A000420 -id:A000420 - cp's OEIS frontend

A367249 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 8.

0, 3, 79, 1323, 18175, 223323, 2555119, 27828363, 292407775, 2990349243, 29943991759, 294872615403, 2864776362175, 27525734996763, 262061152909999, 2475899571994443, 23240879960425375, 216963121865909883, 2015960236625789839, 18656492902684557483, 172056837889322101375

Offset: 1

Stefano Spezia, Nov 11 2023

a(n) is the number of n-digit numbers in A366965.

Index entries for linear recurrences with constant coefficients, signature (24,-191,504).

Cf. A366965.
Cf. A000420, A001018, A001019.
Cf. A367243, A367244, A367245, A367246, A367247, A367248, A367250.

LinearRecurrence[{24,-191,504},{0,3,79},21]

a(n) = 17*9^(n-1) - 31*8^(n-1) + 2*7^n.
a(n) = 24*a(n-1) - 191*a(n-2) + 504*a(n-3) for n > 3.
O.g.f.: x^2*(3 + 7*x)/((1 - 7*x)*(1 - 8*x)*(1 - 9*x)).
E.g.f.: (136*exp(9*x) - 279*exp(8*x) + 144*exp(7*x) - 1)/72.

A036293 a(n) = n*7^n.

Original entry on oeis.org

0, 7, 98, 1029, 9604, 84035, 705894, 5764801, 46118408, 363182463, 2824752490, 21750594173, 166095446412, 1259557135291, 9495123019886, 71213422649145, 531726889113616, 3954718737782519, 29311444762388082, 216579008522089717, 1595845325952240020, 11729463145748964147

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A000420.

[n*7^n: n in [0..20]] ; // Vincenzo Librandi, Aug 09 2011

Table[n 7^n,{n,0,30}] (* or *) LinearRecurrence[{14,-49},{0,7},31] (* Harvey P. Dale, Aug 08 2011 *)

From Harvey P. Dale, Aug 08 2011: (Start)
a(n) = 14*a(n-1) - 49*a(n-2) with a(0)=0 and a(1)=7.
G.f.: (7*x)/(7*x-1)^2. (End)
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(7/6).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(8/7). (End)
From Elmo R. Oliveira, Sep 09 2024: (Start)
E.g.f.: 7*x*exp(7*x).
a(n) = n*A000420(n). (End)

A083528 a(n) = 5^n mod 2*n.

Original entry on oeis.org

1, 1, 5, 1, 5, 1, 5, 1, 17, 5, 5, 1, 5, 25, 5, 1, 5, 1, 5, 25, 41, 25, 5, 1, 25, 25, 53, 9, 5, 25, 5, 1, 59, 25, 45, 1, 5, 25, 47, 65, 5, 1, 5, 9, 35, 25, 5, 1, 19, 25, 23, 1, 5, 1, 45, 81, 11, 25, 5, 25, 5, 25, 125, 1, 5, 49, 5, 81, 125, 65, 5, 1, 5, 25, 125, 17, 3, 25, 5, 65, 161, 25, 5, 1, 65

Offset: 1

Labos Elemer, Apr 30 2003

a(n) = 1 iff n is in A067946. - Robert Israel, Dec 26 2014

			a(3) = 5 because 5^3 = 125 and 125 == 5 mod (2 * 3).
a(4) = 1 because 5^4 = 625 and 625 == 1 mod (2 * 4).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000079, A000244, A000351, A000420, A082511.

Cf. A067946, A083529, A083530.

[Modexp(5, n, 2*n): n in [1..80]]; // Vincenzo Librandi, Oct 19 2018

seq(5 &^n mod (2*n), n = 1 .. 100); # Robert Israel, Dec 26 2014

Mathematica
```
Table[PowerMod[5, w, 2w], {w, 1, 100}]
```

vector(100, n, lift(Mod(5, 2*n)^n)) \\ Michel Marcus, Dec 29 2014

A083529 a(n) = 5^n mod 3*n.

Original entry on oeis.org

2, 1, 8, 1, 5, 1, 5, 1, 26, 25, 5, 1, 5, 25, 35, 1, 5, 1, 5, 25, 62, 25, 5, 1, 50, 25, 80, 37, 5, 55, 5, 1, 26, 25, 80, 1, 5, 25, 8, 25, 5, 1, 5, 97, 80, 25, 5, 1, 68, 25, 125, 1, 5, 1, 155, 25, 125, 25, 5, 145, 5, 25, 188, 1, 5, 181, 5, 13, 125, 205, 5, 1, 5, 25, 125, 169, 80, 181, 5

Offset: 1

Labos Elemer, Apr 30 2003

From Robert Israel, Dec 25 2014: (Start)
a(n) == (-1)^n mod 3.
a(n) = 1 if and only if n is even and in A067946.
For n > 3, a(n) = 5 if and only if n is odd and in A123091. (End)

			a(3) = 8 because 5^3 = 125 and 125 mod (3 * 3) = 8.
a(4) = 1 because 5^4 = 625 and 625 mod (3 * 4) = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000079, A000244, A000351, A000420, A082511, A083528, A083530, A067946, A123091.

[Modexp(5, n, 3*n): n in [1..80]]; // Vincenzo Librandi, Oct 19 2018

seq(5 &^n mod 3*n, n = 1 .. 1000); # Robert Israel, Dec 25 2014

Table[Mod[5^w, 3 * w], {w, 100}]
Table[PowerMod[5, n, 3n], {n, 80}] (* Harvey P. Dale, Jul 26 2014 *)

a(n)=lift(Mod(5,3*n)^n) \\ Charles R Greathouse IV, Oct 03 2016

A083530 a(n) = 7^n mod (2*n).

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 7, 1, 1, 9, 7, 1, 7, 21, 13, 1, 7, 1, 7, 1, 7, 5, 7, 1, 7, 49, 1, 49, 7, 49, 7, 1, 13, 49, 63, 1, 7, 49, 31, 1, 7, 49, 7, 25, 37, 49, 7, 1, 49, 49, 37, 9, 7, 1, 43, 49, 1, 49, 7, 1, 7, 49, 91, 1, 37, 37, 7, 89, 67, 49, 7, 1, 7, 49, 43, 121, 105, 25, 7, 1, 1, 49, 7, 49, 147, 49

Offset: 1

Labos Elemer, Apr 30 2003

			For n = 5, a(5) = 7 because 7^5 = 16807 = 1680*10 + 7, that is 7^5 == 7 (mod 2*5).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000079, A000244, A000351, A000420, A082511, A083528, A083529.

Mathematica
```
Table[Mod[7^w, 2*w], {w, 1, 100}]
```

a(n)=lift(Mod(7,2*n)^n) \\ Charles R Greathouse IV, Oct 03 2016

A151785 a(n) = 7^(wt(n) - 1) where wt(n) is the binary weight of n (A000120).

Original entry on oeis.org

1, 1, 7, 1, 7, 7, 49, 1, 7, 7, 49, 7, 49, 49, 343, 1, 7, 7, 49, 7, 49, 49, 343, 7, 49, 49, 343, 49, 343, 343, 2401, 1, 7, 7, 49, 7, 49, 49, 343, 7, 49, 49, 343, 49, 343, 343, 2401, 7, 49, 49, 343, 49, 343, 343, 2401, 49, 343, 343, 2401, 343, 2401, 2401, 16807, 1, 7, 7, 49, 7, 49

Offset: 1

N. J. A. Sloane, Jun 25 2009

nonn

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

Cf. A000120, A000420.

7^(DigitCount[Range[70],2,1]-1) (* Harvey P. Dale, Feb 15 2015 *)

A151791 a(1)=1; for n > 1, a(n) = 7*6^(wt(n-1)-1).

Original entry on oeis.org

1, 7, 7, 42, 7, 42, 42, 252, 7, 42, 42, 252, 42, 252, 252, 1512, 7, 42, 42, 252, 42, 252, 252, 1512, 42, 252, 252, 1512, 252, 1512, 1512, 9072, 7, 42, 42, 252, 42, 252, 252, 1512, 42, 252, 252, 1512, 252, 1512, 1512, 9072, 42, 252, 252, 1512, 252, 1512, 1512, 9072, 252

Offset: 1

N. J. A. Sloane, Jun 25 2009

wt(n) is the Hamming weight = binary weight of n (A000120).

			From _Omar E. Pol_, Feb 26 2015: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
7;
7,42;
7,42,42,252;
7,42,42,252,42,252,252,1512;
7,42,42,252,42,252,252,1512,42,252,252,1512,252,1512,1512,9072;
7,42,42,252,42,252,252,1512,42,252,252,1512,252,1512,1512,9072,42,252,252,1512,252,1512,1512,9072,252,...
It appears that the right border gives A003949.
It appears that the row sums give A000420.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..8192

Cf. A011782, A000120, A000420, A151792 (partial sums).

a[n_] := 7*6^(Total@ IntegerDigits[n - 1, 2] - 1); a[1] = 1; Array[a, 57] (* Michael De Vlieger, Nov 01 2022 *)

A164783 a(n) = 7^n-6.

Original entry on oeis.org

1, 43, 337, 2395, 16801, 117643, 823537, 5764795, 40353601, 282475243, 1977326737, 13841287195, 96889010401, 678223072843, 4747561509937, 33232930569595, 232630513987201, 1628413597910443, 11398895185373137

Offset: 1

Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009

Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below: - Sequence m (n,t) = (n^t) - (n-1) for t=2 to infinity is called a Mersenne Sequence Rooted on n - If n is prime, this sequence is called a Legitimate Mersenne Sequence - Any j belonging to the sequence m (n,t) is called a Generalized Mersenne Number (n-GMN) - If j belonging to the sequence m (n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP). Note: m (n,t) = n* m (n,t-1) + n^2 - 2*n+1. This sequence related to sequences: A014232 and A014224; A135535 and A059266. These numbers play a role in the context of hyperperfect numbers.

Daniel Minoli, Sufficient Forms For Generalized Perfect Numbers, Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem. Vol. 4, No. 2, Dec 1978, pp. 277-302.
Daniel Minoli, New Results For Hyperperfect Numbers, Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, pp. 561.
Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.
D. Minoli, Structural Issues For Hyperperfect Numbers, Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.
D. Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A000420.

Cf. A014232, A014224, A135535, A059266.

[7^n-6: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

CoefficientList[Series[(1 + 35 x)/((1-x) (1-7 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)
NestList[7 # + 36 & , 1, 18] (* Bruno Berselli, Feb 06 2013 *)
LinearRecurrence[{8,-7},{1,43},30] (* Harvey P. Dale, Nov 27 2014 *)

a(n)=7^n-6 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 7*a(n-1)+36 with n>1, a(1)=1. - Vincenzo Librandi, Nov 30 2010
G.f.: x*(1+35*x)/((1-x)*(1-7*x)). - Colin Barker, Mar 08 2012
a(n) = 8*a(n-1) - 7*a(n-2) for n>2, a(1)=1, a(2)=43. - Vincenzo Librandi, Feb 06 2013
a(n) = A000420(n) - 6 for n>0. - Michel Marcus, Aug 31 2013

More terms a(8)-a(19) from Vincenzo Librandi, Oct 29 2009

A165425 a(1) = 1, a(2) = 7, a(n) = product of the previous terms for n >= 3.

Original entry on oeis.org

1, 7, 7, 49, 2401, 5764801, 33232930569601, 1104427674243920646305299201, 1219760487635835700138573862562971820755615294131238401

Offset: 1

Jaroslav Krizek, Sep 17 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..13

a[1]:= 1; a[2]:= 7; a[n_]:= Product[a[j], {j,1,n-1}]; Table[a[n],{n,1, 12}] (* G. C. Greubel, Oct 19 2018 *)

{a(n) = if(n==1, 1, if(n==2, 7, prod(j=1,n-1, a(j))))};
for(n=1,10, print1(a(n), ", ")) \\ G. C. Greubel, Oct 19 2018

a(1) = 1, a(2) = 7, a(n) = Product_{i=1..n-1} a(i), n >= 3.
a(1) = 1, a(2) = 7, a(n) = A000420(2^(n-3)) = 7^(2^(n-3)), n >= 3.
a(1) = 1, a(2) = 7, a(3) = 7, a(n) = (a(n-1))^2, n >= 4.

More terms from Vincenzo Librandi, Apr 21 2010

A175434 (Digit sum of 2^n) mod n.

Original entry on oeis.org

0, 0, 2, 3, 0, 4, 4, 5, 8, 7, 3, 7, 7, 8, 11, 9, 14, 1, 10, 11, 5, 3, 18, 13, 4, 14, 8, 15, 12, 7, 16, 26, 29, 27, 24, 28, 19, 29, 32, 21, 9, 4, 13, 14, 17, 24, 21, 25, 16, 26, 29, 27, 24, 28, 37, 29, 23, 12, 18, 22, 13, 23, 26, 24, 21, 43, 43, 35, 20, 0, 15, 37, 37, 56, 50, 30, 27, 22, 31, 32, 26, 42, 39, 34, 43, 26, 20, 27, 24, 28, 55, 47, 32, 57, 45, 31, 40, 14, 8, 15

Offset: 1

N. J. A. Sloane, Dec 03 2010

			For n = 1,2,3,4,5,6, the digit-sum of 2^n is 2,4,8,7,5,10, so
a(1) through a(6) are 0,0,2,3,0,4. - _N. J. A. Sloane_, Aug 12 2014

Sum of digits of k^n mod n: (k=2) A000079, A001370, A175434, A175169; (k=3) A000244, A004166, A175435, A067862; (k=5) A000351, A066001, A175456; (k=6) A000400, A066002, A175457, A067864; (k=7) A000420, A066003, A175512, A067863; (k=8) A062933; (k=13) A001022, A175527, A175528, A175525; (k=21) A175589; (k=167) A175558, A175559, A175560, A175552.

Table[Mod[Total[IntegerDigits[2^n]],n],{n,100}] (* Harvey P. Dale, Aug 12 2014 *)

Offset changed to 1 at the suggestion of Harvey P. Dale, Aug 12 2014

cp's OEIS Frontend

A367249 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 8.

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A036293 a(n) = n*7^n.

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Magma

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A083528 a(n) = 5^n mod 2*n.

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Magma

Maple

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A083529 a(n) = 5^n mod 3*n.

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Maple

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A083530 a(n) = 7^n mod (2*n).

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A151785 a(n) = 7^(wt(n) - 1) where wt(n) is the binary weight of n (A000120).

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A151791 a(1)=1; for n > 1, a(n) = 7*6^(wt(n-1)-1).

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A164783 a(n) = 7^n-6.

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