A268354 -id:A268354 - cp's OEIS frontend

A038500 Highest power of 3 dividing n.

1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 81

Offset: 1

N. J. A. Sloane

To construct the sequence: start with 1 and concatenate twice: 1,1,1 then tripling the last term gives: 1,1,3. Concatenating those 3 terms twice gives: 1,1,3,1,1,3,1,1,3, triple the last term -> 1,1,3,1,1,3,1,1,9. Concatenating those 9 terms twice gives: 1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9, triple the last term -> 1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,27 etc. - Benoit Cloitre, Dec 17 2002

Also 3-adic value of 1/n, n >= 1. See the Mahler reference, definition on p. 7. This is a non-archimedean valuation. See Mahler, p. 10. Sometimes also called 3-adic absolute value. - Wolfdieter Lang, Jun 28 2014

Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
Zoran Sunic, Tree morphisms, transducers and integer sequences, arXiv:math/0612080 [math.CO], 2006.

Cf. A001620, A007949, A038502, A060904, A268354, A268357.

a038500 = f 1 where
   f y x = if m == 0 then f (y * 3) x' else y  where (x', m) = divMod x 3
-- Reinhard Zumkeller, Jul 06 2014

[3^Valuation(n,3): n in [1..100]]; // Vincenzo Librandi, Dec 29 2015

A038500 := n -> 3^padic[ordp](n,3): # Peter Luschny, Nov 26 2010

Flatten[{1,1,#}&/@(3^IntegerExponent[#,3]&/@(3*Range[40]))] (* or *) hp3[n_]:=If[Divisible[n,3],3^IntegerExponent[n,3],1]; Array[hp3,90] (* Harvey P. Dale, Mar 24 2012 *)
Table[3^IntegerExponent[n, 3], {n, 100}] (* Vincenzo Librandi, Dec 29 2015 *)

{a(n) = if( n<1, 0, 3^valuation(n, 3))};

Multiplicative with a(p^e) = p^e if p = 3, 1 otherwise. - Mitch Harris, Apr 19 2005
a(n) = n / A038502(n). Dirichlet g.f. zeta(s)*(3^s-1)/(3^s-3). - R. J. Mathar, Jul 12 2012
From Peter Bala, Feb 21 2019: (Start)
a(n) = gcd(n,3^n).
O.g.f.: x/(1 - x) + 2*Sum_{n >= 1} 3^(n-1)*x^(3^n)/ (1 - x^(3^n)). (End)
Sum_{k=1..n} a(k) ~ (2/(3*log(3)))*n*log(n) + (2/3 + 2*(gamma-1)/(3*log(3)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022

A060904 Largest power of 5 that divides n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001

Also the largest power of 5 that divides the n-th Fibonacci number A000045(n).
Multiplicative with a(p^e) = 5^e if p = 5, else a(p^e) = 1. - Mitch Harris, Apr 19 2005
Also 5-adic value of 1/n, n >= 1. See the Mahler reference, definition on p. 7. This is a non-archimedean valuation. See Mahler, p. 10. Sometimes also called 5-adic absolute value. - Wolfdieter Lang, Jun 30 2014

			a(10) = 5 because 10 = 5 * 2.

Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

Cf. A000045, A000203, A001620, A038500, A060865, A060901, A112765, A132739, A268354, A268357.

[5^Valuation(n,5): n in [1..100]]; // Vincenzo Librandi, Dec 29 2015

A060904 := n -> 5^padic[ordp](n,5): # Peter Luschny, Nov 26 2010

Table[5^IntegerExponent[n, 5], {n, 100}] (* Vincenzo Librandi, Dec 29 2015 *)

a(n)=5^valuation(n,5) \\ Charles R Greathouse IV, Aug 05 2015

[5^valuation(i,5) for i in [1..100]] # Tom Edgar, Mar 22 2014

If n is not divisible by 5, then a(n) = 1. If n = 5^k * m where m is not divisible by 5, then a(n) = 5^k.
Dirichlet g.f.: zeta(s)*(5^s-1)/(5^s-5). - R. J. Mathar, Jul 12 2012
a(n) = 5^A112765(n). - Tom Edgar, Mar 22 2014
From Peter Bala, Feb 21 2019: (Start)
a(n) = gcd(n,5^n).
a(n) = n/A132739(n).
O.g.f.: x/(1 - x) + 4*Sum_{n >= 1} 5^(n-1)*x^(5^n)/ (1 - x^(5^n)). (End).
a(n) = (1/5)*(sigma(5*n) - sigma(n))/(sigma(5*n) - 5*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
Sum_{k=1..n} a(k) ~ (4/(5*log(5)))*n*log(n) + (3/5 + 4*(gamma-1)/(5*log(5)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001

Edited by Joerg Arndt and M. F. Hasler, Dec 29 2015

A242603 Largest divisor of n not divisible by 7. Remove factors 7 from n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 13, 2, 15, 16, 17, 18, 19, 20, 3, 22, 23, 24, 25, 26, 27, 4, 29, 30, 31, 32, 33, 34, 5, 36, 37, 38, 39, 40, 41, 6, 43, 44, 45, 46, 47, 48, 1, 50, 51, 52, 53, 54, 55, 8, 57, 58, 59, 60, 61, 62, 9, 64, 65, 66, 67, 68, 69, 10, 71, 72, 73, 74, 75, 76, 11

Offset: 1

Wolfdieter Lang, Jun 18 2014

This is member p = 7 in the p-family of sequences (p a prime).
See A000265, A038502 and A132739 for primes 2, 3 and 5, also for formulas, programs and references.
As well as being multiplicative, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 21 2019

			From _Indranil Ghosh_, Jan 31 2017: (Start)
For n = 12, the divisors of 12 are 1,2,3,4,6 and 12. The largest divisor not divisible by 7 is 12. So, a(12) = 12.
For n = 14, the divisors of 14 are 1,2,7 and 14. The largest divisor not divisible by 7 is 2. So, a(14) = 2. (End)
From _Peter Bala_, Feb 21 2019: (Start)
Sum_{n >= 1} n*a(n)*x^n = G(x) - (6*7)*G(x^7) - (6*49)*G(x^49) - (6*343)*G(x^343) - ..., where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (6/7)*H(x^7) - (6/49)*H(x^49) - (6/343)*H(x^343) - ..., where H(x) = x/(1 - x).
Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (6/7^2)*L(x^7) - (6/49^2)*L(x^49) - (6/343^2)*L(x^343) - ..., where L(x) = Log(1/(1 - x)).
Also, Sum_{n >= 1} (1/a(n))*x^n = L(x) + (6/7)*L(x^7) + (6/7)*L(x^49) + (6/7)*L(x^343) ... . (End)

Indranil Ghosh, Table of n, a(n) for n = 1..20000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.

Cf. A000265, A038502, A132739, A214411.

Table[n/7^IntegerExponent[n, 7], {n, 80}] (* Alonso del Arte, Jun 18 2014 *)

a(n) = f = factor(n);  for (i=1, #f~, if (f[i,1]==7, f[i, 1]=1)); factorback(f); \\ Michel Marcus, Jun 18 2014

a(n) = n \ 7^valuation(n, 7) \\ David A. Corneth, Feb 21 2019

def A242603(n):
    for i in range(n,0,-1):
        if n%i==0 and i%7!=0:
            return i # Indranil Ghosh, Jan 31 2017

Multiplicative with a(p^e) = 1 if p = 7, else p^e.
Dirichlet g.f.: zeta(s-1)*7*(7^(s-1) -  1)/(7^s - 1).
a(n) = n/A268354(n).
From Peter Bala, Feb 21 2019: (Start)
a(n) = n/gcd(n,7^n).
O.g.f.: F(x) - 6*F(x^7) - 6*F(x^49) - 6*F(x^243) - ..., where F(x) = x/(1 - x)^2 is the generating function for the positive integers. More generally, for m >= 1,
Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) - (7^m - 1)( F(m,x^7) + F(m,x^49) + F(m,x^243) + ...), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m_th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.
Repeatedly applying the Euler operator x*d/dx or its inverse operator to the o.g.f. for the sequence a(n) produces generating functions for the sequences (n^m*a(n))n>=1, m in Z. Some examples are given below. (End)
Sum_{k=1..n} a(k) ~ (7/16) * n^2. - Amiram Eldar, Nov 28 2022

A268357 Highest power of 11 dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 121, 1

Offset: 1

Tom Edgar, Feb 02 2016

The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 11.

This first index where this differs from A109014 is 121; a(121) = 121 and A109014(121) = 11.

			Since 22 = 11 * 2, a(22) = 11. Likewise, since 11 does not divide 21, a(21) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Cf. A001620, A006519, A038500, A234957, A234959, A109014, A268354, A060904.

[11^Valuation(n,11): n in [1..130]]; // Vincenzo Librandi, Feb 03 2016

Table[11^IntegerExponent[n, 11], {n, 130}] (* Bruno Berselli, Feb 03 2016 *)

[11^valuation(i, 11) for i in [1..130]]

a(n) = 11^valuation(n,11).
Completely multiplicative with a(11) = 11, a(p) = 1 for prime p and p<>11. - Andrew Howroyd, Jul 20 2018
From Peter Bala, Feb 21 2019: (Start)
a(n) = gcd(n,11^n).
O.g.f.: x/(1 - x) + 10*Sum_{n >= 1} 11^(n-1)*x^(11^n)/ (1 - x^(11^n)). (End)
Sum_{k=1..n} a(k) ~ (10/(11*log(11)))*n*log(n) + (6/11 + 10*(gamma-1)/(11*log(11)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022
Dirichlet g.f.: zeta(s)*(11^s-1)/(11^s-11). - Amiram Eldar, Jan 03 2023

A088841 Numerator of the quotient sigma(7*n)/sigma(n).

Original entry on oeis.org

8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 400, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8

Offset: 1

Labos Elemer, Nov 04 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A088837, A088838, A088839, A088840, A088841, A088842 (denominators).

Cf. A000203, A038712, A080278, A214411, A268354, A283078.

Table[Numerator[DivisorSigma[1, 7*n]/DivisorSigma[1, n]], {n, 1, 128}]

a(n) = numerator(sigma(7*n)/sigma(n)); \\ Amiram Eldar, Mar 22 2024

From Amiram Eldar, Mar 22 2024: (Start)
a(n) = numerator(A283078(n)/A000203(n)).
a(n) = (7^(A214411(n)+2)-1)/6 = (49*A268354(n)-1)/6.
Sum_{k=1..n} a(k) ~ (7/log(7))*n*log(n) + (9/2 + 7*(gamma-1)/log(7))*n, where gamma is Euler's constant (A001620).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A088842(k) = 1 + 36 * Sum_{k>=1} 1/(7^k-1) = 7.87276224676... . (End)

A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 16, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 8, 1, 4, 5, 4, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 9, 64, 1, 6, 1, 64, 9, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 8, 7, 8

Offset: 1

Rémy Sigrist, Jun 09 2018

The array T is completely multiplicative in both parameters.
For any n > 0 and prime number p, T(n, p) is the highest power of p dividing n.
For any function f associating a nonnegative value to any pair of nonnegative values and such that f(0, 0) = 0, we can build an analog of this sequence, say P_f, such that for any prime number p and any n and k > 0 with p-adic valuations i and j, the p-adic valuation of P_f(n, k) equals f(i, j):
   f(i, j)    P_f
   -------    ---
   i * j      T (this sequence)
   i + j      A003991 (product)
   abs(i-j)   A089913
   min(i, j)  A003989 (GCD)
   max(i, j)  A003990 (LCM)
   i AND j    A059895
   i OR j     A059896
   i XOR j    A059897
If log(N) denotes the set {log(n) : n is in N, the set of the positive integers}, one can define a binary operation on log(N): with prime factorizations n = Product p_i^e_i and k = Product p_i^f_i, set log(n) o log(k) = Sum_{i} (e_i*f_i) * log(p_i). o has the premises of a scalar product even if log(N) isn't a vector space. T(n, k) can be viewed as exp(log(n) o log(k)). - Luc Rousseau, Oct 11 2020

			Array T(n, k) begins:
  n\k|    1    2    3    4    5    6    7    8    9   10
  ---+--------------------------------------------------
    1|    1    1    1    1    1    1    1    1    1    1
    2|    1    2    1    4    1    2    1    8    1    2  -> A006519
    3|    1    1    3    1    1    3    1    1    9    1  -> A038500
    4|    1    4    1   16    1    4    1   64    1    4
    5|    1    1    1    1    5    1    1    1    1    5  -> A060904
    6|    1    2    3    4    1    6    1    8    9    2  -> A065331
    7|    1    1    1    1    1    1    7    1    1    1  -> A268354
    8|    1    8    1   64    1    8    1  512    1    8
    9|    1    1    9    1    1    9    1    1   81    1
   10|    1    2    1    4    5    2    1    8    1   10  -> A132741

Cf. A003989, A003990, A003991, A006519, A007947, A038500, A054496, A059895, A059896, A059897, A060904, A065331, A089913, A132741, A268354, A268357.

T[n_, k_] := With[{p = FactorInteger[GCD[n, k]][[All, 1]]}, If[p == {1}, 1, Times @@ (p^(IntegerExponent[n, p] * IntegerExponent[k, p]))]];
Table[T[n-k+1, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 11 2018 *)

T(n, k) = my (p=factor(gcd(n, k))[,1]); prod(i=1, #p, p[i]^(valuation(n, p[i]) * valuation(k, p[i])))

T(n, k) = T(k, n) (T is commutative).
T(m, T(n, k)) = T(T(m, n), k) (T is associative).
T(n, k) = 1 iff gcd(n, k) = 1.
T(n, n) = A054496(n).
T(n, A007947(n)) = n.
T(n, 1) = 1.
T(n, 2) = A006519(n).
T(n, 3) = A038500(n).
T(n, 4) = A006519(n)^2.
T(n, 5) = A060904(n).
T(n, 6) = A065331(n).
T(n, 7) = A268354(n).
T(n, 8) = A006519(n)^3.
T(n, 9) = A038500(n)^2.
T(n, 10) = A132741(n).
T(n, 11) = A268357(n).

cp's OEIS Frontend

A038500 Highest power of 3 dividing n.

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A060904 Largest power of 5 that divides n.

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A242603 Largest divisor of n not divisible by 7. Remove factors 7 from n.

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A268357 Highest power of 11 dividing n.

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A088841 Numerator of the quotient sigma(7*n)/sigma(n).

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A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.

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