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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A064138 Sum of non-unitary divisors of n!.

Original entry on oeis.org

0, 0, 0, 24, 144, 1398, 11184, 97200, 973296, 10950696, 131408352, 1593191808, 22304685312, 333297226080, 5103130001760, 81686161277280, 1470350902991040, 26490792085668288, 529815841713365760, 10635027891469974720
Offset: 1

Views

Author

Labos Elemer, Sep 11 2001

Keywords

Examples

			For n = 6, 6! = 720, the sum of its 30 divisors is 2418, the sum of the 8 unitary divisors is 1020, so the remaining 22 divisors give a(6) = 1398.

Links

Crossrefs

Cf. A034448, A048105, A046656, A056657, A056171, A056172, A000203, A000142, A062569, A063955, A063960.

Programs

  • Mathematica
    f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n!]) - Times @@ f2 @@@ fct; a[1] = 0; Array[a, 20] (* Amiram Eldar, Apr 01 2024 *)
  • PARI
    usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ n=0; f=1; for (n=1, 100, f*=n; write("b064138.txt", n, " ", sigma(f) - usigma(f)); ) } \\ Harry J. Smith, Sep 08 2009

Formula

a(n) = sigma(n!) - usigma(n!) = A000203(n!) - A034448(A000142(n)) = A062569(n) - A034448(n!) = A048105(n!).

Extensions

Term corrected and more terms added by Harry J. Smith, Sep 08 2009

A294023 Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the smaller part prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 26, 30, 34, 38, 42, 46, 50, 54, 59, 64, 69, 74, 80, 86, 92, 98, 104, 110, 116, 122, 129, 136, 143, 150, 158, 166, 174, 182, 190, 198, 206, 214, 223, 232, 241, 250, 259, 268, 277, 286, 295, 304, 313, 322, 332
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 21 2017

Keywords

Comments

Sum of the slopes of the tangent lines along the left side of the parabola b(x) = n*x-x^2 at prime values of x for x in 0 < x <= floor(n/2). For example, d/dx n*x-x^2 = n-2x. So for a(11), x=2,3,5 and so 11-2*2 + 11-2*3 + 11-2*5 = 7 + 5 + 1 = 13. - Wesley Ivan Hurt, Mar 24 2018

Examples

			The partitions of n = 11 into a number and a smaller prime number are 9 + 2, 8 + 3, and 6 + 5, so a(11) = (9 - 2) + (8 - 3) + (6 - 5) = 13. - _Michael B. Porter_, Apr 06 2018

Links

Crossrefs

Cf. A010051, A056172, A294022, A000720.

Programs

  • Maple
    with(numtheory): seq(add(pi(floor(i/2)), i=1..n-1), n=1..100); # Ridouane Oudra, Nov 24 2019
  • Mathematica
    Table[Sum[(n - 2 i) (PrimePi[i] - PrimePi[i - 1]), {i, Floor[n/2]}], {n, 60}]
  • PARI
    a(n) = sum(i=1, n\2, (n - 2*i)*isprime(i)); \\ Michel Marcus, Mar 24 2018
  • PARI
    a(n) = my(res = 0); forprime(p = 2, n >> 1, res += (n - p << 1)); res \\ David A. Corneth, Apr 06 2018

Formula

a(n) = Sum_{i=1..floor(n/2)} (n - 2i)*A010051(i).
First differences are A056172. - David A. Corneth, Apr 06 2018
a(n) = Sum_{i=1..n-1} pi(floor(i/2)), where pi(n) = A000720(n). - Ridouane Oudra, Nov 24 2019

A300951 a(n) = Product_{j=1..floor(n/2)} p(j) where p(j) = j if j is prime else 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 6, 6, 6, 6, 30, 30, 30, 30, 210, 210, 210, 210, 210, 210, 210, 210, 2310, 2310, 2310, 2310, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 510510, 510510, 510510, 510510, 9699690, 9699690, 9699690, 9699690, 9699690, 9699690, 9699690
Offset: 0

Views

Author

Peter Luschny, Mar 16 2018

Keywords

Comments

a(4*n+2)=a(4*n+3)=a(4*n+4)=a(4*n+5) for n >= 1. - Robert Israel, Mar 16 2018
The length of the n-th run is given by 2*A054541(n). - Michel Marcus, Mar 17 2018

Links

Crossrefs

Cf. A002110, A054541, A056172, A089026.

Programs

  • Maple
    a := n -> mul(`if`(isprime(j), j, 1), j=1..iquo(n,2)):
seq(a(n), n=0..44);
# Alternative:
f:= proc(n) option remember;
  if n::even and isprime(n/2) then procname(n-1)*n/2 else procname(n-1) fi
end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Mar 16 2018
  • Mathematica
    {#,#}&/@FoldList[Times,Table[If[PrimeQ[n],n,1],{n,0,30}]]//Flatten (* Harvey P. Dale, Dec 25 2019 *)
  • PARI
    a(n) = prod(i=1, n\2, if(isprime(i), i, 1)); \\ Altug Alkan, Mar 16 2018

Formula

a(n) = A002110(A056172(n)). - Robert Israel, Mar 16 2018
Previous Showing 11-13 of 13 results.

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