A080085 -id:A080085 - cp's OEIS frontend

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

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) ).

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

Paul D. Hanna, Jan 26 2003

nonn

Cf. A080084, A080085, A080087.

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 *)

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

A371151 Numbers k >= 2 such that A362333(k)-A371148(k)/A371149(k) sets a new maximum.

Original entry on oeis.org

2, 80, 224, 5632, 26624, 1114112, 2490368, 24117248

Offset: 1

Pontus von Brömssen, Mar 13 2024

The corresponding maxima are: 0, 2/3, 3/4, 7/8, 9/10, 14/15, 15/16, 18/19, ... .
All terms after a(1) = 2 are in A371150.
Apparently, a(n) = 2^(A080085(n+1)+1)*prime(n+1) for n >= 2, with corresponding maxima 1 - 1/A080085(n+1).

Cf. A080085, A362333, A371148, A371149, A371150.

A379008 Square array A(n, k) = A294898(A246278(n, k)), read by falling antidiagonals; Difference A005187(n)-A000203(n) applied to the prime shift array.

Original entry on oeis.org

0, 0, 0, -2, 3, 2, 0, 2, 16, 3, 0, 10, 19, 38, 7, -6, 7, 88, 54, 104, 9, 1, 8, 33, 280, 113, 151, 14, 0, 16, 96, 65, 1192, 184, 268, 15, -5, 38, 44, 389, 152, 2009, 282, 336, 18, -4, 22, 464, 88, 1279, 207, 4600, 388, 502, 24, 5, 16, 142, 1996, 174, 2445, 345, 6470, 608, 806, 25, -14, 18, 174, 623, 13170, 257, 4834, 497, 11605, 833, 924, 33

Offset: 1

Antti Karttunen, Dec 14 2024

Question: Are all columns increasing, and strictly increasing after the leftmost column?

			The top left corner of the array:
k=  |  1    2    3      4    5      6    7       8      9     10   11      12
2k= |  2    4    6      8   10     12   14      16     18     20   22      24
----+-------------------------------------------------------------------------
1   |  0,   0,  -2,     0,   0,    -6,   1,      0,    -5,    -4,   5,    -14,
2   |  0,   3,   2,    10,   7,     8,  16,     38,    22,    16,  18,     26,
3   |  2,  16,  19,    88,  33,    96,  44,    464,   142,   174,  58,    495,
4   |  3,  38,  54,   280,  65,   389,  88,   1996,   623,   469, 103,   2737,
5   |  7, 104, 113,  1192, 152,  1279, 174,  13170,  1516,  1717, 211,  14102,
6   |  9, 151, 184,  2009, 207,  2445, 257,  26172,  3208,  2756, 328,  31850,
7   | 14, 268, 282,  4600, 345,  4834, 439,  78295,  5406,  5916, 473,  82285,
8   | 15, 336, 388,  6470, 497,  7455, 533, 123071,  9035,  9501, 638, 141745,
9   | 18, 502, 608, 11605, 653, 14081, 784, 267115, 17773, 15097, 870, 324077,
Here 0's occur also after the first row. For example column 30, which corresponds with numbers 60, 315, 1925, 7007, 26741, ..., begins as -52, 0, 868, 4428, 19958, etc. See also A295296.

Cf. A000203, A005187, A246278, A294898, A295296.
Cf. A080085 (column 1, incremented by one).
Cf. also array A378979, and A324348 (another permutation of A294898).

up_to = 11325; \\ = binomial(150+1,2)
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A379008sq(row,col) = A294898(A246278sq(row,col));
A379008list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A379008sq(col,(a-(col-1))))); (v); };
v379008 = A379008list(up_to);
A379008(n) = v379008[n];

A276133 Exponent of highest power of 2 dividing the product of the composite numbers between the n-th prime and the (n+1)-st prime.

Original entry on oeis.org

0, 2, 1, 4, 2, 5, 1, 3, 6, 1, 8, 4, 1, 3, 7, 5, 2, 8, 3, 3, 4, 5, 6, 9, 3, 1, 4, 2, 5, 11, 8, 6, 1, 10, 1, 6, 7, 3, 6, 6, 2, 8, 6, 3, 1, 12, 10, 6, 2, 4, 4, 4, 8, 11, 4, 6, 1, 7, 4, 1, 11, 13, 3, 3, 3, 15, 7, 8, 2, 6, 4, 7, 7, 5, 3, 10, 7, 5, 7

Offset: 1

Juri-Stepan Gerasimov, Aug 29 2016

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

Supersequence of A205649 (Hamming distance between twin primes).
Cf. A000040, A007814, A061214.
First differences of A080085.

A:= Vector(100): q:= 2:
for n from 1 to 100 do
  p:= q; q:= nextprime(q);
  t:= 0;
  for i from p+1 to q-1 do t:= t + padic:-ordp(i,2) od;
  A[n]:= t
od:
convert(A,list); # Robert Israel, Apr 11 2021

IntegerExponent[#,2]&/@(Times@@Range[#[[1]]+1,#[[2]]-1]&/@Partition[ Prime[ Range[ 80]],2,1]) (* Harvey P. Dale, Aug 12 2024 *)

a(n) = valuation(prod(k=prime(n)+1, prime(n+1)-1, k), 2); \\ Michel Marcus, Aug 31 2016

a(n) = my(p=prime(n+1),q=prime(n)); p-hammingweight(p) - (q-hammingweight(q)); \\ Kevin Ryde, Apr 11 2021

from sympy import prime
def A276133(n): return (p:=prime(n+1)-1)-p.bit_count()-(q:=prime(n))+q.bit_count() # Chai Wah Wu, Jul 10 2022

a(n) = A007814(A061214(n)).

a(n+1) = Sum_{k = A000040(n+1)..A000040(n+2)} A007814(k).

a(16) corrected by Robert Israel, Apr 11 2021

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

cp's OEIS Frontend

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

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A371151 Numbers k >= 2 such that A362333(k)-A371148(k)/A371149(k) sets a new maximum.

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A379008 Square array A(n, k) = A294898(A246278(n, k)), read by falling antidiagonals; Difference A005187(n)-A000203(n) applied to the prime shift array.

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A276133 Exponent of highest power of 2 dividing the product of the composite numbers between the n-th prime and the (n+1)-st prime.

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

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