A097251 -id:A097251 - cp's OEIS frontend

A033042 Sums of distinct powers of 5.

0, 1, 5, 6, 25, 26, 30, 31, 125, 126, 130, 131, 150, 151, 155, 156, 625, 626, 630, 631, 650, 651, 655, 656, 750, 751, 755, 756, 775, 776, 780, 781, 3125, 3126, 3130, 3131, 3150, 3151, 3155, 3156, 3250, 3251, 3255, 3256, 3275, 3276, 3280, 3281, 3750, 3751

Offset: 0

Clark Kimberling

Numbers without any base-5 digits larger than 1.
a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 17 2011
Values of k where A008977(k) does not end with 0. - Henry Bottomley, Nov 09 2022

T. D. Noe, Table of n, a(n) for n = 0..1023
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
K. Dilcher and L. Ericksen, Hyperbinary expansions and Stern polynomials, Elec. J. Combin, 22, 2015, #P2.24.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

For generating functions Product_{k>=0} (1 + a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
Cf. A000695, A005836, A008977, A010060, A033043-A033052.
Row 5 of array A104257.

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 2)
        r += b * q
        b *= 5
    end
r end; [a(n) for n in 0:49] |> println # Peter Luschny, Jan 03 2021

a:= proc(n) local m, r, b; m, r, b:= n, 0, 1;
      while m>0 do r:= r+b*irem(m, 2, 'm'); b:= b*5 od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 16 2013

t = Table[FromDigits[RealDigits[n, 2], 5], {n, 1, 100}]
(* Clark Kimberling, Aug 02 2012 *)
FromDigits[#,5]&/@Tuples[{0,1},7] (* Harvey P. Dale, May 22 2018 *)

a(n) = subst(Pol(binary(n)), 'x, 5);
vector(50, i, a(i-1))  \\ Gheorghe Coserea, Sep 15 2015

a(n)=fromdigits(binary(n),5) \\ Charles R Greathouse IV, Jan 11 2017

def A033042(n): return int(bin(n)[2:],5) # Chai Wah Wu, Oct 30 2024

a(n) = Sum_{i=0..m} d(i)*5^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
Numbers j such that the coefficient of x^j is > 0 in Product_{k>=0} (1 + x^(5^k)). - Benoit Cloitre, Jul 29 2003
a(n) = A097251(n)/4.
a(2n) = 5*a(n), a(2n+1) = a(2n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*5^k. - Philippe Deléham, Oct 17 2011
liminf a(n)/n^(log(5)/log(2)) = 1/4 and limsup a(n)/n^(log(5)/log(2)) = 1. - Gheorghe Coserea, Sep 15 2015
G.f.: (1/(1 - x))*Sum_{k>=0} 5^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extended by Ray Chandler, Aug 03 2004

A097262 Numbers whose set of base 16 digits is {0,F}, where F base 16 = 15 base 10.

Original entry on oeis.org

0, 15, 240, 255, 3840, 3855, 4080, 4095, 61440, 61455, 61680, 61695, 65280, 65295, 65520, 65535, 983040, 983055, 983280, 983295, 986880, 986895, 987120, 987135, 1044480, 1044495, 1044720, 1044735, 1048320, 1048335, 1048560, 1048575

Offset: 0

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 16 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001196, A005823, A097251-A097261.

[n: n in [0..1110000] | Set(IntegerToSequence(n, 16)) subset {0, 15}]; // Vincenzo Librandi, Jun 05 2012

f[n_] := FromDigits[ IntegerDigits[n, 2] /. {1 -> 15}, 16]; Array[f, 32, 0] (* or *)
FromDigits[#, 16] & /@ Tuples[{0, 15}, 6] (* Harvey P. Dale, Sep 22 2011 *) (* or much slower *)
fQ[n_] := Union@ Join[{0, 15}, IntegerDigits[n, 16]] == {0, 15}; Select[ Range[0, 11000000 ], fQ] (* Robert G. Wilson v, May 12 2012 *)

a(n) = 15*A033052(n).

a(2n) = 16*a(n), a(2n+1) = a(2n)+15.

A097256 Numbers whose set of base 10 digits is {0,9}.

Original entry on oeis.org

0, 9, 90, 99, 900, 909, 990, 999, 9000, 9009, 9090, 9099, 9900, 9909, 9990, 9999, 90000, 90009, 90090, 90099, 90900, 90909, 90990, 90999, 99000, 99009, 99090, 99099, 99900, 99909, 99990, 99999, 900000, 900009, 900090, 900099, 900900

Offset: 0

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 10 for every i.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A001196, A005823, A097251-A097262.

Cf. A078248, A169965, A169966, A169967, A169964, A204093, A204094, A204095.

a097256 n = a097256_list !! n
a097256_list = map (* 9) a007088_list
-- Reinhard Zumkeller, Jan 10 2012

A097256:=n->(9/2) * add((1-(-1)^floor(n/2^i))*10^i, i=0..n); seq(A097256(n), n=0..30); # Wesley Ivan Hurt, Feb 11 2014

Table[(9/2) Sum[(1 - (-1)^Floor[n/2^i]) 10^i, {i, 0, n}], {n, 0, 30}] (* Wesley Ivan Hurt, Feb 11 2014 *)

a(n) = 9*A007088(n).

a(2n) = 10*a(n), a(2n+1) = a(2n)+9.

A097252 Numbers whose set of base 6 digits is {0,5}.

Original entry on oeis.org

0, 5, 30, 35, 180, 185, 210, 215, 1080, 1085, 1110, 1115, 1260, 1265, 1290, 1295, 6480, 6485, 6510, 6515, 6660, 6665, 6690, 6695, 7560, 7565, 7590, 7595, 7740, 7745, 7770, 7775, 38880, 38885, 38910, 38915, 39060, 39065, 39090, 39095, 39960, 39965

Offset: 0

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 6 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A001196, A005823, A007088, A033043, A097251-A097262.

[n: n in [0..40000] | Set(IntegerToSequence(n, 6)) subset {0, 5}]; // Vincenzo Librandi, May 25 2012

fQ[n_]:=Union@Join[{0,5},IntegerDigits[n,6]]=={0,5};Select[Range[0,40000],fQ] (* Vincenzo Librandi, May 25 2012 *)
FromDigits[#,6]&/@Tuples[{ 0,5},6] (* Harvey P. Dale, Aug 15 2021 *)

def A079252(n): return 5*int(bin(n)[2:],6) # Chai Wah Wu, Apr 04 2025

a(n) = 5*A033043(n).

a(2n) = 6*a(n), a(2n+1) = a(2n)+5.

A097254 Numbers whose set of base 8 digits is {0,7}.

Original entry on oeis.org

0, 7, 56, 63, 448, 455, 504, 511, 3584, 3591, 3640, 3647, 4032, 4039, 4088, 4095, 28672, 28679, 28728, 28735, 29120, 29127, 29176, 29183, 32256, 32263, 32312, 32319, 32704, 32711, 32760, 32767, 229376, 229383, 229432, 229439, 229824

Offset: 1

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 8 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A001196, A005823, A097251-A097262.

[n: n in [0..250000] | Set(IntegerToSequence(n, 8)) subset {0, 7}]; // Vincenzo Librandi, May 25 2012

fQ[n_]:=Union@Join[{0,7},IntegerDigits[n,8]]=={0,7};Select[Range[0,300000],fQ] (* Vincenzo Librandi, May 25 2012 *)
FromDigits[#,8]&/@Tuples[{0,7},6] (* Harvey P. Dale, Aug 10 2021 *)

a[1]:0$ a[n]:=8*a[floor((n+1)/2)]+7*(1+(-1)^n)/2$ makelist(a[n], n, 1, 37); /* Bruno Berselli, May 25 2012 */

a(n) = 7*fromdigits(binary(n-1), 8) \\ Rémy Sigrist, Dec 06 2018

a(n) = 7*A033045(n-1).

a(2n-1) = 8*a(n), a(2n) = 8*a(n)+7.

A097261 Numbers whose set of base 15 digits is {0,E}, where E base 15 = 14 base 10.

Original entry on oeis.org

0, 14, 210, 224, 3150, 3164, 3360, 3374, 47250, 47264, 47460, 47474, 50400, 50414, 50610, 50624, 708750, 708764, 708960, 708974, 711900, 711914, 712110, 712124, 756000, 756014, 756210, 756224, 759150, 759164, 759360, 759374, 10631250

Offset: 0

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 15 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001196, A005823, A097251-A097262.

[n: n in [0..4500000] | Set(IntegerToSequence(n, 15)) subset {0, 14}]; // Vincenzo Librandi, Jun 05 2012

f[n_] := FromDigits[ IntegerDigits[n, 2] /. {1 -> 14}, 15]; Array[f, 33, 0] (* or *)
FromDigits[#, 15] & /@ Tuples[{0, 14}, 6] (* Harvey P. Dale, Sep 22 2011 *) (* or much slower *)
fQ[n_] := Union@ Join[{0, 14}, IntegerDigits[n, 15]] == {0, 14}; Select[ Range[0, 10634414 ], fQ] (* Robert G. Wilson v, May 12 2012 *)

a(n) = 14*A033051(n).

a(2n) = 15*a(n), a(2n+1) = a(2n)+14.

A097253 Numbers whose set of base 7 digits is {0,6}.

Original entry on oeis.org

0, 6, 42, 48, 294, 300, 336, 342, 2058, 2064, 2100, 2106, 2352, 2358, 2394, 2400, 14406, 14412, 14448, 14454, 14700, 14706, 14742, 14748, 16464, 16470, 16506, 16512, 16758, 16764, 16800, 16806, 100842, 100848, 100884, 100890, 101136

Offset: 1

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 7 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A001196, A005823, A097251-A097262.

[n: n in [0..200000] | Set(IntegerToSequence(n, 7)) subset {0, 6}]; // Vincenzo Librandi, May 25 2012

fQ[n_]:=Union@Join[{0,6},IntegerDigits[n,7]]=={0,6};Select[Range[0,140000],fQ] (* Vincenzo Librandi, May 25 2012 *)
FromDigits[#,7]&/@Tuples[{0,6},6] (* This program is several thousand times faster than the first program, above. *) (* Harvey P. Dale, Aug 12 2023 *)

a[0]:0$ a[n]:=7*a[floor(n/2)]+3*(1-(-1)^n)$ makelist(a[n], n, 0, 36); /* Bruno Berselli, May 25 2012 */

a(n) = 6*A033044(n).

a(2n) = 7*a(n), a(2n+1) = a(2n)+6.

Offset corrected by Arkadiusz Wesolowski, Nov 09 2013

A097255 Numbers whose set of base 9 digits is {0,8}.

Original entry on oeis.org

0, 8, 72, 80, 648, 656, 720, 728, 5832, 5840, 5904, 5912, 6480, 6488, 6552, 6560, 52488, 52496, 52560, 52568, 53136, 53144, 53208, 53216, 58320, 58328, 58392, 58400, 58968, 58976, 59040, 59048, 472392, 472400, 472464, 472472, 473040

Offset: 0

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 9 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001196, A005823, A097251-A097262.

[n: n in [0..500000] | Set(IntegerToSequence(n, 9)) subset {0, 8}]; // Vincenzo Librandi, May 25 2012

fQ[n_]:=Union@Join[{0,8},IntegerDigits[n,9]]=={0,8};Select[Range[0,500000],fQ] (* or *) FromDigits[#,9]&/@Tuples[{0,8},6](* Vincenzo Librandi, May 25 2012 *)

a[0]:0$ a[n]:=9*a[floor(n/2)]+4*(1-(-1)^n)$ makelist(a[n], n, 0, 36); /* Bruno Berselli, May 26 2012 */

a(n) = 8*A033046(n).

a(2n) = 9*a(n), a(2n+1) = a(2n)+8.

A097257 Numbers whose set of base 11 digits is {0,A}, where A base 11 = 10 base 10.

Original entry on oeis.org

0, 10, 110, 120, 1210, 1220, 1320, 1330, 13310, 13320, 13420, 13430, 14520, 14530, 14630, 14640, 146410, 146420, 146520, 146530, 147620, 147630, 147730, 147740, 159720, 159730, 159830, 159840, 160930, 160940, 161040, 161050, 1610510

Offset: 0

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 11 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001196, A005823, A097251-A097262.

f[n_] := FromDigits[ IntegerDigits[n, 2] /. {1 -> 10}, 11]; Array[f, 33, 0] (* or much slower *)
fQ[n_] := Union@ Join[{0, 10}, IntegerDigits[n, 11]] == {0, 10}; Select[ Range[0, 1610519], fQ] (* Robert G. Wilson v, May 12 2012 *)
Join[{0},Union[Flatten[Table[FromDigits[#,11]&/@(Join[{10},#]&/@ Tuples[ {10,0},n]),{n,0,5}]]]] (* Harvey P. Dale, Sep 23 2013 *)

{for(vv=0,32,
bvv=binary(vv);
texp=0;btb=0;
forstep(i=length(bvv),1,-1,btb=btb+10*bvv[i]*11^texp;texp++);
print1(btb,", ") )} \\ Douglas Latimer, May 12 2012

a(n) = 10*A033047(n).

a(2n) = 11*a(n), a(2n+1) = a(2n)+10.

A097258 Numbers whose set of base 12 digits is {0,B}, where B base 12 = 11 base 10.

Original entry on oeis.org

0, 11, 132, 143, 1584, 1595, 1716, 1727, 19008, 19019, 19140, 19151, 20592, 20603, 20724, 20735, 228096, 228107, 228228, 228239, 229680, 229691, 229812, 229823, 247104, 247115, 247236, 247247, 248688, 248699, 248820, 248831, 2737152

Offset: 0

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 12 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001196, A005823, A097251-A097262.

[n: n in [0..2800000] | Set(IntegerToSequence(n, 12)) subset {0, 11}]; // Vincenzo Librandi, May 26 2012

f[n_] := FromDigits[ IntegerDigits[n, 2] /. {1 -> 11}, 12]; Array[f, 33, 0] (* or much slower *)
fQ[n_] := Union@ Join[{0, 11}, IntegerDigits[n, 12]] == {0, 11}; Select[ Range[0, 27370162], fQ] (* Robert G. Wilson v, May 12 2012 *)
FromDigits[#,12]&/@Tuples[{0,11},6] (* Vincenzo Librandi, May 26 2012 *)

a(n) = 11*A033048(n).

a(2n) = 12*a(n), a(2n+1) = a(2n)+11.

cp's OEIS Frontend

A033042 Sums of distinct powers of 5.

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A097262 Numbers whose set of base 16 digits is {0,F}, where F base 16 = 15 base 10.

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A097256 Numbers whose set of base 10 digits is {0,9}.

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A097252 Numbers whose set of base 6 digits is {0,5}.

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A097254 Numbers whose set of base 8 digits is {0,7}.

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A097261 Numbers whose set of base 15 digits is {0,E}, where E base 15 = 14 base 10.

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A097253 Numbers whose set of base 7 digits is {0,6}.

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