A033052 -id:A033052 - cp's OEIS frontend

A033048 Sums of distinct powers of 12.

0, 1, 12, 13, 144, 145, 156, 157, 1728, 1729, 1740, 1741, 1872, 1873, 1884, 1885, 20736, 20737, 20748, 20749, 20880, 20881, 20892, 20893, 22464, 22465, 22476, 22477, 22608, 22609, 22620, 22621, 248832, 248833, 248844, 248845, 248976

Offset: 0

Clark Kimberling

Numbers without any base-12 digits greater than 1.

T. D. Noe, Table of n, a(n) for n = 0..1023
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.
Eric Weisstein's World of Mathematics, Duodecimal
Wikipedia, Duodecimal

Subsequence of A102487.
Cf. A000695, A005836, A033042-A033052.
Row 11 of array A104257.

import Data.List (unfoldr)
a033048 n = a033048_list !! (n-1)
a033048_list = filter (all (< 2) . unfoldr (\x ->
   if x == 0 then Nothing else Just $ swap $ divMod x 12)) [1..]
-- Reinhard Zumkeller, Apr 17 2011

With[{k = 12}, Map[FromDigits[#, k] &, Tuples[{0, 1}, 6]]] (* Michael De Vlieger, Oct 28 2022 *)

{maxn=37;
for(vv=0,maxn,
bvv=binary(vv);
ll=length(bvv);texp=0;btod=0;
forstep(i=ll,1,-1,btod=btod+bvv[i]*12^texp;texp++);
print1(btod,", "))}
\\ Douglas Latimer, Apr 16 2012

a(n)=fromdigits(binary(n),12) \\ Charles R Greathouse IV, Jan 11 2017

a(n) = Sum_{i=0..m} d(i)*12^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
a(n) = A097258(n)/11.
a(2n) = 12*a(n), a(2n+1) = a(2n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(k) = 12^k = A001021(k). - Philippe Deléham, Oct 19 2011
G.f.: (1/(1 - x))*Sum_{k>=0} 12^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extended by Ray Chandler, Aug 03 2004

A115845 Numbers n such that there is no bit position where the binary expansions of n and 8n are both 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17, 20, 21, 24, 28, 32, 33, 34, 35, 40, 42, 48, 49, 56, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 84, 85, 96, 97, 98, 99, 112, 113, 128, 129, 130, 131, 132, 133, 134, 135, 136, 138, 140, 142, 160, 161, 162, 163, 168, 170, 192

Offset: 1

Antti Karttunen, Feb 01 2006

nonn

Equivalently, numbers n such that 9*n = 9 X n, i.e., 8*n XOR n = 9*n. Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).
Equivalently, numbers n such that the binomial coefficient C(9n,n) (A169958) is odd. - Zak Seidov, Aug 06 2010
The equivalence of these three definitions follows from Lucas's theorem on binomial coefficients. - N. J. A. Sloane, Sep 01 2010
Clearly all numbers k*2^i for 1 <= k <= 7 have this property. - N. J. A. Sloane, Sep 01 2010
A116361(a(n)) <= 3. - Reinhard Zumkeller, Feb 04 2006

N. J. A. Sloane and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences defined by congruent products between domains N and GF(2)[X]
Index entries for sequences defined by congruent products under XOR

A115846 shows this sequence in binary.
A033052 is a subsequence.
Cf. A003714, A048716, A115847, A116360, A005809, A003714, A048716, A048715.

Reap[Do[If[OddQ[Binomial[9n,n]],Sow[n]],{n,0,400}]][[2,1]] (* Zak Seidov, Aug 06 2010 *)

is(n)=!bitand(n,n<<3) \\ Charles R Greathouse IV, Sep 23 2012

a(n)/n^k is bounded (but does not tend to a limit), where k = 1.44... = A104287. - Charles R Greathouse IV, Sep 23 2012

Edited with a new definition by N. J. A. Sloane, Sep 01 2010, merging this sequence with a sequence submitted by Zak Seidov, Aug 06 2010

A033047 Sums of distinct powers of 11.

Original entry on oeis.org

0, 1, 11, 12, 121, 122, 132, 133, 1331, 1332, 1342, 1343, 1452, 1453, 1463, 1464, 14641, 14642, 14652, 14653, 14762, 14763, 14773, 14774, 15972, 15973, 15983, 15984, 16093, 16094, 16104, 16105, 161051, 161052, 161062, 161063, 161172

Offset: 0

Clark Kimberling

Numbers without any base-11 digits greater than 1.

a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 17 2011

T. D. Noe, Table of n, a(n) for n = 0..1023
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-A033052.

Row 10 of array A104257.

With[{k = 11}, Map[FromDigits[#, k] &, Tuples[{0, 1}, 6]]] (* Michael De Vlieger, Oct 28 2022 *)

{for(vv=0,35,
bvv=binary(vv);
texp=0;btb=0;
forstep(i=length(bvv),1,-1,btb=btb+bvv[i]*11^texp;texp++);
print1(btb,", "))} \\ Douglas Latimer, May 12 2012

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

a(n) = Sum_{i=0..m} d(i)*11^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
a(n) = A097257(n)/10.
a(2n) = 11*a(n), a(2n+1) = a(2n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*11^k. - Philippe Deléham, Oct 17 2011
G.f.: (1/(1 - x))*Sum_{k>=0} 11^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extended by Ray Chandler, Aug 03 2004

A033049 Sums of distinct powers of 13.

Original entry on oeis.org

0, 1, 13, 14, 169, 170, 182, 183, 2197, 2198, 2210, 2211, 2366, 2367, 2379, 2380, 28561, 28562, 28574, 28575, 28730, 28731, 28743, 28744, 30758, 30759, 30771, 30772, 30927, 30928, 30940, 30941, 371293, 371294, 371306, 371307, 371462

Offset: 0

Clark Kimberling

Numbers without any base-13 digits greater than 1.

a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 17 2011

T. D. Noe, Table of n, a(n) for n = 0..1023
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-A033052.

Row 12 of array A104257.

With[{k = 13}, Map[FromDigits[#, k] &, Tuples[{0, 1}, 6]]] (* Michael De Vlieger, Oct 28 2022 *)

A033049(n,b=13)=subst(Pol(binary(n)),'x,b) \\ M. F. Hasler, Feb 01 2016

a(n) = Sum_{i=0..m} d(i)*13^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
a(n) = A097259(n)/12.
a(2n) = 13*a(n), a(2n+1) = a(2n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*13^k. - Philippe Deléham, Oct 17 2011
G.f.: (1/(1 - x))*Sum_{k>=0} 13^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extended by Ray Chandler, Aug 03 2004

A033051 Numbers whose set of base 15 digits is {0,1}.

Original entry on oeis.org

0, 1, 15, 16, 225, 226, 240, 241, 3375, 3376, 3390, 3391, 3600, 3601, 3615, 3616, 50625, 50626, 50640, 50641, 50850, 50851, 50865, 50866, 54000, 54001, 54015, 54016, 54225, 54226, 54240, 54241, 759375, 759376, 759390, 759391, 759600

Offset: 0

Clark Kimberling

Sums of distinct powers of 15.

a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 17 2011.

T. D. Noe, Table of n, a(n) for n = 0..1023
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-A033052.

Row 14 of array A104257.

With[{k = 15}, Map[FromDigits[#, k] &, Tuples[{0, 1}, 6]]] (* Michael De Vlieger, Oct 28 2022 *)
FromDigits[#,15]&/@Tuples[{0,1},6] (* Harvey P. Dale, Sep 15 2024 *)

A033051(n, b=15)=subst(Pol(binary(n)),'x,b) \\ M. F. Hasler, Feb 01 2016

a(n) = Sum_{i=0..m} d(i)*15^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
a(n) = A097261(n)/14.
a(2n) = 15*a(n), a(2n+1) = a(2n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*15^k. - Philippe Deléham, Oct 17 2011.
G.f.: (1/(1 - x))*Sum_{k>=0} 15^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extended by Ray Chandler, Aug 03 2004

A033050 Numbers whose set of base 14 digits is {0,1}.

Original entry on oeis.org

0, 1, 14, 15, 196, 197, 210, 211, 2744, 2745, 2758, 2759, 2940, 2941, 2954, 2955, 38416, 38417, 38430, 38431, 38612, 38613, 38626, 38627, 41160, 41161, 41174, 41175, 41356, 41357, 41370, 41371, 537824, 537825, 537838, 537839, 538020

Offset: 0

Clark Kimberling

Sums of distinct powers of 14.

The base-14 digits may comprise zero, one, or both. - Harvey P. Dale, May 12 2014

T. D. Noe, Table of n, a(n) for n = 0..1023
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-A033052.

Row 13 of array A104257.

Select[Range[0,540000],Max[IntegerDigits[#,14]]<2&] (* Harvey P. Dale, May 12 2014 *)
FromDigits[#,14]&/@Tuples[{0,1},6] (* Harvey P. Dale, Jun 18 2021 *)

A033050(n,b=14)=subst(Pol(binary(n)),'x,b) \\ M. F. Hasler, Feb 01 2016

a(n) = Sum_{i=0..m} d(i)*14^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
a(n) = A097260(n)/13.
a(2n) = 14*a(n), a(2n+1) = a(2n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*14^k. - Philippe Deléham, Oct 20 2011
G.f.: (1/(1 - x))*Sum_{k>=0} 14^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extended by Ray Chandler, Aug 03 2004

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

A351995 Square array A(n, k), n, k >= 0, read by antidiagonals upwards; A(n, k) = Sum_{ i >= 0 } b_i * 2^(ki) where n = Sum_{ i >= 0 } b_i 2^i.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 1, 3, 4, 1, 0, 2, 4, 5, 8, 1, 0, 2, 5, 16, 9, 16, 1, 0, 3, 6, 17, 64, 17, 32, 1, 0, 1, 7, 20, 65, 256, 33, 64, 1, 0, 2, 8, 21, 72, 257, 1024, 65, 128, 1, 0, 2, 9, 64, 73, 272, 1025, 4096, 129, 256, 1, 0, 3, 10, 65, 512, 273, 1056, 4097, 16384, 257, 512, 1, 0

Offset: 0

Rémy Sigrist, Feb 27 2022

In other words, in binary expansion of n, replace 2^i by 2^(k*i).

			Square array A(n, k) begins:
  n\k|  0  1   2   3    4     5     6      7      8       9       10
  ------------------------------------------------------------------
    0|  0  0   0   0    0     0     0      0      0       0        0
    1|  1  1   1   1    1     1     1      1      1       1        1
    2|  1  2   4   8   16    32    64    128    256     512     1024
    3|  2  3   5   9   17    33    65    129    257     513     1025
    4|  1  4  16  64  256  1024  4096  16384  65536  262144  1048576
    5|  2  5  17  65  257  1025  4097  16385  65537  262145  1048577
    6|  2  6  20  72  272  1056  4160  16512  65792  262656  1049600
    7|  3  7  21  73  273  1057  4161  16513  65793  262657  1049601

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Index entries for sequences related to binary expansion of n

Cf. A000120, A000695, A033045, A033052, A352001.

A351995[n_, k_] := If[n <= 1, n, Total[2^(k*(Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]] - 1))]];
Table[A351995[n - k, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Aug 26 2025 *)

A(n,k) = { my (v=0, e); while (n, n-=2^e=valuation(n, 2); v+=2^(k*e)); v }

A(A(n, k), k') = A(n, k*k') for k, k' > 0.
A(n, 0) = A000120(n).
A(n, 1) = n.
A(n, 2) = A000695(n).
A(n, 3) = A033045(n).
A(n, 4) = A033052(n).
A(0, k) = 0.
A(1, k) = 1.
A(2, k) = 2^k.
A(3, k) = 2^k + 1.

A147845 Odd positive integers a(n) such that for every odd integer m>=7 there exists a unique representation of the form m=a(p)+2a(q)+4a(r).

Original entry on oeis.org

1, 3, 17, 19, 129, 131, 145, 147, 1025, 1027, 1041, 1043, 1153, 1155, 1169, 1171, 8193, 8195, 8209, 8211, 8321, 8323, 8337, 8339, 9217, 9219, 9233, 9235, 9345, 9347, 9361, 9363, 65537, 65539, 65553, 65555

Offset: 1

Vladimir Shevelev, Nov 15 2008

nonn

Since, e.g., 27=17+2*3+4*1 and 17=a(3),3=a(2),1=a(1), then 27 has "coordinates" (3,2,1). Thus we have a one-to-one map of odd integers >=7 to the positive lattice points in the three-dimensional space.

A145812 A145818 A033045 A033052 [From Vladimir Shevelev, Nov 19 2008]

a(n)=2A033045(n-1)+1

cp's OEIS Frontend

A033048 Sums of distinct powers of 12.

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A115845 Numbers n such that there is no bit position where the binary expansions of n and 8n are both 1.

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A033047 Sums of distinct powers of 11.

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A033049 Sums of distinct powers of 13.

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A033051 Numbers whose set of base 15 digits is {0,1}.

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A033050 Numbers whose set of base 14 digits is {0,1}.

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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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A351995 Square array A(n, k), n, k >= 0, read by antidiagonals upwards; A(n, k) = Sum_{ i >= 0 } b_i * 2^(ki) where n = Sum_{ i >= 0 } b_i 2^i.