n ] ], {n, 1, 45} ]
Table[ n^2 Times@@(1+1/Select[ Range[ 1, n ], (Mod[ n, #1 ]==0&&PrimeQ[ #1 ])& ]), {n, 1, 45} ] (* Olivier Gérard, Aug 15 1997 *)
f[p_, e_] := (p+1)*p^(2*e - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 23 2020 *)
  • PARI
    a(n)=if(n<1,0,direuler(p=2,n,(1+p*X)/(1-p^2*X))[n])

Formula

a(n) = n * A001615(n).
Dirichlet g.f.: zeta(s-1)*zeta(s-2)/zeta(2*s-2).
Dirichlet convolution: Sum_{d|n} mu(n/d)*sigma(d^2). - Vladeta Jovovic, Nov 16 2001
Multiplicative with a(p^e) = p^(2*e-1)*(p+1). - David W. Wilson, Aug 01 2001
a(n) = A181797(n)*A003557(n). - R. J. Mathar, Mar 30 2011
a(n) = A001615(n^2). - Enrique Pérez Herrero, Mar 06 2012
Sum_{k=1..n} a(k) ~ 5*n^3 / Pi^2. - Vaclav Kotesovec, Jan 11 2019
Sum_{n>=1} 1/a(n) = A335762. - Amiram Eldar, Jun 23 2020

Extensions

Additional comments from Michael Somos, May 19 2000

A367987 The number of square divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Dec 07 2023

Keywords

Comments

Also, the number of divisors of the square root of the largest unitary divisor of n that is a square.

Links

Crossrefs

Cf. A000005, A046951, A071974, A268335, A335762, A350388, A365401, A367988.

Programs

  • Mathematica
    f[p_, e_] := If[EvenQ[e], e/2 + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> if(x%2, 1, x/2+1), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = e/2 + 1 if e is even and 1 otherwise.
a(n) = A046951(A350388(n)).
a(n) = A000005(A071974(n)).
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + p/((p-1)*(p+1)^2)) = 1.450032... (A335762).

A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 28 2020

Keywords

Comments

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/(zeta(2*n))^2 = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.81(0781476121562952243125904486251808972503617945007235890014471780028943
5600578871201157742402315484804630969609261939218523878437047756874095
5137481910274963820549927641099855282199710564399421128798842257597684
51519536903039073806).

Examples

			1.8107814761215629522431259...

Crossrefs

Cf. A000040, A005361, A084920, A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340066.

Programs

  • Mathematica
    RealDigits[N[5005/2764,105]][[1]]
  • PARI
    default(realprecision,105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2))

Formula

Equals 5005/2764 = 5*7*11*13/(2^2*691).
Equals Product_{n>=1} 1+A000040(n)^2/A084920(n)^2.
Equals (13/9)*A340066.
From Vaclav Kotesovec, Dec 29 2020: (Start)
Equals 3/2 * (Product_{p prime} (p^6+1)/(p^6-1)) * (Product_{p prime} (p^4+1)/(p^4-1)).
Equals 7*zeta(6)^2 / (4*zeta(12)).
Equals -7*binomial(12, 6) * Bernoulli(6)^2 / (8*Bernoulli(12)). (End)
Equals Sum_{k>=1} A005361(k)/k^2. - Amiram Eldar, Jan 23 2024

A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 28 2020

Keywords

Comments

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/zeta^2(2*n) = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.25(3617945007235890014471780028943560057887120115774240231548480463096960
9261939218523878437047756874095513748191027496382054992764109985528219
9710564399421128798842257597684515195369030390738060781476121562952243
12590448625180897250).

Examples

			1.25361794500723589001447178...

Crossrefs

Cf. A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340065.

Programs

  • Mathematica
    RealDigits[N[3465/2764, 105]][[1]]
  • PARI
    default(realprecision, 105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2),1,3)

Formula

Equals 3465/2764 = 3^2*5*7*11/(2^2*691).
Equals Product_{n>=2} 1+A000040(n)^2/A084920(n)^2.
Equals (9/13)*A340065.
Showing 1-4 of 4 results.

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

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