A033046 -id:A033046 - cp's OEIS frontend

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

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.

A077723 Primes which can be expressed as sum of distinct powers of 9.

Original entry on oeis.org

739, 811, 6571, 59779, 65701, 532261, 538093, 591301, 597133, 597781, 4783699, 4789621, 4842109, 4849399, 5314411, 5314501, 5373469, 5374279, 5380831, 43047541, 43112341, 43113061, 43643773, 43643863, 47837071, 47888821

Offset: 1

Amarnath Murthy, Nov 19 2002

nonn

Primes whose base 9 representation contains only zeros and 1's.

Zak Seidov, Table of n, a(n) for n = 1..3000

Cf. A020449, A000695, A033046, A077717, A077718, A077719, A077720, A077721, A077722.

Select[Prime[Range[3000000]],Union[Most[Rest[DigitCount[#,9]]]]=={0}&] (* Harvey P. Dale, Jul 31 2013 *)

lista(nn) = {forprime(p=2, nn, if (vecmax(digits(p, 9)) <= 1, print1(p, ", ")););} \\ Michel Marcus, Oct 10 2014

More terms from Sascha Kurz, Jan 03 2003

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

A077724 a(n) = smallest prime which can be expressed as a sum of distinct powers of n.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 73, 739, 11, 11, 13, 13, 197, 241, 17, 17, 19, 19, 401, 463, 23, 23, 577, 10171901, 677, 757, 29, 29, 31, 31, 32801, 1123, 1336337, 44101, 37, 37, 1483, 59359, 41, 41, 43, 43, 85229, 93151, 47, 47, 110641, 13847169701, 2551, 345157903, 53, 53

Offset: 2

Amarnath Murthy, Nov 19 2002

nonn

a(n) = smallest prime whose base n representation contains only zeros and 1's.

Values of n at which a(n) reach record values are: 2, 3, 4, 6, 8, 9, 25, 49, 91, 121, 187, 201, 301, 721, 799, 841... Notably, many of them are squares of primes. - Ivan Neretin, Sep 20 2017

Ivan Neretin, Table of n, a(n) for n = 2..10000

Cf. A020449, A000695, A033046, A077717, A077718, A077719, A077720, A077721, A077722, A077723.

Table[i = p = 1; While[! PrimeQ[p], p = FromDigits[IntegerDigits[i++, 2], n]]; p, {n, 2, 53}] (* Ivan Neretin, Sep 20 2017 *)

from itertools import count
from sympy import isprime
def A077724(n): return next(filter(isprime,(sum(n**i for i, j in enumerate(bin(m)[-1:1:-1]) if j=='1') for m in count(1)))) # Chai Wah Wu, Apr 04 2025

More terms from Sascha Kurz, Jan 03 2003

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.

A341907 T(n, k) is the result of replacing 2^e with k^e in the binary expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 2, 2, 1, 0, 1, 1, 3, 3, 1, 0, 0, 2, 4, 4, 4, 1, 0, 1, 2, 5, 9, 5, 5, 1, 0, 0, 3, 6, 10, 16, 6, 6, 1, 0, 1, 1, 7, 12, 17, 25, 7, 7, 1, 0, 0, 2, 8, 13, 20, 26, 36, 8, 8, 1, 0, 1, 2, 9, 27, 21, 30, 37, 49, 9, 9, 1, 0, 0, 3, 10, 28, 64, 31, 42, 50, 64, 10, 10, 1, 0

Offset: 0

Rémy Sigrist, Jun 04 2021

For any n >= 0, the n-th row, k -> T(n, k), corresponds to a polynomial in k with coefficients in {0, 1}.

For any k > 1, the k-th column, n -> T(n, k), corresponds to sums of distinct powers of k.

			Array T(n, k) begins:
  n\k|  0  1   2   3   4    5    6    7    8    9    10    11    12
  ---+-------------------------------------------------------------
    0|  0  0   0   0   0    0    0    0    0    0     0     0     0
    1|  1  1   1   1   1    1    1    1    1    1     1     1     1
    2|  0  1   2   3   4    5    6    7    8    9    10    11    12
    3|  1  2   3   4   5    6    7    8    9   10    11    12    13
    4|  0  1   4   9  16   25   36   49   64   81   100   121   144
    5|  1  2   5  10  17   26   37   50   65   82   101   122   145
    6|  0  2   6  12  20   30   42   56   72   90   110   132   156
    7|  1  3   7  13  21   31   43   57   73   91   111   133   157
    8|  0  1   8  27  64  125  216  343  512  729  1000  1331  1728
    9|  1  2   9  28  65  126  217  344  513  730  1001  1332  1729
   10|  0  2  10  30  68  130  222  350  520  738  1010  1342  1740
   11|  1  3  11  31  69  131  223  351  521  739  1011  1343  1741
   12|  0  2  12  36  80  150  252  392  576  810  1100  1452  1872

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

See A342707 for a similar sequence.

Cf. A000035, A000120, A000583, A000695, A001093, A002061, A002378, A002522, A002523, A005836, A007088, A011379, A027444, A033042, A033043, A033044, A033045, A033046, A033047, A033048, A033049, A034262, A053698, A071568, A098547, A104258.

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

T(n, n) = A104258(n).
T(n, 0) = A000035(n).
T(n, 1) = A000120(n).
T(n, 2) = n.
T(n, 3) = A005836(n).
T(n, 4) = A000695(n).
T(n, 5) = A033042(n).
T(n, 6) = A033043(n).
T(n, 7) = A033044(n).
T(n, 8) = A033045(n).
T(n, 9) = A033046(n).
T(n, 10) = A007088(n).
T(n, 11) = A033047(n).
T(n, 12) = A033048(n).
T(n, 13) = A033049(n).
T(0, k) = 0.
T(1, k) = 1.
T(2, k) = k.
T(3, k) = k + 1.
T(4, k) = k^2.
T(5, k) = k^2 + 1 = A002522(k).
T(6, k) = k^2 + k = A002378(k).
T(7, k) = k^2 + k + 1 = A002061(k).
T(8, k) = k^3.
T(9, k) = k^3 + 1 = A001093(k).
T(10, k) = k^3 + k = A034262(k).
T(11, k) = k^3 + k + 1 = A071568(k).
T(12, k) = k^3 + k^2 = A011379(k).
T(13, k) = k^3 + k^2 + 1 = A098547(k).
T(14, k) = k^3 + k^2 + k = A027444(k).
T(15, k) = k^3 + k^2 + k + 1 = A053698(k).
T(16, k) = k^4 = A000583(k).
T(17, k) = k^4 + 1 = A002523(k).
T(m + n, k) = T(m, k) + T(n, k) when m AND n = 0 (where AND denotes the bitwise AND operator).

A037414 Positive numbers having the same set of digits in base 2 and base 9.

Original entry on oeis.org

1, 9, 81, 82, 90, 729, 730, 738, 739, 810, 811, 819, 6561, 6562, 6570, 6571, 6642, 6643, 6651, 6652, 7290, 7291, 7299, 7300, 7371, 7372, 7380, 59049, 59050, 59058, 59059, 59130, 59131, 59139, 59140, 59778, 59779, 59787, 59788, 59859, 59860, 59868, 59869

Offset: 1

Clark Kimberling

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A033046.

isok(n) = Set(binary(n)) == Set(digits(n, 9)); \\ Michel Marcus, Jan 21 2017

Name edited by John Cerkan, Jan 20 2017

a(28)-a(43) from John Cerkan, Jan 20 2017

A190598 Maximal digit in base-9 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 2, 2, 2, 3, 4, 5, 6, 7, 8, 3, 3, 3, 3, 4, 5, 6, 7, 8, 4, 4, 4, 4, 4, 5, 6, 7, 8, 5, 5, 5, 5, 5, 5, 6, 7, 8, 6, 6, 6, 6, 6, 6, 6, 7, 8, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 2, 2, 2, 3, 4, 5

Offset: 0

N. J. A. Sloane, May 13 2011

Cf. A007095 (base 9), A033046 (indices of 1's), A338090 (indices of 8's).

Cf. A054055 (maximal digit in decimal).

a[n_] := Max[IntegerDigits[n, 9]]; (* Matej Veselovac, Jul 23 2021 *)

a(n) = if (n, vecmax(digits(n, 9)), 0); \\ Michel Marcus, Jul 19 2020

from sympy.ntheory.digits import digits
def a(n): return max(digits(n, 9)[1:])
print([a(n) for n in range(105)]) # Michael S. Branicky, Jul 23 2021

From Matej Veselovac, Jul 23 2021: (Start)
a(n) = 1 iff n is in A033046.
a(n) = 8 iff n is in A338090. (End)

cp's OEIS Frontend

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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A077723 Primes which can be expressed as sum of distinct powers of 9.

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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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Maple

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A077724 a(n) = smallest prime which can be expressed as a sum of distinct powers of n.

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

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Magma

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A341907 T(n, k) is the result of replacing 2^e with k^e in the binary expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

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A037414 Positive numbers having the same set of digits in base 2 and base 9.

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