A080086 -id:A080086 - cp's OEIS frontend

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

Paul D. Hanna, Jan 26 2003

nonn

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

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

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

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 *)

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

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

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

Paul D. Hanna, Jan 26 2003

nonn

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

Robert Israel, Table of n, a(n) for n = 1..10000

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

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

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 *)

a(n) = valuation(prime(n)!, 5); \\ Michel Marcus, Jan 15 2015

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

A080084 Number of prime factors in the factorial of the n-th prime, counted with multiplicity.

Original entry on oeis.org

1, 2, 5, 8, 16, 20, 29, 33, 41, 56, 60, 76, 85, 89, 98, 114, 129, 134, 151, 160, 166, 180, 192, 207, 229, 240, 244, 254, 260, 271, 308, 321, 338, 342, 369, 374, 391, 409, 418, 435, 451, 457, 484, 492, 502, 507, 541, 572, 585, 590, 601, 616, 623, 653, 674, 689

Offset: 1

Paul D. Hanna, Jan 26 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A080085, A080086, A080087.

b:= proc(n) option remember; `if`(n=1, 0,
      b(n-1)+numtheory[bigomega](n))
    end:
a:= n-> b(ithprime(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Jul 05 2019

PrimeOmega[#!]&/@Prime[Range[60]] (* Harvey P. Dale, Nov 09 2011 *)

a(n) = Sum_{m=1..n} Sum_{k=1..L} floor( p_n /(p_m)^k ), where L = ceiling( log(p_n)/log(p_m) ).

A163464 Cumulative sum of a repeated shift-and-add operation on the base-7 representation of prime(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 16, 16, 17, 17, 18, 20, 20, 21, 21, 24, 24, 25, 26, 26, 27, 28, 28, 30, 30, 32, 32, 34, 35, 36, 36, 37, 38, 38, 40, 41, 42, 43, 43, 44, 45, 45, 46, 49, 50, 50, 51, 53, 54, 57, 57, 58, 59, 60, 61, 62

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 28 2009

Starting from the base-7 representation of prime(n) = d_m*7^m + ... + d_3*7^3 + d_2*7^2 + d_1*7 + d_0, the least-significant digit is recursively removed (a shift-right operation in base 7), and the intermediate numbers are all added up:
a(n) =   (d_m*7^(m-1) + ... + d_3*7^2 + d_2*7 + d_1)
       + (d_m*7^(m-2) + ... + d_4*7^2 + d_3*7 + d_2)
       + (d_m*7^(m-3) +      ...      + d_4*7 + d_3)
       +                     ...              + d_m
     = Sum_{j=1..m} d_j*(7^j - 1)/6.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A080085, A080086, A080087.

shiftadd := proc(n,b) dgs := convert(n,base,b) ; add( op(i,dgs)*(b^(i-1)-1),i=2..nops(dgs))/(b-1) ; end:
A163464 := proc(n) shiftadd(ithprime(n),7) ; end:
seq(A163464(n),n=1..40) ; # R. J. Mathar, Aug 02 2009

lst={}; Do[p=Prime[n]; s=0; While[p>1,p=IntegerPart[p/7];s+=p;]; AppendTo[lst, s],{n,6!}]; lst

Definition rewritten by R. J. Mathar, Aug 02 2009

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

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

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A080084 Number of prime factors in the factorial of the n-th prime, counted with multiplicity.

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Author

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Crossrefs

Programs

Maple

Mathematica

Formula

A163464 Cumulative sum of a repeated shift-and-add operation on the base-7 representation of prime(n).

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Maple

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