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.

A080085 Number of factors of 2 in the factorial of the n-th prime, counted with multiplicity.

Original entry on oeis.org

1, 1, 3, 4, 8, 10, 15, 16, 19, 25, 26, 34, 38, 39, 42, 49, 54, 56, 64, 67, 70, 74, 79, 85, 94, 97, 98, 102, 104, 109, 120, 128, 134, 135, 145, 146, 152, 159, 162, 168, 174, 176, 184, 190, 193, 194, 206, 216, 222, 224, 228, 232, 236, 244, 255, 259, 265, 266, 273, 277
Offset: 1

Views

Author

Paul D. Hanna, Jan 26 2003

Keywords

Comments

n-th prime minus number of 1's in binary representation of n-th prime. [Juri-Stepan Gerasimov, May 17 2010]

Links

Crossrefs

Cf. A276133 (first differences).
Cf. A000040, A080084, A080086, A080087, A294898, A371151.
Column 1 of array A379008, incremented by one.

Programs

Formula

a(n) = Sum_{k=1..L} floor( p_n /2^k ), where L = log(p_n)/log(2), where p_n is the n-th prime.
a(n) = A000040(n) - A014499(n). - Juri-Stepan Gerasimov, May 17 2010
a(n) = 1+A294898(A000040(n)). - Antti Karttunen, Dec 14 2024

A080087 Number of factors of 5 in the factorial of the n-th prime, counted with multiplicity.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 6, 7, 8, 9, 9, 10, 12, 13, 14, 15, 16, 16, 18, 19, 20, 22, 24, 24, 25, 25, 26, 31, 32, 33, 33, 35, 37, 38, 39, 40, 41, 43, 44, 46, 46, 47, 47, 51, 53, 55, 55, 56, 57, 58, 62, 63, 64, 65, 66, 68, 69, 69, 71, 75, 76, 76, 77, 81, 82, 84, 84, 86, 87, 89, 90
Offset: 1

Views

Author

Paul D. Hanna, Jan 26 2003

Keywords

Comments

Highest power of 5 dividing prime(n)! = A039716(n), or also the number of trailing end 0's in A039716(n). - Lekraj Beedassy, Oct 31 2010

Links

Crossrefs

Cf. A080084, A080085, A080086, A039716, A112765, A027868.

Programs

  • Maple
    R:= NULL: v:= 0: p:= 0:
for i from 1 to 100 do
   q:= p;
   p:= nextprime(p);
   v:= v + add(1+padic:-ordp(x,5), x = 1+floor(q/5) .. floor(p/5));
   R:= R,v;
od:
R; # Robert Israel, Sep 27 2023
  • Mathematica
    lst={};Do[p=Prime[n];s=0;While[p>1,p=IntegerPart[p/5];s+=p;];AppendTo[lst,s],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 28 2009 *)
  • PARI
    a(n) = valuation(prime(n)!, 5); \\ Michel Marcus, Jan 15 2015

Formula

a(n) = Sum_{k=1..L} floor(prime(n)/5^k), where L = log(p_n)/log(5).
a(n) = A112765(A039716(n)). - Michel Marcus, Sep 28 2023

A080086 Number of factors of 3 in the factorial of the n-th prime, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 4, 5, 6, 8, 9, 13, 14, 17, 18, 19, 21, 23, 27, 28, 31, 32, 34, 36, 40, 42, 46, 48, 49, 50, 53, 54, 61, 62, 66, 67, 71, 72, 75, 80, 81, 84, 86, 88, 93, 94, 95, 97, 102, 108, 110, 111, 112, 115, 116, 123, 126, 129, 131, 134, 136, 138, 139, 143, 151, 152, 153
Offset: 1

Views

Author

Paul D. Hanna, Jan 26 2003

Keywords

Crossrefs

Cf. A080084, A080085, A080087.

Programs

  • Mathematica
    lst={};Do[p=Prime[n];s=0;While[p>1,p=IntegerPart[p/3];s+=p;];AppendTo[lst,s],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 28 2009 *)
Join[{0},FactorInteger[#][[2,2]]&/@(Prime[Range[2,70]]!)] (* Harvey P. Dale, Sep 05 2014 *)

Formula

a(n) = sum_{k=1..L} floor(prime(n) / 3^k), where L = log(p_n)/log(3).
Showing 1-3 of 3 results.

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

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