A104257 -id:A104257 - cp's OEIS frontend

A033047 Sums of distinct powers of 11.

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

A197351 a(0)=0, a(1)=1, a(2n)=17*a(n), a(2n+1)=a(2n)+1.

Original entry on oeis.org

0, 1, 17, 18, 289, 290, 306, 307, 4913, 4914, 4930, 4931, 5202, 5203, 5219, 5220, 83521, 83522, 83538, 83539, 83810, 83811, 83827, 83828, 88434, 88435, 88451, 88452, 88723, 88724, 88740, 88741, 1419857, 1419858, 1419874, 1419875

Offset: 0

Philippe Deléham, Oct 14 2011

Numbers whose set of base 17 digits is {0,1}.
Sums of distinct powers of 17.
a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 17 2011.

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. A001026, A104257.

[n: n in [0..1500000] | Set(IntegerToSequence(n, 17)) subset {0, 1}]; // Vincenzo Librandi, Jun 05 2012

Take[Union[Total/@Subsets[17^Range[0,20],5]],40] (* Harvey P. Dale, Dec 17 2011 *)
FromDigits[#,17]&/@Tuples[{0,1},5] (* Vincenzo Librandi, Jun 05 2012 *)

a(n)=Sum_k>=0 {A030308(n,k)*17^k}.

G.f.: (1/(1 - x))*Sum_{k>=0} 17^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

A197352 a(0)=0, a(1)=1, a(2n)=18*a(n), a(2n+1)=a(2n)+1.

Original entry on oeis.org

0, 1, 18, 19, 324, 325, 342, 343, 5832, 5833, 5850, 5851, 6156, 6157, 6174, 6175, 104976, 104977, 104994, 104995, 105300, 105301, 105318, 105319, 110808, 110809, 110826, 110827, 111132, 111133, 111150, 111151, 1889568, 1889569, 1889586, 1889587

Offset: 0

Philippe Deléham, Oct 14 2011

Numbers whose set of base 18 digits is {0,1}.

Sums of distinct powers of 18.

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. A001027, A104257, A197351.

[n: n in [0..2000000] | Set(IntegerToSequence(n, 18)) subset {0, 1}]; // Vincenzo Librandi, Jun 05 2012

FromDigits[#,18]&/@Tuples[{0,1},5] (* Vincenzo Librandi, Jun 05 2012 *)

a(n) = Sum_{k>=0} A030308(n,k)*18^k.

G.f.: (1/(1 - x))*Sum_{k>=0} 18^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

A197353 a(0)=0, a(1)=1, a(2n)=19*a(n), a(2n+1)=a(2n)+1.

Original entry on oeis.org

0, 1, 19, 20, 361, 362, 380, 381, 6859, 6860, 6878, 6879, 7220, 7221, 7239, 7240, 130321, 130322, 130340, 130341, 130682, 130683, 130701, 130702, 137180, 137181, 137199, 137200, 137541, 137542, 137560, 137561, 2476099, 2476100, 2476118, 2476119

Offset: 0

Philippe Deléham, Oct 14 2011

Numbers whose set of base 19 digits is {0,1}.
Sums of distinct powers of 19.
a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 17 2011

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. A001029, A010060, A104257, A030308, A197352.

[n: n in [0..2500000] | Set(IntegerToSequence(n, 19)) subset {0, 1}]; // Vincenzo Librandi, Jun 05 2012

FromDigits[#,19]&/@Tuples[{0,1},5] (* Vincenzo Librandi, Jun 05 2012 *)

a(n) = Sum_{k>=0} A030308(n,k)*19^k.

G.f.: (1/(1 - x))*Sum_{k>=0} 19^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

A356274 a(n) is the number whose base-(n+1) expansion equals the binary expansion of n.

Original entry on oeis.org

1, 3, 5, 25, 37, 56, 73, 729, 1001, 1342, 1741, 2366, 2941, 3615, 4369, 83521, 104977, 130340, 160021, 194922, 234741, 280393, 332377, 406250, 474553, 551151, 636637, 732511, 837901, 954304, 1082401, 39135393, 45435425, 52521910, 60466213, 69345326, 79236613

Offset: 1

Thomas Scheuerle, Aug 02 2022

If the binary expansion of n is n = bit0*2^0 + bit1*2^1 + bit2*2^2 + ... then a(n) = bit0*(n+1)^0 + bit1*(n+1)^1 + bit2*(n+1)^2 + ... . In other words: Interpret the binary expansion of n as digits in base n+1.

			n=4 in binary is 100 and interpreting those digits as base n+1 = 5 is a(4) = 25.

Cf. A000523, A007814, A104257, A104258, A128889, A136516.

a[n_] := FromDigits[IntegerDigits[n, 2], n + 1]; Array[a, 40] (* Amiram Eldar, Aug 19 2022 *)

PARI
```
a(n) = fromdigits(digits(n, 2), n+1)
```

def a(n): return sum((n+1)**i*int(bi) for i, bi in enumerate(bin(n)[2:][::-1]))
print([a(n) for n in range(1, 39)]) # Michael S. Branicky, Aug 02 2022

a(2^n) = (2^n + 1)^n = A136516(n).
a(2^n - 1) = (2^(n^2) - 1)/(2^n - 1) = A128889(n).
a(2^n + 1) = 1 + (2^n + 2)^n.
a(n) = A104257(n+1, n).
a(n) = (1/n)*Sum_{j>=1} floor((n + 2^(j-1))/2^j) * ((n-1)*(n+1)^(j-1) + 1).
a(n) = (1/n)*Sum_{j=1..n} ((n-1)*(n+1)^A007814(j) + 1).
a(2*n) = A104258(2*n+1) - 1.
(2*m+1)^n divides a((2*m+1)^n-1) for positive m and n > 0.

cp's OEIS Frontend

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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A197351 a(0)=0, a(1)=1, a(2n)=17*a(n), a(2n+1)=a(2n)+1.

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A197352 a(0)=0, a(1)=1, a(2n)=18*a(n), a(2n+1)=a(2n)+1.

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A197353 a(0)=0, a(1)=1, a(2n)=19*a(n), a(2n+1)=a(2n)+1.

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