A055457 -id:A055457 - cp's OEIS frontend

A191610 Possible number of trailing zeros in k!.

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 124, 125, 126, 127, 128, 130, 131, 132, 133, 134, 136

Offset: 1

José María Grau Ribas, Jun 09 2011

nonn

Equivalently, possible values of 10-adic valuation of k!. - Joerg Arndt, Sep 21 2020

			3 is a term because 15! = 1307674368000 has 3 trailing 0's.
5 is not a term because 24! has 4 trailing 0's, but 25! has 6 trailing 0's.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A027868, A000351, A055457 (first differences).
Complement of A000966.
Cf. A380662, A381239.

a191610 1 = 0
a191610 n = sum $ takeWhile (> 0) $ map ((n - 1) `div`) a000351_list
-- Reinhard Zumkeller, Oct 31 2012

zOF[n_Integer?Positive]:=Module[{maxpow=0},While[5^maxpow<=n,maxpow++];Plus@@Table[ Quotient[n,5^i],{i,maxpow-1}]]; Attributes[zOF]={Listable}; zOF[Range[1000]]//Union (* Harvey P. Dale, Dec 06 2023 *)
Table[Sum[Floor[(n - 1)/5^k], {k, 0, Floor[Log[5, n]]}], {n, 1, 200}] (* Clark Kimberling, Feb 17 2025 *)

# requires Python 3.2 and higher
from itertools import accumulate
from sympy import multiplicity
A191610 = [0]+list(accumulate(multiplicity(5,n) for n in range(5,10**3,5)))
# Chai Wah Wu, Sep 05 2014

a(n) ~ 5*n/4. - Vaclav Kotesovec, Sep 21 2020
G.f.: 1/(1-x) * Sum_{k>=0} x^(5^k)/(1-x^5^k). - Joerg Arndt, Sep 21 2020
a(n) = Sum_{k>=0} floor((n-1)/5^k). - Clark Kimberling, Feb 17 2025

A373217 Expansion of Sum_{k>=0} x^(7^k) / (1 - x^(7^k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Seiichi Manyama, May 28 2024

The number of powers of 7 that divide n. - Amiram Eldar, Mar 29 2025

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001511, A051064, A055457, A115362, A373216.

Cf. A000005, A008683, A214411, A373221.

a[n_] := 1 + IntegerExponent[n, 7]; Array[a, 100] (* Amiram Eldar, May 29 2024 *)

PARI
```
a(n) = valuation(n, 7)+1;
```

G.f. A(x) satisfies A(x) = x/(1 - x) + A(x^7).
a(7*n+1) = a(7*n+2) = ... = (7*n+6) = 1 and a(7*n+7) = 1 + a(n+1) for n >= 0.
Multiplicative with a(p^e) = e+1 if p = 7, 1 otherwise.
a(n) = -Sum_{d|n} mu(7*d) * tau(n/d).
a(n) = A214411(n) + 1.
From Amiram Eldar, May 29 2024: (Start)
Dirichlet g.f.: (7^s/(7^s-1)) * zeta(s).
Sum_{k=1..n} a(k) ~ (7/6) * n. (End)
G.f.: Sum_{i>=1, j>=0} x^(i*7^j). - Seiichi Manyama, Mar 23 2025
a(n) = A214411(7*n). - R. J. Mathar, Jun 28 2025

A359100 a(n) = (1/4) * Sum_{d|n} phi(5 * d).

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 7, 8, 9, 12, 11, 12, 13, 14, 18, 16, 17, 18, 19, 24, 21, 22, 23, 24, 31, 26, 27, 28, 29, 36, 31, 32, 33, 34, 42, 36, 37, 38, 39, 48, 41, 42, 43, 44, 54, 46, 47, 48, 49, 62, 51, 52, 53, 54, 66, 56, 57, 58, 59, 72, 61, 62, 63, 64, 78, 66, 67, 68, 69, 84, 71, 72, 73, 74, 93, 76

Offset: 1

Seiichi Manyama, Dec 16 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A129527, A327625, A359099, A373188.

Cf. A000010, A055457.

Array[DivisorSum[#, EulerPhi[5 #] &]/4 &, 76] (* Michael De Vlieger, Dec 16 2022 *)
f[p_, e_] := If[p == 5, (5^(e + 1) - 1)/4, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 17 2022 *)

PARI
```
a(n) = sumdiv(n, d, eulerphi(5*d))/4;
```

my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(5*k)*x^k/(1-x^k))/4)

G.f.: Sum_{k>=1} phi(5 * k) * x^k / (4 * (1 - x^k)).
G.f.: Sum_{k>=0} x^(5^k) / (1 - x^(5^k))^2.
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(5^e) = (5^(e+1)-1)/4, and a(p^e) = p if p != 5.
Dirichlet g.f.: zeta(s-1)*(1+1/(5^s-1)).
Sum_{k=1..n} a(k) ~ (25/48) * n^2. (End)
From Seiichi Manyama, Jun 04 2024: (Start)
G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^5).
If n == 0 (mod 5), a(n) = n + a(n/5) otherwise a(n) = n. (End)

A373219 Expansion of Product_{i>=1, j>=0} (1 + x^(i * 5^j)).

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 6, 8, 10, 14, 17, 21, 26, 32, 42, 50, 60, 74, 88, 109, 130, 154, 184, 218, 263, 309, 363, 428, 500, 593, 690, 802, 936, 1084, 1266, 1463, 1687, 1948, 2242, 2590, 2970, 3403, 3900, 4456, 5107, 5820, 6626, 7546, 8572, 9755, 11059, 12523, 14180, 16028, 18133, 20455, 23056, 25978

Offset: 0

Seiichi Manyama, May 28 2024

nonn

Cf. A000041, A174065, A327726, A373220, A373221.

Cf. A000009, A055457.

my(N=60, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(valuation(k, 5)+1)))

G.f.: Product_{k>=1} (1 + x^k)^A055457(k).

Let A(x) be the g.f. of this sequence, and B(x) be the g.f. of A000009, then B(x) = A(x)/A(x^5).

A373281 Expansion of Sum_{k>=0} x^(5^k) / (1 - 5*x^(5^k)).

Original entry on oeis.org

1, 5, 25, 125, 626, 3125, 15625, 78125, 390625, 1953130, 9765625, 48828125, 244140625, 1220703125, 6103515650, 30517578125, 152587890625, 762939453125, 3814697265625, 19073486328250, 95367431640625, 476837158203125, 2384185791015625, 11920928955078125

Offset: 1

Seiichi Manyama, May 30 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A187767, A373279, A373280, A373282, A373283.

Cf. A000351, A054662, A055457.

b(n, k) = sumdiv(n, d, (gcd(d, k)==1)*(moebius(d)*k^(n/d)))/(k*n);
a(n, k=5) = sumdiv(n, d, d*b(d, k));

G.f. A(x) satisfies A(x) = x/(1 - 5*x) + A(x^5).
If n == 0 (mod 5), a(n) = 5^n + a(n/5) otherwise a(n) = 5^n.
a(n) = Sum_{d|n} d * A054662(d).

A038510 Composite numbers with smallest prime factor >= 7.

Original entry on oeis.org

49, 77, 91, 119, 121, 133, 143, 161, 169, 187, 203, 209, 217, 221, 247, 253, 259, 287, 289, 299, 301, 319, 323, 329, 341, 343, 361, 371, 377, 391, 403, 407, 413, 427, 437, 451, 469, 473, 481, 493, 497, 511, 517, 527, 529, 533, 539, 551, 553, 559, 581, 583

Offset: 1

Jeff Burch

nonn

Let A = set of numbers of form 6n + 1, B = numbers of form 6n - 1. Eliminating numbers of form 25 + 30s from A and those of form 35 + 30s from B we obtain sets A* and B*. Removing all terms of the sequence from the union of A* and B*, only prime numbers remain. - Hisanobu Shinya (ilikemathematics(AT)hotmail.com), Jul 14 2002
Divide n by a*b*c where a = 2^(A001511(n)-1), b = 3^(A051064(n)-1) and c = 5^(A055457(n) -1). Then the resulting sequence includes only primes and a(n). - Alford Arnold, Sep 08 2003
Composite numbers not divisible by 2, 3 or 5. - Lekraj Beedassy, Jun 30 2004
Composite numbers k such that k^4 mod 30 = 1. - Gary Detlefs, Dec 09 2012
Composite numbers congruent to 1, 7, 11, 13, -13, -11, -7, -1 (mod 30). Since asymptotically, 100% of integers are composite, we have a(n)/n ~ 30/phi(30) = 30/8 = 3.75. - Daniel Forgues, Mar 16 2013
"John [Conway] recommends the more refined partition [of the positive numbers]: 1, prime, trivially composite, or nontrivially composite. Here, a composite integer is trivially composite if it is divisible by 2, 3, or 5." See link to (van der Poorten, Thomsen, and Wiebe; 2006) pp. 73-74. - Daniel Forgues, Jan 30 2015, Feb 04 2015
For the eight congruences coprime to 30, we can use one byte to encode the "primality/non-primality (unit or composite)" for each [30*n, 30*(n+1)[, n >= 0, closed-open interval, either as little endian binary sequence {01111111, 11111011, 11110111, 01111110, ...}, or as big endian binary sequence {11111110, 11011111, 11101111, 01111110, ...}, which we may then express in base 10. - Daniel Forgues, Feb 05 2015

J. H. Silverman, A Friendly Introduction to Number Theory, 2nd Edn. "Appendix A: Factorization of Small Composite Integers", Prentice Hall NY 2001.

T. D. Noe, Table of n, a(n) for n = 1..10000
Alf van der Poorten, Kurt Thomsen, and Mark Wiebe, A Curious Cubic Identity and Self-similar Sums of Squares, 2006, pp. 73-74.

Cf. A001511, A051064, A055457, A038509, A071904.
Cf. A070884, A038511.
Intersection of A002808 and A007775.

for n from 1 to 583 do if n^4 mod 30 = 1 and not isprime(n) then print(n)fi od; # Gary Detlefs, Dec 09 2012

Select[Range[1000], ! PrimeQ[#] && FactorInteger[#][[1, 1]] >= 7 &] (* T. D. Noe, Mar 16 2013 *)

is(n)=gcd(n,30)==1 && !isprime(n) \\ Charles R Greathouse IV, Dec 09 2012

a(n) ~ 3.75n. - Charles R Greathouse IV, Dec 09 2012

Corrected by Ralf Stephan, Apr 04 2003

A382373 Expansion of 1/( 1 - Sum_{k>=0} x^(5^k) / (1 - x^(5^k)) ).

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 34, 69, 140, 284, 578, 1173, 2382, 4837, 9822, 19948, 40508, 82261, 167050, 339233, 688896, 1398964, 2840926, 5769169, 11715654, 23791402, 48314044, 98113049, 199241660, 404607125, 821650100, 1668554099, 3388392198, 6880928638, 13973346686

Offset: 0

Seiichi Manyama, Mar 23 2025

nonn

Cf. A327736, A382367, A382372, A382378.

Cf. A055457.

a(0) = 1; a(n) = Sum_{k=1..n} A055457(k) * a(n-k).
G.f.: 1/(1 - Sum_{i>=1, j>=0} x^(i*5^j)).
G.f. A(x) satisfies A(x) = 1/( 1/A(x^5) - x/(1-x) ).

A364223 Expansion of Sum_{k>=0} 5^k * x^(5^k) / (1 - x^(5^k))^2.

Original entry on oeis.org

1, 2, 3, 4, 10, 6, 7, 8, 9, 20, 11, 12, 13, 14, 30, 16, 17, 18, 19, 40, 21, 22, 23, 24, 75, 26, 27, 28, 29, 60, 31, 32, 33, 34, 70, 36, 37, 38, 39, 80, 41, 42, 43, 44, 90, 46, 47, 48, 49, 150, 51, 52, 53, 54, 110, 56, 57, 58, 59, 120, 61, 62, 63, 64, 130, 66, 67, 68, 69, 140, 71, 72, 73, 74, 225

Offset: 1

Seiichi Manyama, Jul 14 2023

Cf. A055457, A112765, A359100.

Cf. A091512, A364222, A364224.

a[n_] := n * (IntegerExponent[n, 5] + 1); Array[a, 100] (* Amiram Eldar, Jul 14 2023 *)

PARI
```
a(n) = n*(valuation(n, 5)+1);
```

a(n) = n * A055457(n).
If n == 0 (mod 5), a(n) = n + 5 * a(n/5) otherwise a(n) = n.
From Amiram Eldar, Jul 14 2023: (Start)
Multiplicative with a(5^e) = (e+1)*5^e and a(p^e) = p*e if p != 5.
Dirichlet g.f.: (5^s/(5^s-5)) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (5/8)*n^2. (End)

cp's OEIS Frontend

A191610 Possible number of trailing zeros in k!.

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A373217 Expansion of Sum_{k>=0} x^(7^k) / (1 - x^(7^k)).

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A359100 a(n) = (1/4) * Sum_{d|n} phi(5 * d).

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A373219 Expansion of Product_{i>=1, j>=0} (1 + x^(i * 5^j)).

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A373281 Expansion of Sum_{k>=0} x^(5^k) / (1 - 5*x^(5^k)).

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A038510 Composite numbers with smallest prime factor >= 7.

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A382373 Expansion of 1/( 1 - Sum_{k>=0} x^(5^k) / (1 - x^(5^k)) ).

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A364223 Expansion of Sum_{k>=0} 5^k * x^(5^k) / (1 - x^(5^k))^2.

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