A024049 -id:A024049 - cp's OEIS frontend

A064081 Zsigmondy numbers for a = 5, b = 1: Zs(n, 5, 1) is the greatest divisor of 5^n - 1^n (A024049) that is relatively prime to 5^m - 1^m for all positive integers m < n.

4, 3, 31, 13, 781, 7, 19531, 313, 15751, 521, 12207031, 601, 305175781, 13021, 315121, 195313, 190734863281, 5167, 4768371582031, 375601, 196890121, 8138021, 2980232238769531, 390001, 95397958987501, 203450521, 3814699218751, 234750601, 46566128730773925781, 464881, 1164153218269348144531

Offset: 1

Jens Voß, Sep 04 2001

nonn

By Zsigmondy's theorem, the n-th Zsigmondy number for bases a and b is not 1 except in the three cases (1) a = 2, b = 1, n = 1, (2) a = 2, b = 1, n = 6, (3) n = 2 and a+b is a power of 2.

Robert Israel, Table of n, a(n) for n = 1..133
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. f. Math. 3 (1892) 265-284.

Cf. A024049, A064078, A064079, A064080, A064082, A064083.

More terms from Vladeta Jovovic, Sep 06 2001
Definition corrected by Jerry Metzger, Nov 04 2009
More terms from Robert Israel, Feb 21 2025

A034474 a(n) = 5^n + 1.

Original entry on oeis.org

2, 6, 26, 126, 626, 3126, 15626, 78126, 390626, 1953126, 9765626, 48828126, 244140626, 1220703126, 6103515626, 30517578126, 152587890626, 762939453126, 3814697265626, 19073486328126, 95367431640626, 476837158203126

Offset: 0

N. J. A. Sloane

a(n) is the deficiency of 3*5^n (see A033879). - Patrick J. McNab, May 28 2017

			G.f. = 2 + 6*x + 26*x^2 + 126*x^3 + 626*x^4 + 3126*x^5 + 15626*x^6 + ...

T. D. Noe, Table of n, a(n) for n = 0..200
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000051, A000351, A007689, A024049, A033879, A034472, A034478, A034491, A034524, A052539, A062394, A062395, A062396, A062397, A063376, A063481, A074600-A074624, A178248, A228081, A279396.

[5^n+1: n in [0..30]]; // Vincenzo Librandi, Jan 11 2017

Table[5^n + 1, {n, 0, 25}]
LinearRecurrence[{6,-5},{2,6},30] (* Harvey P. Dale, Jul 29 2015 *)

a(n)=5^n+1 \\ Charles R Greathouse IV, Sep 24 2015

[lucas_number2(n,6,5) for n in range(25)] # Zerinvary Lajos, Jul 08 2008

[sigma(5,n) for n in range(25)] # Zerinvary Lajos, Jun 04 2009

[5^n+1 for n in range(30)] # Bruno Berselli, Jan 11 2017

a(n) = 5*a(n-1) - 4 with a(0) = 2.
a(n) = 6*a(n-1) - 5*a(n-2) for n > 1.
From Mohammad K. Azarian, Jan 02 2009: (Start)
G.f.: 1/(1-x) + 1/(1-5*x) = (2-6*x)/((1-x)*(1-5*x)).
E.g.f.: exp(x) + exp(5*x). (End)
a(n) = A279396(n+5,5). - Wolfdieter Lang, Jan 10 2017
From Elmo R. Oliveira, Dec 06 2023: (Start)
a(n) = A000351(n) + 1.
a(n) = 2*A034478(n). (End)

A002278 a(n) = 4*(10^n - 1)/9.

Original entry on oeis.org

0, 4, 44, 444, 4444, 44444, 444444, 4444444, 44444444, 444444444, 4444444444, 44444444444, 444444444444, 4444444444444, 44444444444444, 444444444444444, 4444444444444444, 44444444444444444, 444444444444444444, 4444444444444444444, 44444444444444444444, 444444444444444444444

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A002276, A002277, A002279, A002280, A002281, A002282, A002283, A075415, A178632.

Cf. A007091, A010785, A017209, A024049.

LinearRecurrence[{11, -10}, {0, 4}, 20] (* Robert G. Wilson v, Jul 06 2013 *)

a(n)=4*(10^n-1)/9 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A075415(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
From Vincenzo Librandi, Jul 22 2010: (Start)
a(n) = a(n-1) + 4*10^(n-1) with a(0)=0;
a(n) = 11*a(n-1) - 10*a(n-2) with a(0)=0, a(1)=4. (End)
G.f.: 4*x/((1 - x)*(1 - 10*x)). - Ilya Gutkovskiy, Feb 24 2017
E.g.f.: 4*exp(x)*(exp(9*x) - 1)/9. - Stefano Spezia, Sep 13 2023
a(n) = A007091(A024049(n)). - Michel Marcus, Jun 16 2024
From Elmo R. Oliveira, Jul 19 2025: (Start)
a(n) = 4*A002275(n).
a(n) = A010785(A017209(n-1)) for n >= 1. (End)

A057651 a(n) = (3*5^n - 1)/2.

Original entry on oeis.org

1, 7, 37, 187, 937, 4687, 23437, 117187, 585937, 2929687, 14648437, 73242187, 366210937, 1831054687, 9155273437, 45776367187, 228881835937, 1144409179687, 5722045898437, 28610229492187, 143051147460937, 715255737304687, 3576278686523437, 17881393432617187, 89406967163085937

Offset: 0

N. J. A. Sloane, Oct 13 2000

Sum of n-th row of triangle of powers of 5: 1; 1 5 1; 1 5 25 5 1 ; 1 5 25 125 25 5 1; ... - Philippe Deléham, Feb 23 2014

			a(0) = 1;
a(1) = 1 + 5 + 1 = 7;
a(2) = 1 + 5 + 25 + 5 + 1 = 37;
a(3) = 1 + 5 + 25 + 125 + 25 + 5 + 1 = 187; etc. - _Philippe Deléham_, Feb 23 2014
G.f. = 1 + 7*x + 37*x^2 + 187*x^3 + 937*x^4 + 4687*x^5 + 23437*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A024049, A081655, A097162, A198762.

Cf. A020989, A061801, A112468, A112739.

[(3*5^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Oct 30 2011

G.f=(1+x)/(1-5*x)/(1-x): gser:=series(g, x=0, 43): seq(coeff(gser, x, n), n=0..30); # Zerinvary Lajos, Jan 11 2009

Table[(3*5^n-1)/2,{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)

a(n)=3*5^n\2 \\ Charles R Greathouse IV, Dec 22 2011

G.f.: (1+x)/(1 - 6*x + 5*x^2).
a(0)=1, a(n) = 5*a(n-1) + 2; a(n) = a(n-1) + 6*(5^(n-1)). - Amarnath Murthy, May 27 2001
a(n) = 6*a(n-1) - 5*a(n-2), n > 1. - Vincenzo Librandi, Oct 30 2011
a(n) = Sum_{k=0..n} A112468(n,k)*6^k. - Philippe Deléham, Feb 23 2014
From Elmo R. Oliveira, Mar 29 2025: (Start)
E.g.f.: exp(x)*(3*exp(4*x) - 1)/2.
a(n) = A097162(2*n) = A198762(n)/2. (End)

A059379 Array of values of Jordan function J_k(n) read by antidiagonals (version 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 2, 8, 7, 1, 4, 12, 26, 15, 1, 2, 24, 56, 80, 31, 1, 6, 24, 124, 240, 242, 63, 1, 4, 48, 182, 624, 992, 728, 127, 1, 6, 48, 342, 1200, 3124, 4032, 2186, 255, 1, 4, 72, 448, 2400, 7502, 15624, 16256, 6560, 511, 1, 10, 72, 702, 3840

Offset: 1

N. J. A. Sloane, Jan 28 2001

			Array begins:
  1,  1,  2,   2,   4,    2,    6,    4,   6,  4, 10, 4, ...
  1,  3,  8,  12,  24,   24,   48,   48,  72, 72, ...
  1,  7, 26,  56, 124,  182,  342,  448, 702, ...
  1, 15, 80, 240, 624, 1200, 2400, 3840, ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Columns give A000225, A024023, A020522, A024049, A059387, etc.

Main diagonal gives A067858.

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end;
#alternative
A059379 := proc(n,k)
    add(d^k*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
end proc:
seq(seq(A059379(d-k,k),k=1..d-1),d=2..12) ; # R. J. Mathar, Nov 23 2018

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
A059379[n_]:=JordanTotient[A004736[n],A002260[n]]; (* Enrique Pérez Herrero, Dec 19 2010 *)

jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);
A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;
A059379(n)=jordantot(A004736(n),A002260(n)); \\ Enrique Pérez Herrero, Jan 08 2011

from functools import cache
def MoebiusTrans(a, i):
    @cache
    def mb(n, d = 1):
          return d % n and -mb(d, n % d < 1) + mb(n, d + 1) or 1 // n
    def mob(m, n): return mb(m // n) if m % n == 0 else 0
    return sum(mob(i, d) * a(d) for d in range(1, i + 1))
def Jrow(n, size):
    return [MoebiusTrans(lambda m: m ** n, k) for k in range(1, size)]
for n in range(1, 8): print(Jrow(n, 13))
# Alternatively:
from sympy import primefactors as prime_divisors
from fractions import Fraction as QQ
from math import prod as product
def J(n: int, k: int) -> int:
    t = QQ(pow(k, n), 1)
    s = product(1 - QQ(1, pow(p, n)) for p in prime_divisors(k))
    return (t * s).numerator  # the denominator is always 1
for n in range(1, 8): print([J(n, k) for k in range(1, 13)])
# Peter Luschny, Dec 16 2023

J_k(n) = Sum_{d|n} d^k*mu(n/d). - Benoit Cloitre and Michael Orrison (orrison(AT)math.hmc.edu), Jun 07 2002
From Amiram Eldar, Jun 07 2025: (Start)
For a given k, J_k(n) is multiplicative with J_k(p^e) = p^(k*e) - p^(k*e-k).
For a given k, Dirichlet g.f. of J_k(n): zeta(s-k)/zeta(s).
Sum_{i=1..n} J_k(i) ~ n^(k+1) / ((k+1)*zeta(k+1)).
Sum_{n>=1} 1/J_k(n) = Product_{p prime} (1 + p^k/(p^k-1)^2) for k >= 2. (End)

A027872 a(n) = Product_{i=1..n} (5^i - 1).

Original entry on oeis.org

1, 4, 96, 11904, 7428096, 23205371904, 362560730628096, 28324694519589371904, 11064305472020078810628096, 21609960560733744406929189371904, 211034749490954911990173458030810628096

Offset: 0

N. J. A. Sloane

nonn

Given probability p = 1/5^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1 - a(n)/A109345(n+1) is the probability that the outcome has occurred at or before the n-th iteration. The limiting ratio is 1-A100222 ~ 0.2396672. - Bob Selcoe, Mar 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A109345, A100222.

[1] cat [&*[ 5^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

A027872 := proc(n)
        mul( 5^i-1, i=1..n) ;
end proc: # R. J. Mathar, Mar 12 2013

Table[Product[(5^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[5, 5, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)
Join[{1},FoldList[Times,5^Range[10]-1]] (* Harvey P. Dale, Dec 28 2021 *)

a(n) = prod(i=1, n, 5^i-1); \\ Michel Marcus, Nov 21 2015

4^n|a(n) for n >= 1. - G. C. Greubel, Nov 21 2015
a(n) ~ c * 5^(n*(n+1)/2), where c = Product_{k>=1} (1-1/5^k) = A100222 . - Vaclav Kotesovec, Nov 21 2015
a(n) = 5^(binomial(n+1,2))*(1/5; 1/5){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015
a(n) = Product_{i=1..n} A024049(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 5^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 5^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A100222. - Amiram Eldar, May 07 2023

A059380 Array of values of Jordan function J_k(n) read by antidiagonals (version 2).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 7, 8, 2, 1, 15, 26, 12, 4, 1, 31, 80, 56, 24, 2, 1, 63, 242, 240, 124, 24, 6, 1, 127, 728, 992, 624, 182, 48, 4, 1, 255, 2186, 4032, 3124, 1200, 342, 48, 6, 1, 511, 6560, 16256, 15624, 7502, 2400, 448, 72, 4, 1, 1023, 19682

Offset: 1

N. J. A. Sloane, Jan 28 2001

			Array begins:
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, ...
1, 3, 8, 12, 24, 24, 48, 48, 72, 72, ...
1, 7, 26, 56, 124, 182, 342, 448, 702, ...
1, 15, 80, 240, 624, 1200, 2400, 3840, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, The many facets of Euler's totient. II. Generalizations and analogues, Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Columns give A000225, A024023, A020522, A024049, A059387, etc.

Main diagonal gives A067858.

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end;

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
A059380[n_]:=JordanTotient[A002260[n],A004736[n]]; (* Enrique Pérez Herrero, Dec 19 2010 *)

jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);
A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;
A059380(n)=jordantot(A002260(n),A004736(n)); \\ Enrique Pérez Herrero, Jan 08 2011

A067946 Numbers k that divide 5^k - 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 42, 48, 52, 54, 64, 72, 84, 96, 104, 108, 126, 128, 144, 156, 162, 168, 186, 192, 208, 216, 252, 256, 272, 288, 294, 312, 324, 336, 342, 372, 378, 384, 416, 432, 468, 486, 504, 512, 544, 558, 576, 588, 624, 648, 672, 676

Offset: 1

Benoit Cloitre, Mar 05 2002

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A024049.

Join[{1},Select[Range[700],PowerMod[5,#,#]==1&]] (* Harvey P. Dale, Feb 25 2015 *)

is(n)=Mod(5,n)^n==1 \\ Charles R Greathouse IV, Nov 04 2016

A057956 Number of prime factors of 5^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 4, 3, 6, 4, 7, 3, 8, 5, 7, 3, 10, 3, 7, 7, 11, 4, 11, 5, 11, 6, 8, 4, 13, 8, 7, 9, 10, 5, 14, 4, 14, 6, 8, 9, 16, 5, 10, 6, 15, 4, 16, 4, 12, 12, 8, 3, 17, 4, 13, 8, 12, 5, 19, 10, 13, 7, 9, 4, 21, 5, 9, 11, 18, 8, 15, 7, 14, 9, 16, 4, 22, 5, 10, 16, 14, 7, 14, 5, 20, 11, 10, 5, 22, 9, 10

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..502 (first 448 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), this sequence (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A024049, A074479, A085030.

PrimeOmega[5^Range[90]-1] (* Harvey P. Dale, Dec 16 2017 *)

Mobius transform of A085030. - T. D. Noe, Jun 19 2003

a(n) = A001222(A024049(n)). - Amiram Eldar, Feb 01 2020

A295502 a(n) = phi(5^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

2, 8, 60, 192, 1400, 4320, 39060, 119808, 894240, 2912000, 24414060, 62208000, 610351560, 1959874560, 13154400000, 44043337728, 380537036928, 997843069440, 9485297382000, 25606963200000, 230106651919200, 748687423334400, 5959800062798400, 15138938880000000

Offset: 1

Seiichi Manyama, Nov 22 2017

nonn

Faye et al. prove that no term is of the form 5^k-1. - Michel Marcus, Jun 16 2024

Max Alekseyev, Table of n, a(n) for n = 1..502
Bernadette Faye, Florian Luca, and Amadou Tall, On the equation phi(5^m-1)=5^n-1, Bull. Korean Math. Soc. 2015; 52(2): 513-524.
Eric Weisstein's World of Mathematics, Totient Function
Wikipedia, Euler's totient function

Cf. A000010, A024049.

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), this sequence (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

EulerPhi[5^Range[25] - 1] (* Paolo Xausa, Jun 18 2024 *)

PARI
```
{a(n) = eulerphi(5^n-1)}
```

a(n) = n*A027741(n).

a(n) = A000010(A024049(n)). - Michel Marcus, Jun 16 2024

cp's OEIS Frontend

A064081 Zsigmondy numbers for a = 5, b = 1: Zs(n, 5, 1) is the greatest divisor of 5^n - 1^n (A024049) that is relatively prime to 5^m - 1^m for all positive integers m < n.

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A034474 a(n) = 5^n + 1.

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A002278 a(n) = 4*(10^n - 1)/9.

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A057651 a(n) = (3*5^n - 1)/2.

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A059379 Array of values of Jordan function J_k(n) read by antidiagonals (version 1).

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A027872 a(n) = Product_{i=1..n} (5^i - 1).

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A059380 Array of values of Jordan function J_k(n) read by antidiagonals (version 2).

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