A080278 -id:A080278 - cp's OEIS frontend

A080333 Partial sums of A080278.

1, 2, 6, 7, 8, 12, 13, 14, 27, 28, 29, 33, 34, 35, 39, 40, 41, 54, 55, 56, 60, 61, 62, 66, 67, 68, 108, 109, 110, 114, 115, 116, 120, 121, 122, 135, 136, 137, 141, 142, 143, 147, 148, 149, 162, 163, 164, 168, 169, 170, 174, 175, 176, 216, 217, 218, 222, 223, 224, 228, 229

Offset: 1

N. J. A. Sloane, Mar 19 2003

nonn

Klaus Brockhaus, Illustration of A080278 and A080333
B. Dearden, J. Iiams, and J. Metzger, A Function Related to the Rumor Sequence Conjecture , J. Int. Seq. 14 (2011) # 11.2.3.

Cf. A080277, A080278, A333979.

a(n) = fromdigits(Vec(Pol(digits(3*n,3))'),3); \\ Kevin Ryde, Apr 29 2021

a(n) = Sum_{k=0..log_3(n)} 3^k*floor(n/3^k).
a(3^k) = (k+1)*3^k.
a(n) is conjectured to be asymptotic to n*log(n)/log(3). - Klaus Brockhaus, Mar 23 2003 [This follows from the asymptotics of A333979. - Pontus von Brömssen, Sep 06 2020]
a(n) = n + 3*a(floor(n/3)), a(0)=0. - Vladeta Jovovic, Aug 06 2003
G.f.: (1/(1 - x))*Sum_{k>=0} 3^k*x^(3^k)/(1 - x^(3^k)). - Ilya Gutkovskiy, Mar 15 2018
a(n) = A333979(3*n,3). - Pontus von Brömssen, Sep 06 2020

A007949 Greatest k such that 3^k divides n. Or, 3-adic valuation of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1

Offset: 1

R. Muller

Obeys the general recurrences for p-adic valuation discussed in A214411. - Redjan Shabani, Jul 17 2012
Lexicographically earliest cubefree sequence, which also (conjecturally) appears in the construction of the lexicographically earliest cubefree {0,1}-sequence A282317, cf. Example section of A286940. - M. F. Hasler, May 21 2017
The sequence is invariant under the "lower trim" operator: remove all zeros, and subtract one from each remaining term. - Franklin T. Adams-Watters, May 25 2017

F. Q. Gouvea, p-Adic Numbers, Springer-Verlag, 1993; see p. 23.

T. D. Noe, Table of n, a(n) for n = 1..1000
K. Atanassov, On the 61-st, 62-nd and the 63-rd Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 4 (1998), No. 4, 175-182.
K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.
S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
F. Smarandache, Only Problems, Not Solutions!.
M. Vassilev-Missana and K. Atanassov, Some Representations related to n!, Notes on Number Theory and Discrete Mathematics, Vol. 4 (1998), No. 4, 148-153.
Index entries for sequences that are fixed points of mappings

Partial sums give A054861.
Cf. A000005, A038500, A080278, A001511, A122841, A007814, A112765, A253786.
One less than A051064.

a007949 n = if m > 0 then 0 else 1 + a007949 n'
            where (n', m) = divMod n 3
-- Reinhard Zumkeller, Jun 23 2013, May 14 2011

% Input:
%  n: an integer
% Output:
%  m: max power of 3 such that 3^m divides n
%  M: 1-by-K matrix where M(i) is the max power of 3 such that 3^M(i) divides n
function [m,M] = Omega3(n)
  M = NaN*zeros(1,n);
  M(1)=0; M(2)=0; M(3)=0;
    for k=4:n
      if M(k-3)~=0
        M(k)=M(k-k/3)+1;
      else
        M(k)=0;
      end
    end
    m=M(end);
end
% Redjan Shabani, Jul 17 2012

[Valuation(n, 3): n in [1..110]]; // Bruno Berselli, Aug 05 2013

A007949 := proc(n) option remember; if n mod 3 > 0 then 0 else procname(n/3)+1; fi; end;
# alternative by R. J. Mathar, Mar 29 2017
A007949 := proc(n)
    padic[ordp](n,3) ;
end proc:

p=3; Array[ If[ Mod[ #, p ]==0, Select[ FactorInteger[ # ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 81 ]
Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 0, 1}, 1 -> {0, 0, 2}, 2 -> {0, 0, 3}, 3 -> {0, 0, 4}}) ]}], {0}, 5] (* Robert G. Wilson v, Mar 03 2005 *)
IntegerExponent[Range[200], 3] (* Zak Seidov, Apr 15 2010 *)
Table[If[Mod[n, 3] > 0, 0, 1 + b[n/3]], {n, 200}] (* Zak Seidov, Apr 15 2010 *)

PARI
```
a(n)=valuation(n,3)
```

def a(n):
    k = 0
    while n > 0 and n%3 == 0: n //= 3; k += 1
    return k
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Aug 06 2021

[valuation(n, 3) for n in (1..106)]  # Peter Luschny, Nov 16 2012

(define (A007949 n) (let loop ((n n) (k 0)) (cond ((not (zero? (modulo n 3))) k) (else (loop (/ n 3) (+ 1 k)))))) ;; Antti Karttunen, Oct 06 2017

a(n) = 0 if n != 0 (mod 3), otherwise a(n) = 1 + a(n/3). - Reinhard Zumkeller, Aug 12 2001, edited by M. F. Hasler, Aug 11 2015
From Ralf Stephan, Apr 12 2002: (Start)
a(n) = A051064(n) - 1.
G.f.: Sum_{k>=1} x^3^k/(1 - x^3^k). (End)
Fixed point of the morphism: 0 -> 001; 1 -> 002; 2 -> 003; 3 -> 004; 4 -> 005; etc.; starting from a(1) = 0. - Philippe Deléham, Mar 29 2004
a(n) mod 2 = 1 - A014578(n). - Reinhard Zumkeller, Oct 04 2008
Totally additive with a(p) = 1 if p = 3, 0 otherwise.
v_{m}(n) = Sum_{r>=1} (r/m^(r+1)) Sum_{j=1..m-1} Sum_{k=0..m^(r+1)-1} exp((2*k*Pi*i*(n+(m-j)*m^r)) / m^(r+1)). This formula is for the general case; for this specific one, set m=3. - A. Neves, Oct 04 2010
a(3n) = A051064(n), a(2n) = a(n), a(2n-1) = A253786(n). - Cyril Damamme, Aug 04 2015
a(3n) = a(n) + 1, a(pn) = a(n) for any other prime p != 3. - M. F. Hasler, Aug 11 2015
3^a(n) = A038500(n). - Antti Karttunen, Oct 09 2017
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. - Amiram Eldar, Jul 11 2020
a(n) = tau(n)/(tau(3*n) - tau(n)) - 1, where tau(n) = A000005(n). - Peter Bala, Jan 06 2021
a(n) = 3*Sum_{j=1..floor(log_3(n))} frac(binomial(n,3^j)*3^(j-1)/n). - Dario T. de Castro, Jul 10 2022
a(n) = A080342(gcd(n, 3^A080342(n))). - Alan Michael Gómez Calderón, Jul 28 2024

A088837 Numerator of sigma(2*n)/sigma(n). Denominator see in A038712.

Original entry on oeis.org

3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 63, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 127, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 63, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 255, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 63, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3

Offset: 1

Labos Elemer, Nov 04 2003

In general sigma(2^k*n) / sigma(n) = ((2^k*n) XOR (2^k*n-1)) / (n XOR (n-1)), see link. Jon Maiga, Dec 10 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
Jon Maiga, Efficient computation of ratios between divisor sums, 2018.
Index entries for sequences related to sigma(n).

Cf. A000203, A038712 (denominators), A062731, A088838, A088839, A088840, A080278, A220466.

Cf. A001620, A065442.

nmax:=93: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 2^(p+2)-1 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 09 2013

k=2; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]
Table[BitXor[2*n, 2*n - 1], {n, 128}] (* Jon Maiga, Dec 10 2018 *)

A088837(n) = numerator(sigma(n<<1)/sigma(n)); \\ Antti Karttunen, Nov 01 2018

a(n) = 4*2^A007814(n)-1 = 4*A006519(n)-1 = A059159(n)-1 = 2*A038712(n) + 1.
a((2*n-1)*2^p) = 2^(p+2)-1, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 09 2013
a(n) = (2n) XOR (2n-1). - Jon Maiga, Dec 10 2018
From Amiram Eldar, Jan 06 2023: (Start)
a(n) = numerator(A062731(n)/A000203(n)).
Sum_{k=1..n} a(k) ~ (log_2(n) + (gamma-1)/log(2) + 1)*2*n, where gamma is Euler's constant (A001620).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A038712(k) = A065442 + 1 = 2.606695... . (End).

A088838 Numerator of the quotient sigma(3n)/sigma(n).

Original entry on oeis.org

4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 121, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 121, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 364, 4, 4, 13, 4, 4, 13, 4, 4, 40

Offset: 1

Labos Elemer, Nov 04 2003

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A007949, A038712, A088837, A088839, A088840, A080278 (denominator), A144613.

Cf. A001620, A214369.

A088838 := proc(n)
    numtheory[sigma](3*n)/numtheory[sigma](n) ;
    numer(%) ;
end proc:
seq(A088838(n),n=1..100) ; # R. J. Mathar, Nov 19 2017
seq((3^(2+padic:-ordp(n,3))-1)/2, n=1..100); # Robert Israel, Nov 19 2017

k=3; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]

a(n) = numerator(sigma(3*n)/sigma(n)) \\ Felix Fröhlich, Nov 19 2017

From Robert Israel, Nov 19 2017: (Start)
a(n) = (3^(2+A007949(n))-1)/2.
G.f.: Sum_{k>=0} (3^(k+2)-1)*(x^(3^k)+x^(2*3^k))/(2*(1-x^(3^(k+1)))). (End)
a(n) = sigma(3*n)/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
From Amiram Eldar, Jan 06 2023: (Start)
a(n) = numerator(A144613(n)/A000203(n)).
Sum_{k=1..n} a(k) ~ (3/log(3))*n*log(n) + (1/2 + 3*(gamma-1)/log(3))*n, where gamma is Euler's constant (A001620).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A080278(k) = 4*A214369 + 1 = 3.728614... . (End)

A088839 Numerator of sigma(4n)/sigma(n).

Original entry on oeis.org

7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 127, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 85, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 127, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 511, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 127, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31

Offset: 1

Labos Elemer, Nov 04 2003

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n).

For denominator see A088840.
Cf. A000203, A038712, A088837, A088838, A088841, A080278, A193553.
Cf. A006519, A065442, A096268.

f:= proc(n) local m;
  m:= padic:-ordp(n,2);
  if m::odd then (2^(m+3)-1)/3 else 2^(m+3)-1 fi
end proc:
map(f, [$1..200]); # Robert Israel, Nov 19 2017

k=4; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]

A088839(n) = numerator(sigma(4*n)/sigma(n)); \\ Antti Karttunen, Nov 18 2017

a(n) = (8*A006519(n)-1)/(1+2*A096268(n)). - Robert Israel, Nov 19 2017
From Amiram Eldar, Jan 06 2023: (Start)
a(n) = numerator(A193553(n)/A000203(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A088840(k) = 3*A065442 + 1 = 5.820085... . (End)

Typo in definition corrected by Antti Karttunen, Nov 18 2017

A088840 Denominator of sigma(4n)/sigma(n).

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 21, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 127, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 21, 1, 1, 1, 7, 1, 1

Offset: 1

Labos Elemer, Nov 04 2003

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n).

See A088839 for numerator.

Cf. A088837, A088838, A088841, A038712, A080278, A000203, A001620, A007814, A193553, A213243.

Table[Denominator[DivisorSigma[1, 4*n]/DivisorSigma[1, n]], {n, 1, 128}]
a[n_] := Module[{e = IntegerExponent[n, 2]}, (((-1)^e+2)*(2^(e+1)-1))/3]; Array[a, 100] (* Amiram Eldar, Oct 03 2023 *)

A088840(n) = denominator(sigma(4*n)/sigma(n)); \\ Antti Karttunen, Nov 18 2017

a(n) = {my(e = valuation(n, 2)); (((-1)^e+2) * (2^(e+1)-1))/3;} \\ Amiram Eldar, Oct 03 2023

From Amiram Eldar, Oct 03 2023: (Start)
Multiplicative with a(2^e) = (((-1)^e+2)*(2^(e+1)-1))/3 = A213243(e+1), and a(p^e) = 1 for an odd prime p.
a(n) = A213243(A007814(n+1)).
Dirichlet g.f.: ((8^s + 4^s + 2^(s+1))/(8^s + 4^s - 2^(s+2) - 4)) * zeta(s).
Sum_{k=1..n} a(k) = (2*n/(3*log(2))) * (log(n) + gamma - 1 + 7*log(2)/12), where gamma is Euler's constant (A001620). (End)

Typo in definition corrected by Antti Karttunen, Nov 18 2017

A088842 Denominator of the quotient sigma(7n)/sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 8

Offset: 1

Labos Elemer, Nov 04 2003

Sum of powers of 7 dividing n. - Amiram Eldar, Nov 27 2022

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

Cf. A000203 (sigma), A001620, A088841 (numerators), A283078 (sigma(7n)).

Cf. A080278, A038712, A088837, A088838, A088839, A088840.

Table[Denominator[DivisorSigma[1, 7*n]/DivisorSigma[1, n]], {n, 1, 128}] (* corrected by Ilya Gutkovskiy, Dec 15 2020 *)
a[n_] := (7^(IntegerExponent[n, 7] + 1) - 1)/6; Array[a, 100] (* Amiram Eldar, Nov 27 2022 *)

a(n) = denominator(sigma(7*n)/sigma(n)); \\ Michel Marcus, Dec 15 2020

a(n) = (7^(valuation(n, 7) + 1) - 1)/6; \\ Amiram Eldar, Nov 27 2022

G.f.: Sum_{k>=0} 7^k * x^(7^k) / (1 - x^(7^k)). - Ilya Gutkovskiy, Dec 15 2020
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(7^e) = (7^(e+1)-1)/6, and a(p^e) = 1 for p != 7.
Dirichlet g.f.: zeta(s) / (1 - 7^(1 - s)).
Sum_{k=1..n} a(k) ~ n*log_7(n) + (1/2 + (gamma - 1)/log(7))*n, where gamma is Euler's constant (A001620). (End)

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)

A323921 a(n) = (4^(valuation(n, 4) + 1) - 1) / 3.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 85, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5

Offset: 1

Ilya Gutkovskiy, Dec 15 2020

Sum of powers of 4 dividing n.

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

Cf. A001620, A038712, A080278, A088842, A234957, A235127, A339747, A339748.

Table[(4^(IntegerExponent[n, 4] + 1) - 1)/3, {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[4^k x^(4^k)/(1 - x^(4^k)), {k, 0, Floor[Log[4, nmax]] + 1}], {x, 0, nmax}], x] // Rest

a(n) = (4^(valuation(n, 4) + 1) - 1) / 3; \\ Michel Marcus, Jul 09 2022

def A323921(n): return ((1<<((~n&n-1).bit_length()&-2)+2)-1)//3 # Chai Wah Wu, Jul 09 2022

G.f.: Sum_{k>=0} 4^k * x^(4^k) / (1 - x^(4^k)).
L.g.f.: -log(Product_{k>=0} (1 - x^(4^k))).
Dirichlet g.f.: zeta(s) / (1 - 4^(1 - s)).
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(2^e) = (4^floor((e+2)/2)-1)/3, and a(p^e) = 1 for p != 2.
Sum_{k=1..n} a(k) ~ n*log_4(n) + (1/2 + (gamma - 1)/log(4))*n, where gamma is Euler's constant (A001620). (End)

A339747 a(n) = (5^(valuation(n, 5) + 1) - 1) / 4.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31

Offset: 1

Ilya Gutkovskiy, Dec 15 2020

Sum of powers of 5 dividing n.

Denominator of the quotient sigma(5*n) / sigma(n).

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

Cf. A000203, A001620, A038712, A060904, A080278, A088842, A112765, A323921, A339748.

Table[(5^(IntegerExponent[n, 5] + 1) - 1)/4, {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[5^k x^(5^k)/(1 - x^(5^k)), {k, 0, Floor[Log[5, nmax]] + 1}], {x, 0, nmax}], x] // Rest

a(n) = (5^(valuation(n, 5) + 1) - 1)/4; \\ Amiram Eldar, Nov 27 2022

G.f.: Sum_{k>=0} 5^k * x^(5^k) / (1 - x^(5^k)).
L.g.f.: -log(Product_{k>=0} (1 - x^(5^k))).
Dirichlet g.f.: zeta(s) / (1 - 5^(1 - s)).
a(n) = sigma(n)/(sigma(5*n) - 5*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(5^e) = (5^(e+1)-1)/4, and a(p^e) = 1 for p != 5.
Sum_{k=1..n} a(k) ~ n*log_5(n) + (1/2 + (gamma - 1)/log(5))*n, where gamma is Euler's constant (A001620). (End)

cp's OEIS Frontend

A080333 Partial sums of A080278.

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A007949 Greatest k such that 3^k divides n. Or, 3-adic valuation of n.

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A088837 Numerator of sigma(2*n)/sigma(n). Denominator see in A038712.

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A088838 Numerator of the quotient sigma(3n)/sigma(n).

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A088839 Numerator of sigma(4n)/sigma(n).

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A088840 Denominator of sigma(4n)/sigma(n).

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A088842 Denominator of the quotient sigma(7n)/sigma(n).

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