A000400 -id:A000400 - cp's OEIS frontend

A165827 Totally multiplicative sequence with a(p) = 6.

1, 6, 6, 36, 6, 36, 6, 216, 36, 36, 6, 216, 6, 36, 36, 1296, 6, 216, 6, 216, 36, 36, 6, 1296, 36, 36, 216, 216, 6, 216, 6, 7776, 36, 36, 36, 1296, 6, 36, 36, 1296, 6, 216, 6, 216, 216, 36, 6, 7776, 36, 216

Offset: 1

Jaroslav Krizek, Sep 28 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000400, A001222.

6^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 09 2016 *)

a(n) = A000400(A001222(n)) = 6^bigomega(n) = 6^A001222(n).

Dirichlet g.f.: Product_{p prime} 1 / (1 - 6 * p^(-s)). - Ilya Gutkovskiy, Oct 30 2019

A175169 Numbers k that divide the sum of digits of 2^k.

Original entry on oeis.org

1, 2, 5, 70

Offset: 1

N. J. A. Sloane, Dec 03 2010

No other terms <= 200000. - Harvey P. Dale, Dec 16 2010
No other terms <= 1320000. - Robert G. Wilson v, Dec 18 2010
There are almost certainly no further terms.

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.

Select[Range[1000],Divisible[Total[IntegerDigits[2^#]],#]&] (* Harvey P. Dale, Dec 16 2010 *)

is(n)=sumdigits(2^n)%n==0 \\ Charles R Greathouse IV, Sep 06 2016

A175525 Numbers k that divide the sum of digits of 13^k.

Original entry on oeis.org

1, 2, 5, 140, 158, 428, 788, 887, 914, 1814, 1895, 1976, 2579, 2732, 3074, 3299, 3641, 4658, 4874, 5378, 5423, 5504, 6170, 6440, 6944, 8060, 8249, 8915, 9041, 9158, 9725, 9824, 10661, 11291, 13820, 15305, 17051, 17393, 18716, 19589, 20876, 21641, 23756, 24188, 25961, 28409, 30632, 31307, 32387, 33215, 34970, 35240, 36653, 36977, 41558, 43970, 44951, 47444, 51764, 52655, 53375, 53852, 54104, 56831, 57506, 59153, 66479, 68063, 73562, 78485, 79286, 87908, 92093, 102029, 106934, 114854, 116321, 134051, 139397, 184037, 192353, 256469, 281381, 301118, 469004

Offset: 1

T. D. Noe, Dec 03 2010

Almost certainly there are no further terms.
Comments from Donovan Johnson on the computation of this sequence, Dec 05 2010 (Start):
The number of digits of 13^k is approximately 1.114*k, so I defined an array d() that is a little bigger than 1.114 times the maximum k value to be checked. The elements of d() each are the value of a single digit of the decimal expansion of 13^k with d(1) being the least significant digit.
It's easier to see how the program works if I start with k = 2.
For k = 1, d(2) would have been set to 1 and d(1) would have been set to 3.
k = 2:
x = 13*d(1) = 13*3 = 39
y = 39\10 = 3 (integer division)
x-y*10 = 39-30 = 9, d(1) is set to 9
x = 13*d(2)+y = 13*1+3 = 16, y is the carry from previous digit
y = 16\10 = 1
x-y*10 = 16-10 = 6, d(2) is set to 6
x = 13*d(3)+y = 13*0+1 = 1, y is the carry from previous digit
y = 1\10 = 0
x-y*10 = 1-0 = 1, d(3) is set to 1
These steps would of course be inside a loop and that loop would be inside a k loop. A pointer to the most significant digit increases usually by one and sometimes by two for each successive k value checked. The number of steps of the inner loop is the size of the pointer. A scan is done from the first element to the pointer element to get the digit sum.
(End)
No other terms < 3*10^6. - Donovan Johnson, Dec 07 2010

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.

Select[Range[1000], Mod[Total[IntegerDigits[13^#]], #] == 0 &]

a(47)-a(79) from N. J. A. Sloane, Dec 04 2010

a(80)-a(85) from Donovan Johnson, Dec 05 2010

A013600 a(n) = nextprime(6^n)-6^n.

Original entry on oeis.org

1, 1, 1, 7, 1, 13, 7, 5, 11, 25, 5, 35, 35, 35, 5, 91, 47, 35, 5, 17, 11, 7, 103, 61, 7, 13, 23, 7, 25, 47, 7, 73, 5, 41, 133, 77, 101, 103, 193, 61, 47, 187, 71, 35, 215, 83, 121, 95, 37, 95, 145, 35, 77, 13, 7, 5, 25, 77, 47, 283, 235, 23, 137, 137, 47

Offset: 0

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 0..1000

Cf. A000400, A013632, A063766.

Maple
```
seq(nextprime(6^i)-6^i,i=0..100);
```

(NextPrime[#]-#)&/@(6^Range[0,70]) (* Harvey P. Dale, May 04 2024 *)

a(n) = nextprime(6^n)-6^n; \\ Michel Marcus, Jan 08 2020

a(n) = A063766(n)-6^n = A013632(6^n). - R. J. Mathar, Nov 28 2016

A025622 Numbers of form 5^i*6^j, with i, j >= 0.

Original entry on oeis.org

1, 5, 6, 25, 30, 36, 125, 150, 180, 216, 625, 750, 900, 1080, 1296, 3125, 3750, 4500, 5400, 6480, 7776, 15625, 18750, 22500, 27000, 32400, 38880, 46656, 78125, 93750, 112500, 135000, 162000, 194400, 233280, 279936, 390625, 468750, 562500, 675000

Offset: 1

David W. Wilson

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

Cf. A000351 (subset), A000400 (another subset), A025651 (exponent of 5).

n = 10^6; Flatten[Table[5^i*6^j, {i, 0, Log[5, n]}, {j, 0, Log[6, n/5^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

Sum_{n>=1} 1/a(n) = (5*6)/((5-1)*(6-1)) = 3/2. - Amiram Eldar, Sep 25 2020
a(n) ~ exp(sqrt(2*log(5)*log(6)*n)) / sqrt(30). - Vaclav Kotesovec, Sep 25 2020
a(n) = 5^A025651(n) *6^A025659(n). - R. J. Mathar, Jul 06 2025

A049304 Numbers k such that k is a substring of 6^k.

Original entry on oeis.org

6, 7, 9, 13, 21, 22, 23, 29, 39, 40, 42, 44, 45, 48, 53, 55, 56, 60, 63, 64, 65, 67, 68, 69, 70, 73, 74, 75, 76, 77, 79, 82, 83, 87, 89, 92, 93, 94, 98, 105, 107, 127, 129, 131, 134, 137, 143, 147, 152, 163, 165, 167, 174, 179, 184, 189, 197, 224, 226, 227, 234, 240

Offset: 1

David W. Wilson

			9 is in the sequence because 6^9 = 10077696 contains 9 as a substring. - _David A. Corneth_, Aug 13 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000400, A032740, A049301, A049302, A049303, A049305, A049306, A049307.

Select[Range[250],SequenceCount[IntegerDigits[6^#],IntegerDigits[#]]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 03 2018 *)

is(n) = { my(digs6n, digsn, streak, i, j); digs6n = digits(6^n); digsn = digits(n); for(i = 1, #digs6n + 1 - #digsn, streak = 0; for(j = 1, #digsn, if(digs6n[i + j - 1] == digsn[j], streak++ , next(2) ) ); if(streak == #digsn, return(1) ) ); 0 } \\ David A. Corneth, Aug 13 2021

def ok(n): return str(n) in str(6**n)
print(list(filter(ok, range(241)))) # Michael S. Branicky, Aug 13 2021

A124508 a(n) = 2^BigO(n) * 3^omega(n), where BigO = A001222 and omega = A001221, the numbers of prime factors of n with and without repetitions.

Original entry on oeis.org

1, 6, 6, 12, 6, 36, 6, 24, 12, 36, 6, 72, 6, 36, 36, 48, 6, 72, 6, 72, 36, 36, 6, 144, 12, 36, 24, 72, 6, 216, 6, 96, 36, 36, 36, 144, 6, 36, 36, 144, 6, 216, 6, 72, 72, 36, 6, 288, 12, 72, 36, 72, 6, 144, 36, 144, 36, 36, 6, 432, 6, 36, 72, 192, 36, 216, 6, 72, 36, 216, 6, 288, 6

Offset: 1

Reinhard Zumkeller, Nov 04 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.
Eric Weisstein's World of Mathematics, Smooth number.

Cf. A000079, A000244, A000400, A001221, A001222, A003586, A005117, A007283, A061142, A074816, A124509, A124510, A124511, A124512.

Table[2^PrimeOmega[n] 3^PrimeNu[n],{n,80}] (* Harvey P. Dale, Mar 26 2013 *)

a(n) = my(f = factor(n)); 2^bigomega(f) * 3^omega(f); \\ Amiram Eldar, Jul 11 2023

Multiplicative with p^e -> 3*2^e, p prime and e>0.
a(n) = A061142(n)*A074816(n) = A000079(A001222(n))*A000244(A001221(n)).
A124509 gives the range: A124509(n) = a(A124510(n)) and a(m) <> a(A124510(n)) for m < A124510(n).
For primes p, q with p <> q: a(p) = 6; a(p*q) = 36; a(p^k) = 3*2^k, k>0.
For squarefree numbers m: a(m) = 6^omega(m).
A001222(a(n)) = A001222(n)+1; A001221(a(n)) = 2 for n > 1.
A124511(n) = a(a(n)); A124512(n) = a(a(a(n))).

A154626 a(n) = 2^n*A001519(n).

Original entry on oeis.org

1, 2, 8, 40, 208, 1088, 5696, 29824, 156160, 817664, 4281344, 22417408, 117379072, 614604800, 3218112512, 16850255872, 88229085184, 461973487616, 2418924584960, 12665653559296, 66318223015936, 347246723858432, 1818207451086848, 9520257811087360

Offset: 0

Paul Barry, Jan 13 2009

Hankel transform of 1,1,3,11,45,... (see A026375). Binomial transform of A015448.
From Gary W. Adamson, Jul 22 2016: (Start)
A production matrix for the sequence is M =
  1, 1, 0, 0, 0, ...
  1, 0, 5, 0, 0, ...
  1, 0, 0, 5, 0, ...
  1, 0, 0, 0, 5, ...
  ...
Take powers of M, extracting the upper left terms; getting
the sequence starting (1, 1, 2, 8, 40, 208, ...). (End)
The sequence is N=5 in an infinite set of INVERT transforms of powers of N prefaced with a "1". (1, 2, 8, 40, ...) is the INVERT transform of (1, 1, 5, 25, 125, ...). The first six of such sequences are shown in A006012 (N=3). - Gary W. Adamson, Jul 24 2016
From Gary W. Adamson, Jul 27 2016: (Start)
The sequence is the first in an infinite set in which we perform the operation for matrix M (Cf. Jul 22 2016), but change the left border successively from (1, 1, 1, 1, ...) then to (1, 2, 2, 2, ...), then (1, 3, 3, 3, ...) ...; generally (1, N, N, N, ...). Extracting the upper left terms of each matrix operation, we obtain the infinite set beginning:
  N=1  (A154626): 1,  2,  8,  40,  208,  1088, ...
  N=2  (A084120): 1,  3, 15,  81,  441,  1403, ...
  N=3  (A180034): 1,  4, 22, 124,  700,  3952, ...
  N=4  (A001653): 1,  5, 29, 169,  985,  5741, ...
  N=5  (A000400): 1,  6, 36, 216, 1296,  7776, ...
  N=6  (A015451): 1,  7, 43, 265, 1633, 10063, ...
  N=7  (A180029): 1,  8, 50, 316, 1996, 12608, ...
  N=8  (A180028): 1,  9, 57, 369, 1285, 15417, ...
  N=9  (.......): 1, 10, 64, 424, 2800, 18496, ...
  N=10 (A123361): 1, 11, 71, 481, 3241, 21851, ...
  N=11 (.......): 1, 12, 78, 540, 3708, 25488, ...
... Each of the sequences begins (1, (N+1), (7*N + 1),
  (40*N + (N-1)^2), ... (End)
The set of infinite sequences shown (Cf. comment of Jul 27 2016), can be generated from the matrices P = [(1,N; 1,5]^n, (N=1,2,3,...) by extracting the upper left terms. Example: N=6 sequence (A015451): (1, 7, 43, 265, ...) can be generated from the matrix P = [(1,6); (1,5)]^n. - Gary W. Adamson, Jul 28 2016

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Index entries for linear recurrences with constant coefficients, signature (6,-4).

Cf. A006012, A084120, A180034, A001653, A000400, A015451, A180029, A180028, A123362.

[n le 2 select (n) else 6*Self(n-1)-4*Self(n-2): n in [1..25]]; // Vincenzo Librandi, May 15 2015

LinearRecurrence[{6, -4}, {1, 2}, 30] (* Vincenzo Librandi, May 15 2015 *)

Vec((1-4*x) / (1-6*x+4*x^2) + O(x^30)) \\ Colin Barker, Sep 22 2017

G.f.: (1 - 4*x) / (1 - 6*x + 4*x^2).
a(n) = A084326(n+1) - 4*A084326(n). - R. J. Mathar, Jul 19 2012
From Colin Barker, Sep 22 2017: (Start)
a(n) = (((3-sqrt(5))^n*(1+sqrt(5)) + (-1+sqrt(5))*(3+sqrt(5))^n)) / (2*sqrt(5)).
a(n) = 6*a(n-1) - 4*a(n-2) for n>1. (End)
E.g.f.: exp(3*x)*(5*cosh(sqrt(5)*x) - sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, Aug 26 2025

A175552 Numbers k such that the digit sum of 167^k is divisible by k.

Original entry on oeis.org

1, 2, 5, 7, 22, 490, 724, 778, 868, 994, 1109, 1390, 1415, 1462, 1642, 1739, 1829, 2146, 2362, 3136, 4954, 6437, 6628, 7103, 11200, 12424, 12863, 14242, 14249, 15059, 15203, 16222, 17140, 18353, 19192, 21233, 22853, 24106, 24574, 24833, 26896, 27652, 28253, 30323, 31306, 31594, 32386, 33790, 34985, 36184, 36310, 40673, 42196, 43931, 45911, 45983

Offset: 1

N. J. A. Sloane, Dec 03 2010

From Donovan Johnson, Dec 03 2010: (Start)
To generate the additional terms I used PFGW.exe to get the decimal expansion for each number of the form 167^n (n <= 50000). Then I wrote a program in powerbasic to read the pfgw.out file and get the digit sums.
The digit sum is 10 times the n value for terms a(5) to a(56). (End)
I believe that this sequence is finite. - N. J. A. Sloane, Dec 05 2010

Donovan Johnson, Table of n, a(n) for n = 1..129

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.

Select[Range[10000], Mod[Total[IntegerDigits[167^#]], #] == 0 &]

a(25)-a(56) from Donovan Johnson, Dec 03 2010

A180032 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5x-7x^2).

Original entry on oeis.org

1, 6, 37, 227, 1394, 8559, 52553, 322678, 1981261, 12165051, 74694082, 458625767, 2815987409, 17290317414, 106163498933, 651849716563, 4002393075346, 24574913392671, 150891318490777, 926480986202582, 5688644160448349

Offset: 0

Johannes W. Meijer, Aug 09 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner or side square (m = 1, 3, 7, 9; 2, 4, 6, 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a white chess queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen.
On a 3 X 3 chessboard there are 2^9 = 512 ways to explode with fury on the central square (we assume here that a red queen might behave like a white queen). The red queen is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the corner and side squares the 512 red queens lead to 17 red queen sequences, see the cross-references for the complete set.
The sequence above corresponds to 8 red queen vectors, i.e., A[5] vectors, with decimal values 239, 367, 431, 463, 487, 491, 493 and 494. The central square leads for these vectors to A152240.
This sequence belongs to a family of sequences with g.f. (1+x)/(1 - 5*x - k*x^2). The members of this family that are red queen sequences are A180030 (k=8), A180032 (k=7; this sequence), A000400 (k=6), A180033 (k=5), A126501 (k=4), A180035 (k=3), A180037 (k=2) A015449 (k=1) and A003948 (k=0). Other members of this family are A030221 (k=-1), A109114 (k=-3), A020989 (k=-4), A166060 (k=-6).
Inverse binomial transform of A054413.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wikipedia, Alice in Wonderland (2010 film)
Index entries for linear recurrences with constant coefficients, signature (5, 7).

Cf. A180028 (Central square).

Cf. Red queen sequences corner and side squares [decimal value A[5]]: A090018 [511], A135030 [255], A180030 [495], A005668 [127], A180032 [239], A000400 [63], A180033 [47], A001109 [31], A126501 [15], A154244 [23], A180035 [7], A138395 [19], A180037 [3], A084326 [17], A015449 [1], A003463 [16], A003948 [0].

I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

with(LinearAlgebra): nmax:=20; m:=1; A[5]:= [1,1,1,1,0,1,1,1,0]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{5,7},{1,6},40] (* Vincenzo Librandi, Nov 15 2011 *)
CoefficientList[Series[(1+x)/(1-5x-7x^2),{x,0,30}],x] (* Harvey P. Dale, Apr 04 2024 *)

G.f.: (1+x)/(1 - 5*x - 7*x^2).
a(n) = 5*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 6.
a(n) = ((7+9*A)*A^(-n-1) + (7+9*B)*B^(-n-1))/53 with A = (-5+sqrt(53))/14 and B = (-5-sqrt(53))/14.

cp's OEIS Frontend

A165827 Totally multiplicative sequence with a(p) = 6.

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A175169 Numbers k that divide the sum of digits of 2^k.

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A175525 Numbers k that divide the sum of digits of 13^k.

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A013600 a(n) = nextprime(6^n)-6^n.

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Maple

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A025622 Numbers of form 5^i*6^j, with i, j >= 0.

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A049304 Numbers k such that k is a substring of 6^k.

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A124508 a(n) = 2^BigO(n) * 3^omega(n), where BigO = A001222 and omega = A001221, the numbers of prime factors of n with and without repetitions.

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A154626 a(n) = 2^n*A001519(n).

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Magma

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A180032 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5x-7x^2).