A276133 -id:A276133

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A080085

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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]

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 *)

PROG

(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

CROSSREFS

Cf. A276133 (first differences).

Cf. A080084, A080086, A080087.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 26 2003

STATUS

approved

A276376

Exponent of highest power of 3 dividing product of composite numbers between n-th prime and (n+1)-st prime.

+10
0

0, 0, 1, 2, 1, 1, 2, 1, 4, 1, 3, 1, 1, 2, 2, 4, 1, 3, 1, 2, 2, 4, 2, 4, 2, 1, 1, 3, 1, 7, 1, 4, 1, 4, 1, 3, 5, 1, 3, 2, 2, 5, 1, 1, 2, 5, 6, 2, 1, 1, 3, 1, 7, 3, 3, 2, 3, 2, 2, 1, 4, 8, 1, 1, 2, 8, 3, 4, 1, 3, 2, 4, 3, 4, 1, 3, 4, 1, 6, 4, 1, 4, 3, 2, 2, 2, 4, 3, 1, 1, 6, 7, 1, 4, 1, 3, 6

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..97.

FORMULA

a(n) = A007949(A061214(n)).

a(n) = Sum_{k = A000040(n)..A000040(n + 1)} A007949(k) for n > 2.

MATHEMATICA

Table[IntegerExponent[Times@@Range[Prime[n]+1, Prime[n+1]-1], 3], {n, 100}] (* Harvey P. Dale, Mar 23 2021 *)

CROSSREFS

Cf. A007849, A061214, A276133.