A033042 -id:A033042 - cp's OEIS frontend

A097251 Numbers whose set of base 5 digits is {0,4}.

0, 4, 20, 24, 100, 104, 120, 124, 500, 504, 520, 524, 600, 604, 620, 624, 2500, 2504, 2520, 2524, 2600, 2604, 2620, 2624, 3000, 3004, 3020, 3024, 3100, 3104, 3120, 3124, 12500, 12504, 12520, 12524, 12600, 12604, 12620, 12624, 13000, 13004, 13020

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 5 for every i.

The first 2^n terms of the sequence could be obtained using the Cantor-like process for the segment [0,5^n-1]. For example, for n=1 we have [0, {1,2,3},4] such that numbers outside of braces are the first 2 terms of the sequence; for n=2 we have [0, {1,2,3}, 4, {5,...,19}, 20, {21,22,23}, 24] such that the numbers outside of braces are the first 4 terms of the sequence, etc. - Vladimir Shevelev, Dec 17 2012

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

Cf. A001196, A005823, A097252-A097262.

[n: n in [0..20000] | Set(IntegerToSequence(n, 5)) subset {0, 4}]; // Vincenzo Librandi, May 25 2012

fQ[n_]:=Union@Join[{0,4},IntegerDigits[n,5]]=={0,4};Select[Range[0,20000],fQ] (* Vincenzo Librandi, May 25 2012 *)
FromDigits[#,5]&/@Tuples[{0,4},6] (* Harvey P. Dale, Feb 01 2015 *)

a[0]:0$ a[n]:=5*a[floor(n/2)]+2*(1-(-1)^n)$ makelist(a[n], n, 0, 42); /* Bruno Berselli, May 25 2012 */

a(n) = 4*fromdigits(binary(n),5); \\ Kevin Ryde, Jun 03 2020

a(n) = 4*A033042(n).

a(2n) = 5*a(n), a(2n+1) = a(2n)+4.

A146025 Numbers that can be written in bases 2, 3, 4, and 5 using only the digits 0 and 1.

Original entry on oeis.org

0, 1, 82000

Offset: 1

Daniel Mondot, Oct 26 2008

Originally checked to 2^65520 (or about 3*10^19723) on Nov 07 2008. - Daniel Mondot, Jan 17 2016
Conjectured to be complete. a(4), if it exists, is greater than 10^15. - Charles R Greathouse IV, Apr 06 2012
Checked to 3125 (5^5) base-5 digits in just under 1/2 hour using a minor modification of the PARI program at A230360.  Interestingly, with 5 replaced by 9 and the digits 2 and 3 permitted, it appears the complete set is--somewhat coincidental with this--{0, 1, 2, 3, 8281, 8282, 8283}, see A146026. - James G. Merickel, Dec 01 2013
Checked to 11 million decimal digits in 1 week using an algorithm that, upon finding that the current guess has non-{0,1} digits in a particular base, increases the guess to only have {0,1} digits in that base. C code in links. - Alex P. Klinkhamer, Aug 29 2015
It is a plausible conjecture that there are no more terms, but this has not been proved. - N. J. A. Sloane, Feb 06 2016

			82000 = 10100000001010000 (2) = 11011111001 (3) = 110001100 (4) = 10111000 (5).

Stuart A. Burrell and Han Yu, Digit expansions of numbers in different bases, arXiv:1905.00832 [math.NT], 2019.
Daniel Glasscock, Joel Moreira, and Florian K. Richter, Additive transversality of fractal sets in the reals and the integers, arXiv:2007.05480 [math.NT], 2020. See p. 5.
James Grime and Brady Haran, Why 82,000 is an extraordinary number, Numberphile video (2015)
Alex P. Klinkhamer, Digits of 82000, search algorithm with code and analysis.

Intersection of A005836, A000695, and A033042.
Cf. A258981 (bases 2,3,4), A258107 (bases 2..n).
Cf. A131646, A146026, A146027, A230360, A275600.

f[n_] := Total[Total@ Drop[RotateRight[DigitCount[n, #]], 2] & /@ Range[3, 5]]; Select[Range[0, 100000], f@ # == 0 &] (* Michael De Vlieger, Aug 29 2015 *)

is(n)=vecmax(digits(n,5))<2 && vecmax(digits(n,4))<2 && vecmax(digits(n,3))<2 \\ Charles R Greathouse IV, Aug 31 2015

Edited by Charles R Greathouse IV, Nov 01 2009
Search limit extended to astronomical odds by James G. Merickel, Dec 03 2013
Search limit increased again with example code by Alex P. Klinkhamer, Aug 29 2015
Removed keywords "fini" and "full", since it is only a conjecture that there are no further terms. - N. J. A. Sloane, Feb 06 2016

A077719 Primes which can be expressed as sum of distinct powers of 5.

Original entry on oeis.org

5, 31, 131, 151, 631, 751, 3251, 3881, 16381, 19381, 19501, 19531, 78781, 78901, 81281, 81401, 81901, 82031, 93901, 94531, 97001, 97501, 97651, 390751, 390781, 393901, 394501, 406381, 468781, 469501, 471901, 472631, 484531, 485131, 487651, 1953151, 1953901

Offset: 1

Amarnath Murthy, Nov 19 2002

nonn

Primes whose base 5 representation contains only zeros and 1's.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A020449, A000695, A033042, A077717, A077718, A077720, A077721, A077722.

from sympy import isprime
def aupton(terms):
  k, alst = 0, []
  while len(alst) < terms:
    k += 1
    t = sum(5**i*int(di) for i, di in enumerate((bin(k)[2:])[::-1]))
    if isprime(t): alst.append(t)
  return alst
print(aupton(37)) # Michael S. Branicky, May 31 2021

More terms from Sascha Kurz, Jan 03 2003

a(36) and beyond from Michael S. Branicky, May 31 2021

A033048 Sums of distinct powers of 12.

Original entry on oeis.org

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

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

A277548 Numbers k such that k/5^m == 4 (mod 5), where 5^m is the greatest power of 5 that divides k.

Original entry on oeis.org

4, 9, 14, 19, 20, 24, 29, 34, 39, 44, 45, 49, 54, 59, 64, 69, 70, 74, 79, 84, 89, 94, 95, 99, 100, 104, 109, 114, 119, 120, 124, 129, 134, 139, 144, 145, 149, 154, 159, 164, 169, 170, 174, 179, 184, 189, 194, 195, 199, 204, 209, 214, 219, 220, 224, 225, 229

Offset: 1

Clark Kimberling, Oct 20 2016

Positions of 4 in A277543. Numbers that have 4 as their rightmost nonzero digit when written in base 5.
This is one sequence in a 4-way splitting of the positive integers; the other three are indicated in the Mathematica program. All these sequences have the same density of 1/4.
Is there some n with a 3 or a 4 in base 5 such that a(n) = 4n + 1? - David A. Corneth, Oct 23 2016

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A033042, A277543, A277550, A277551, A277555.

z = 200; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[5, 1] (* A277550 *)
p[5, 2] (* A277551 *)
p[5, 3] (* A277555 *)
p[5, 4] (* A277548 *)

isok(n) = n/5^valuation(n, 5) % 5 == 4; \\ Michel Marcus, Oct 21 2016

Conjecture: a(n) = 4*n if and only if n is in A033042. - David A. Corneth, Oct 23 2016

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

cp's OEIS Frontend

A097251 Numbers whose set of base 5 digits is {0,4}.

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A146025 Numbers that can be written in bases 2, 3, 4, and 5 using only the digits 0 and 1.

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A077719 Primes which can be expressed as sum of distinct powers of 5.

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A033048 Sums of distinct powers of 12.

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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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