A054999 -id:A054999 - cp's OEIS frontend

A027868 Number of trailing zeros in n!; highest power of 5 dividing n!.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 18, 18, 18, 18, 18, 19

Offset: 0

Warut Roonguthai

Also the highest power of 10 dividing n! (different from A054899). - Hieronymus Fischer, Jun 18 2007
Alternatively, a(n) equals the expansion of the base-5 representation A007091(n) of n (i.e., where successive positions from right to left stand for 5^n or A000351(n)) under a scale of notation whose successive positions from right to left stand for (5^n - 1)/4 or A003463(n); for instance, n = 7392 has base-5 expression 2*5^5 + 1*5^4 + 4*5^3 + 0*5^2 + 3*5^1 + 2*5^0, so that a(7392) = 2*781 + 1*156 + 4*31 + 0*6 + 3*1 + 2*0 = 1845. - Lekraj Beedassy, Nov 03 2010
Partial sums of A112765. - Hieronymus Fischer, Jun 06 2012
Also the number of trailing zeros in A000165(n) = (2*n)!!. - Stefano Spezia, Aug 18 2024

			a(100)  = 24.
a(10^3) = 249.
a(10^4) = 2499.
a(10^5) = 24999.
a(10^6) = 249998.
a(10^7) = 2499999.
a(10^8) = 24999999.
a(10^9) = 249999998.
a(10^n) = 10^n/4 - 3 for 10 <= n <= 15 except for a(10^14) = 10^14/4 - 2. - _M. F. Hasler_, Dec 27 2019

M. Gardner, "Factorial Oddities." Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, 1978, pp. 50-65.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
David S. Hart, James E. Marengo, Darren A. Narayan, and David S. Ross, On the number of trailing zeros in n!, College Math. J., 39(2):139-145, 2008.
Enrique Pérez Herrero, Trailing Zeros in n!, Psychedelic Geometry Blogspot.
Soichi Ikeda and Kaneaki Matsuoka, On transcendental numbers generated by certain integer sequences, Siauliai Math. Semin., 8 (16) 2013, 63-69.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Shu-Chung Liu and Jean C.-C. Yeh, Catalan numbers modulo 2^k, J. Int. Seq. 13 (2010), 10.5.4, eq (5).
Antonio M. Oller-Marcén, A new look at the trailing zeros of n!, arXiv:0906.4868v1 [math.NT], 2009.
Antonio M. Oller-Marcén and J. Maria Grau, On the Base-b Expansion of the Number of Trailing Zeros of b^k!, J. Int. Seq. 14 (2011) 11.6.8
Eric Weisstein's World of Mathematics, Factorial.
Index entries for sequences related to factorial numbers

See A000966 for the missing numbers. See A011371 and A054861 for analogs involving powers of 2 and 3.
Cf. A054899, A007953, A112765, A067080, A098844, A132027, A067080, A098844, A132029, A054999, A112765, A191610, A000351.
Cf. also A000142, A004154.
Cf. A000165, A008904.

a027868 n = sum $ takeWhile (> 0) $ map (n `div`) $ tail a000351_list
-- Reinhard Zumkeller, Oct 31 2012

[Valuation(Factorial(n), 5): n in [0..80]]; // Bruno Berselli, Oct 11 2021

0, seq(add(floor(n/5^i),i=1..floor(log[5](n))), n=1..100); # Robert Israel, Nov 13 2014

Table[t = 0; p = 5; While[s = Floor[n/p]; t = t + s; s > 0, p *= 5]; t, {n, 0, 100} ]
Table[ IntegerExponent[n!], {n, 0, 80}] (* Robert G. Wilson v *)
zOF[n_Integer?Positive]:=Module[{maxpow=0},While[5^maxpow<=n,maxpow++];Plus@@Table[Quotient[n,5^i],{i,maxpow-1}]]; Attributes[zOF]={Listable}; Join[{0},zOF[ Range[100]]] (* Harvey P. Dale, Apr 11 2022 *)

a(n)={my(s);while(n\=5,s+=n);s} \\ Charles R Greathouse IV, Nov 08 2012, edited by M. F. Hasler, Dec 27 2019

a(n)=valuation(n!,5) \\ Charles R Greathouse IV, Nov 08 2012

apply( A027868(n)=(n-sumdigits(n,5))\4, [0..99]) \\ M. F. Hasler, Dec 27 2019

from sympy import multiplicity
A027868, p5 = [0,0,0,0,0], 0
for n in range(5,10**3,5):
    p5 += multiplicity(5,n)
    A027868.extend([p5]*5) # Chai Wah Wu, Sep 05 2014

def A027868(n): return 0 if n<5 else n//5 + A027868(n//5) # David Radcliffe, Jun 26 2016

from sympy.ntheory.factor_ import digits
def A027868(n): return n-sum(digits(n,5)[1:])>>2 # Chai Wah Wu, Oct 18 2024

a(n) = Sum_{i>=1} floor(n/5^i).
a(n) = (n - A053824(n))/4.
From Hieronymus Fischer, Jun 25 2007 and Aug 13 2007, edited by M. F. Hasler, Dec 27 2019: (Start)
G.f.: g(x) = Sum_{k>0} x^(5^k)/(1-x^(5^k))/(1-x).
a(n) = Sum_{k=5..n} Sum_{j|k, j>=5} (floor(log_5(j)) - floor(log_5(j-1))).
G.f.: g(x) = L[b(k)](x)/(1-x) where L[b(k)](x) = Sum_{k>=0} b(k)*x^k/(1-x^k) is a Lambert series with b(k) = 1, if k>1 is a power of 5, else b(k) = 0.
G.f.: g(x) = Sum_{k>0} c(k)*x^k/(1-x), where c(k) = Sum_{j>1, j|k} floor(log_5(j)) - floor(log_5(j - 1)).
Recurrence:
a(n) = floor(n/5) + a(floor(n/5));
a(5*n) = n + a(n);
a(n*5^m) = n*(5^m-1)/4 + a(n).
a(k*5^m) = k*(5^m-1)/4, for 0 <= k < 5, m >= 0.
Asymptotic behavior:
a(n) = n/4 + O(log(n)),
a(n+1) - a(n) = O(log(n)), which follows from the inequalities below.
a(n) <= (n-1)/4; equality holds for powers of 5.
a(n) >= n/4 - 1 - floor(log_5(n)); equality holds for n = 5^m-1, m > 0.
lim inf (n/4 - a(n)) = 1/4, for n -> oo.
lim sup (n/4 - log_5(n) - a(n)) = 0, for n -> oo.
lim sup (a(n+1) - a(n) - log_5(n)) = 0, for n -> oo.
(End)
a(n) <= A027869(n). - Reinhard Zumkeller, Jan 27 2008
10^a(n) = A000142(n) / A004154(n). - Reinhard Zumkeller, Nov 24 2012
a(n) = Sum_{k=1..floor(n/2)} floor(log_5(n/k)). - Ammar Khatab, Feb 01 2025

Examples added by Hieronymus Fischer, Jun 06 2012

A054996 Integers that can be expressed as the sum of consecutive primes in exactly 1 way.

Original entry on oeis.org

2, 3, 7, 8, 10, 11, 12, 13, 15, 18, 19, 24, 26, 28, 29, 30, 37, 39, 42, 43, 47, 48, 49, 52, 56, 58, 61, 68, 73, 75, 77, 78, 79, 84, 88, 89, 95, 98, 102, 103, 107, 113, 121, 124, 128, 129, 132, 137, 144, 149, 150, 151, 155, 156, 157, 158, 159, 160, 161, 162, 163

Offset: 1

Jud McCranie, May 30 2000

nonn

			8=3+5, so 8 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054997, A054998, A054999, A055000, A055001.

A054845(a(n)) = 1. - Ray Chandler, Sep 20 2023

A054998 Integers that can be expressed as the sum of consecutive primes in exactly 3 ways.

Original entry on oeis.org

41, 83, 197, 199, 223, 240, 251, 281, 287, 340, 371, 401, 439, 491, 510, 593, 660, 733, 803, 857, 864, 883, 931, 941, 961, 983, 990, 991, 1012, 1060, 1061, 1099, 1104, 1187, 1236, 1283, 1313, 1361, 1381, 1392, 1433, 1439, 1493, 1511, 1523, 1524, 1553

Offset: 1

Jud McCranie, May 30 2000

nonn

			41 can be expressed as 41 or 11+13+17 or 2+3+5+7+11+13, so 41 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Robert Israel, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054997, A054999, A055500, A055001.

N:= 10^4: # to get all terms <= N
P:= [0,op(select(isprime, [2,seq(i,i=3..N,2)]))]:
nP:= nops(P);
S:= ListTools:-PartialSums(P):
V:= Vector(N):
for i from 1 to nP-1 do
  for j from i+1 to nP while S[j] - S[i] <= N do
     V[S[j]-S[i]]:= V[S[j]-S[i]]+1
od od:
select(t -> V[t] = 3, [$1..N]): # Robert Israel, Apr 05 2017

Module[{nn = 300, s}, s = Array[Prime, nn]; Keys@ Take[Select[KeySort@ Merge[Table[PositionIndex@ Map[Total, Partition[s, k, 1]], {k, nn/2}], Identity], Length@ # == 3 &], Floor[nn/6]]] (* Michael De Vlieger, Apr 06 2017, Version 10 *)

A054845(a(n)) = 3. - Ray Chandler, Sep 20 2023

A054997 Integers that can be expressed as the sum of consecutive primes in exactly 2 ways.

Original entry on oeis.org

5, 17, 23, 31, 36, 53, 59, 60, 67, 71, 72, 90, 97, 100, 101, 109, 112, 119, 120, 127, 131, 138, 139, 143, 152, 173, 180, 181, 187, 204, 210, 211, 221, 228, 233, 258, 263, 269, 271, 276, 300, 304, 323, 330, 331, 349, 353, 372, 373, 379, 384, 390, 395, 408

Offset: 1

Jud McCranie, May 30 2000

nonn

			5 can be expressed as 5 or 2+3, so 5 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054998, A054999, A055500, A055001.

A054845(a(n)) = 2. - Ray Chandler, Sep 20 2023

A055001 Integers that can be expressed as the sum of consecutive primes in exactly 6 ways.

Original entry on oeis.org

34421, 130638, 229841, 235493, 271919, 295504, 345011, 347856, 358446, 358877, 414221, 429804, 434669, 480951, 488603, 532423, 532823, 543625, 561375, 621937, 626852, 655561, 687496, 703087, 734069, 746829, 810418, 824099, 888793

Offset: 1

Jud McCranie, May 30 2000

nonn

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..1000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054997, A054998, A054999, A055000.

A054845(a(n)) = 6. - Ray Chandler, Sep 20 2023

A055000 Integers that can be expressed as the sum of consecutive primes in exactly 5 ways.

Original entry on oeis.org

311, 863, 14369, 14699, 15329, 16277, 19717, 20272, 25416, 28500, 29033, 36467, 37607, 40433, 41074, 42463, 45101, 46660, 48731, 49253, 49499, 50560, 53424, 55813, 59068, 67141, 68787, 70104, 70429, 70692, 71548, 76423, 78756, 78791

Offset: 1

Jud McCranie, May 30 2000

nonn

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054997, A054998, A054999, A055001.

A054845(a(n)) = 5. - Ray Chandler, Sep 20 2023

A067374 Integers expressible as the sum of (at least two) consecutive primes in at least 4 ways.

Original entry on oeis.org

311, 863, 1164, 1320, 1650, 1854, 2856, 2867, 3198, 3264, 3754, 4200, 4920, 5100, 5770, 5999, 6504, 8152, 10134, 10320, 10536, 10649, 11058, 12294, 12438, 12762, 12820, 12954, 12990, 14369, 14699, 14826, 15329, 15610, 15762, 16199, 16277

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

			E.g., 311 = 101 + 103 + 107 = 53 + 59 + 61 + 67 + 71 = 31 + 37 + 41 + 43 + 47 + 53 + 59 = 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47.

Donovan Johnson, Table of n, a(n) for n = 1..1000
P. De Geest, WONplate 122
C. Rivera, Puzzle 46

Cf. A050936, A067372-A067381, A054999.

Clear[lst,lst1,m,n,p,a,b] m=2*6!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[pVladimir Joseph Stephan Orlovsky, Aug 15 2009 *)

A084143(a(n)) > 3. - Ray Chandler, Sep 20 2023

Offset corrected by Donovan Johnson, Nov 14 2013

cp's OEIS Frontend

A027868 Number of trailing zeros in n!; highest power of 5 dividing n!.

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A054996 Integers that can be expressed as the sum of consecutive primes in exactly 1 way.

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A054998 Integers that can be expressed as the sum of consecutive primes in exactly 3 ways.

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A054997 Integers that can be expressed as the sum of consecutive primes in exactly 2 ways.

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A055001 Integers that can be expressed as the sum of consecutive primes in exactly 6 ways.

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A055000 Integers that can be expressed as the sum of consecutive primes in exactly 5 ways.

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A067374 Integers expressible as the sum of (at least two) consecutive primes in at least 4 ways.

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