A033051 -id:A033051 - cp's OEIS frontend

A033052 a(1) = 1, a(2n) = 16a(n), a(2n+1) = a(2n)+1.

0, 1, 16, 17, 256, 257, 272, 273, 4096, 4097, 4112, 4113, 4352, 4353, 4368, 4369, 65536, 65537, 65552, 65553, 65792, 65793, 65808, 65809, 69632, 69633, 69648, 69649, 69888, 69889, 69904, 69905, 1048576, 1048577, 1048592, 1048593, 1048832

Offset: 0

Clark Kimberling

Numbers whose set of base 16 digits is {0,1}.
a(n) = Xpower(n,4). - Antti Karttunen, Apr 26 1999
Sums of distinct powers of 16.
For every nonnegative n, A000695(n) is a unique sum of the form a(k) + 4a(l). Thus every nonnegative n is a unique sum of the form a(p) + 2a(q) + 4a(r) + 8a(s). This gives a one-to-one map of the set N_0 of all nonnegative integers to (N_0)^4. Furthermore, if, for a fixed positive integer m, to consider all sums of distinct powers of 4^m, then one can obtain a one-to-one map of the set N_0 to (N_0)^(2^m). - Vladimir Shevelev, Nov 14 2008

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

Cf. A000695, A005836, A033042-A033051.

Column 4 of A048723. Row 15 of array A104257.

int a_next(int a_n) { return (a_n + 0xeeeeeeef) & 0x11111111; } /* Falk Hüffner, Jan 24 2022 */

[n: n in [1..1050000] | Set(IntegerToSequence(n, 16)) subset {0, 1}]; // Vincenzo Librandi, May 04 2012

FromDigits[#,16]&/@Tuples[{0,1},5] (* Vincenzo Librandi, Jun 04 2012 *)

a(n)=n=Vecrev(binary(n));sum(i=1,#n,n[i]<<(4*i))>>4 \\ Charles R Greathouse IV, Sep 23 2012

a(n)=fromdigits(binary(n),16); \\ Alan Michael Gómez Calderón, Mar 23 2025

a(n) = Sum_{i=0..m} d(i)*16^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
a(n) = A097262(n)/15.
a(2n) = 16*a(n), a(2n+1) = a(2n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*16^k. - Philippe Deléham, Oct 19 2011
G.f.: (1/(1 - x))*Sum_{k>=0} 16^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017
a(n) = A000695(A000695(n)). - Alan Michael Gómez Calderón, Mar 23 2025

Extended by Ray Chandler, Aug 03 2004

Simpler definition from Ralf Stephan, Jun 18 2005

A104257 Square array T(a,n) read by antidiagonals: replace 2^i with a^i in binary representation of n, where a,n >= 2.

Original entry on oeis.org

2, 3, 3, 4, 4, 4, 5, 5, 9, 5, 6, 6, 16, 10, 6, 7, 7, 25, 17, 12, 7, 8, 8, 36, 26, 20, 13, 8, 9, 9, 49, 37, 30, 21, 27, 9, 10, 10, 64, 50, 42, 31, 64, 28, 10, 11, 11, 81, 65, 56, 43, 125, 65, 30, 11, 12, 12, 100, 82, 72, 57, 216, 126, 68, 31, 12, 13, 13, 121, 101, 90, 73, 343

Offset: 2

Ralf Stephan, Mar 05 2005

Sums of distinct powers of a. Numbers having only {0,1} in a-ary representation.

			Array begins:
  2,  3,  4,  5,  6,  7,   8,   9, ...
  3,  4,  9, 10, 12, 13,  27,  28, ...
  4,  5, 16, 17, 20, 21,  64,  65, ...
  5,  6, 25, 26, 30, 31, 125, 126, ...
  6,  7, 36, 37, 42, 43, 216, 217, ...
  7,  8, 49, 50, 56, 57, 343, 344, ...
  8,  9, 64, 65, 72, 73, 512, 513, ...
  9, 10, 81, 82, 90, 91, 729, 730, ...
  ...

John Tyler Rascoe, Antidiagonals n = 2..140, flattened

Rows include (essentially) A005836, A000695, A033042, A033043, A033044, A033045, A033046, A033047, A033048, A033049, A033050, A033051, A033052.
Columns include A000290, A002522, A002378, A000578, A001093, A034262, A071568, A011379, A098547, A027444.
Main diagonal is A104258.

T[, 0] = 0; T[2, n] := n; T[a_, 2] := a;
T[a_, n_] := T[a, n] = If[EvenQ[n], a T[a, n/2], a T[a, (n-1)/2]+1];
Table[T[a-n+2, n], {a, 2, 13}, {n, 2, a}] // Flatten (* Jean-François Alcover, Feb 09 2021 *)

T(a, n) = fromdigits(binary(n), a); \\ Michel Marcus, Aug 19 2022

def T(a, n): return n if n < 2 else (max(a, n) if min(a, n) == 2 else a*T(a, n//2) + n%2)
print([T(a-n+2, n) for a in range(2, 14) for n in range(2, a+1)]) # Michael S. Branicky, Aug 02 2022

T(a, n) = (1/(a-1))*Sum_{j>=1} floor((n+2^(j-1))/2^j) * ((a-2)*a^(j-1) + 1).
T(a, n) = (1/(a-1))*Sum_{j=1..n} ((a-2)*a^A007814(j) + 1).
G.f. of a-th row: (1/(1-x)) * Sum_{k>=0} a^k*x^2^k/(1+x^2^k).
Recurrence: T(a, 2n) = a*T(a, n), T(a, 2n+1) = a*T(a, n) + 1, T(a, 0) = 0.

A063012 Sum of distinct powers of 20; i.e., numbers with digits in {0,1} base 20; i.e., write n in base 2 and read as if written in base 20.

Original entry on oeis.org

0, 1, 20, 21, 400, 401, 420, 421, 8000, 8001, 8020, 8021, 8400, 8401, 8420, 8421, 160000, 160001, 160020, 160021, 160400, 160401, 160420, 160421, 168000, 168001, 168020, 168021, 168400, 168401, 168420, 168421, 3200000, 3200001, 3200020, 3200021, 3200400, 3200401

Offset: 0

Henry Bottomley, Jul 04 2001

			a(5) = 401 since 5 written in base 2 is 101 so a(5) = 1*20^2 + 0*20^1 + 1*20^0 = 400 + 0 + 1 = 401.

Harry J. Smith, Table of n, a(n) for n = 0..1000
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.

A001477, A005836, A000695, A033042, A033043, A033044, A033045, A033046, A007088, A033047, A033048, A033049, A033050, A033051, A033052 are similar sequences for 2-16.

A063013 is similar in a different way.

a:= proc(n) `if`(n<2, n, irem(n, 2, 'r')+20*a(r)) end:
seq(a(n), n=0..37);  # Alois P. Heinz, Apr 04 2025

Table[FromDigits[IntegerDigits[n,2],20],{n,0,40}] (* Harvey P. Dale, Jul 21 2014 *)

baseE(x, b)= { local(d, e, f); e=0; f=1; while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) }
baseI(x, b)= { local(d, e, f); e=0; f=1; while (x>0, d=x-10*(x\10); x\=10; e+=d*f; f*=b); return(e) }
{ for (n=0, 1000, write("b063012.txt", n, " ", baseI(baseE(n, 2), 20)) ) } \\ Harry J. Smith, Aug 15 2009

def A063012(n): return int(bin(n)[2:],20) # Chai Wah Wu, Apr 04 2025

a(n) = a(n-2^floor(log_2(n))) + 20^floor(log_2(n)). a(2n) = 20*a(n); a(2n+1) = a(2n)+1 = 20*a(n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*A009964(k). - Philippe Deléham, Oct 15 2011
G.f.: (1/(1 - x))*Sum_{k>=0} 20^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Edited by Charles R Greathouse IV, Aug 02 2010

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.

cp's OEIS Frontend

A033052 a(1) = 1, a(2n) = 16a(n), a(2n+1) = a(2n)+1.

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A104257 Square array T(a,n) read by antidiagonals: replace 2^i with a^i in binary representation of n, where a,n >= 2.

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A063012 Sum of distinct powers of 20; i.e., numbers with digits in {0,1} base 20; i.e., write n in base 2 and read as if written in base 20.

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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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Programs

Magma

Mathematica

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