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.

A181798 Terms of A181797 arranged in ascending order.

Original entry on oeis.org

1, 6, 12, 12, 24, 30, 36, 48, 56, 72, 96, 108, 132, 144, 150, 180, 182, 192, 216, 288, 306, 324, 336, 360, 360, 380, 384, 392, 432, 552, 576, 648, 672, 672, 720, 750, 768, 792, 864, 870, 900, 972, 992, 1080, 1092, 1152, 1296, 1344, 1406, 1440, 1452, 1536, 1584, 1584
Offset: 1

Views

Author

Matthew Vandermast, Dec 05 2010

Keywords

Comments

The natural density of positive integers m such that A003557(m) = A181799(n) equals 6/(a(n)*Pi^2).

Links

Formula

sum{n>=1} 1/a(n) = pi^2/6 = A013661.

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

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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

A304411 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*k_j).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 9, 8, 18, 12, 24, 14, 24, 24, 12, 18, 24, 20, 36, 32, 36, 24, 36, 12, 42, 12, 48, 30, 72, 32, 15, 48, 54, 48, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 48, 48, 16, 36, 72, 84, 54, 36, 72, 72, 80, 90, 60, 144, 62, 96, 64, 18, 84, 144, 68, 108, 96, 144, 72, 72
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2018

Keywords

Examples

			a(24) = a(2^3*3) = (2 + 1)*3 * (3 + 1)*1 = 36.

Links

Crossrefs

Cf. A000005, A000026, A000040, A000203, A003959, A001615, A003961, A005117, A005361, A007947, A008864, A045965, A048250, A064478, A081294 (numbers n such that a(n) is odd), A181797, A304407, A304408, A304409, A304412.
Cf. A072691, A330596.

Programs

  • Mathematica
    a[n_] := Times @@ ((#[[1]] + 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 72}]
Table[Total[Select[Divisors[n], SquareFreeQ]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 72}]
  • PARI
    a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p+1)*e)} \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = A005361(n)*A048250(n) = A000005(n/A007947(n))*A000203(A007947(n)).
a(p^k) = (p + 1)*k where p is a prime and k > 0.
a(n) = Product_{p|n} (p + 1) if n is a squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A072691 * A330596 = 0.6156455744... . - Amiram Eldar, Nov 30 2022

A181799 List of positive integers, ordered by asymptotic frequency of appearance in A003557; ties broken by listing smaller numbers first.

Original entry on oeis.org

1, 2, 3, 4, 8, 5, 9, 16, 7, 6, 32, 27, 11, 12, 25, 10, 13, 64, 18, 24, 17, 81, 14, 15, 20, 19, 128, 49, 36, 23, 48, 54, 21, 28, 40, 125, 256, 22, 72, 29, 50, 243, 31, 45, 26, 96, 108, 56, 37, 80, 121, 512, 33, 44, 35, 41, 144, 100, 34, 43, 162, 63, 30, 39, 52, 47, 38, 192, 98, 169
Offset: 1

Views

Author

Matthew Vandermast, Dec 05 2010

Keywords

Comments

A permutation of the natural numbers.
The natural density of positive integers m such that A003557(m) = a(n) equals 6/(A181798(n)*Pi^2). If m is coprime to 6, 3m and 4m appear in A003557 equally often.

Links

Formula

The natural density of positive integers m such that A003557(m) = n equals 6/(A181797(n)*Pi^2). A181797(n) = n*A048250(n).
Showing 1-4 of 4 results.

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

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