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

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

Author

Paul D. Hanna, Jan 26 2003

Keywords

• nonn

Comments

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Crossrefs

Cf. A276133 (first differences).

Cf. A000040, A080084, A080086, A080087, A294898, A371151.

Column 1 of array A379008, incremented by one.

Programs

Mathematica

lst={};Do[p=Prime[n];s=0;While[p>1,p=IntegerPart[p/2];s+=p;];AppendTo[lst,s],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 28 2009 *)

PARI

vector(58, n, valuation(prime(n)!, 2)) \\ Arkadiusz Wesolowski, Feb 22 2014

PARI

a(n) = prime(n) - hammingweight(prime(n)); \\ Joerg Arndt, Feb 22 2014

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