A032533 -id:A032533 - cp's OEIS frontend

A032534 Quotient of 'base-2' division described in A032533.

1, 5, 25, 125, 101, 625, 505, 481, 3125, 2525, 2405, 15625, 12625, 12025, 11001, 78125, 63125, 60125, 55005, 390625, 366337, 315625, 300625, 275025, 1953125, 1831685, 1578125, 1503125, 1375125, 1111101, 9765625, 9158425, 7890625

Offset: 1

Patrick De Geest, Apr 15 1998

nonn

Cf. A032532, A032533.

Array[If[Mod[#2, #1] == 0, #2/#1, Nothing] & @@ {#, FromDigits[IntegerDigits[#, 2]]} &, 10^3] (* Michael De Vlieger, Oct 06 2019 *)

Offset 1 from Michael De Vlieger, Oct 06 2019

A032532 Integer part of decimal 'base-2 looking' numbers divided by their actual base-2 values (denominator of a(n) is n, numerator is n written in binary but read in decimal).

Original entry on oeis.org

1, 5, 3, 25, 20, 18, 15, 125, 111, 101, 91, 91, 84, 79, 74, 625, 588, 556, 526, 505, 481, 459, 439, 458, 440, 423, 407, 396, 382, 370, 358, 3125, 3030, 2941, 2857, 2780, 2705, 2634, 2566, 2525, 2463, 2405, 2349, 2297, 2246, 2198, 2151, 2291

Offset: 1

Patrick De Geest, Apr 15 1998

			1_2 / 1_10 = 1/1 = 1;
10_2 / 2_10 = 10/2 = 5;
11_2 / 3_10 = 11/3 = 3.666666666...;
100_2 / 4_10 = 100/4 = 25;
101_2 / 5_10 = 101/5 = 20.2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A032533, A032534, A007088.

Array[Floor[FromDigits[IntegerDigits[#, 2]]/#] &, 48] (* Michael De Vlieger, Oct 06 2019 *)

a(n) = floor(A007088(n)/n).

A062845 When expressed in base 2 and then interpreted in base 3, is a multiple of the original number.

Original entry on oeis.org

0, 1, 5, 6, 10, 12, 30, 36, 60, 120, 180, 215, 216, 252, 360, 430, 432, 1080, 2730, 3276, 13710, 14724, 16380, 20520, 24624, 24840, 27125, 27420, 32760, 38880, 48606, 49091, 54250, 54840, 97212, 98280

Offset: 1

Erich Friedman, Jul 21 2001

The numbers 2*m, 4*m and 8*m are also terms of the sequence for m=a(122). - Dimiter Skordev, Mar 29 2020

			30 = 11110_2; 11110_3 = 120 = 4*30.

Dimiter Skordev, Table of n, a(n) for n = 1..122 (terms < 10^15, terms 1..36 from Erich Friedman, 37..111 from Dimiter Skordev, 112..120 from Giovanni Resta)
Dimiter Skordev, Pascal program
Dimiter Skordev, Python script

Cf. A062846, A062847, A062848, A062849, A062850, A032533, A062853.

Cf. A005836.

[0] cat [k:k in [1..100000]|Seqint(Intseq(Seqint(Intseq(k, 2))),3) mod k eq 0]; // Marius A. Burtea, Dec 29 2019

{0} ~Join~ Select[Range[10^5], Mod[ FromDigits[ IntegerDigits[#, 2], 3], #] == 0 &] (* Giovanni Resta, Dec 10 2019 *)

isok(m) = (m==0) || fromdigits(digits(m, 2), 3) % m == 0; \\ Michel Marcus, Feb 15 2020

def BaseUp(n,b):
    up, b1 = 0, 1
    while n > 0:
        up, b1, n = up+(n%b)*b1, b1*(b+1), n//b
    return up
n, k = 1, 0
print(1,0)
while n < 35:
    n, k = n+1, k+1
    while BaseUp(k,2)%k != 0:
        k = k+1
    print(n,k) # A.H.M. Smeets, Mar 31 2020

A062847 When expressed in base 2 and then interpreted in base 6, is a multiple of the original number.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 66, 72, 96, 128, 132, 144, 174, 186, 192, 216, 252, 256, 264, 288, 348, 372, 384, 396, 432, 462, 504, 512, 528, 576, 696, 744, 768, 792, 864, 924, 1008, 1024, 1056, 1152, 1296, 1392, 1488, 1512, 1536, 1584, 1728

Offset: 1

Erich Friedman, Jul 21 2001

There are only five odd terms of the sequence that are less than 10^7, namely 1, 12025, 233285, 863395 and 9545429. - Dimiter Skordev, Feb 02 2020

			12 in base 2 is 1100, which interpreted in base 6 is 252=21*12.

Dimiter Skordev, Table of n, a(n) for n = 1..550 (terms less than 10^7)
Dimiter Skordev, Python script

Cf. (with base 2 and b): A062845 (b=3), A062846 (b=4), A331841 (b=5), A062848 (b=7), A062849 (b=8), A062850 (b=9), A032533 (b=10).

[0] cat [k:k in [1..1750]|Seqint(Intseq(Seqint(Intseq(k, 2))), 6) mod k eq 0]; // Marius A. Burtea, Feb 02 2020

isok(n) = (n==0) || ((fromdigits(digits(n, 2), 6) % n) == 0); \\ Michel Marcus, Feb 01 2020

A331841 When expressed in base 2 and then interpreted in base 5, is a multiple of the original number.

Original entry on oeis.org

0, 1, 3, 6, 9, 10, 18, 21, 27, 30, 54, 57, 60, 63, 89, 90, 108, 114, 126, 130, 178, 180, 189, 228, 300, 356, 378, 390, 630, 712, 780, 900, 1170, 1299, 1300, 1890, 1953, 2340, 2370, 2730, 3510, 3900, 3906, 4740, 7020, 7110, 7410, 7800, 8100, 8190, 9261, 11700

Offset: 1

Dimiter Skordev, Jan 29 2020

			30 = 11110_2; 11110_5 = 780 = 26*30.

Dimiter Skordev, Table of n, a(n) for n = 1..138 (terms less than 10^7).
Dimiter Skordev, Python script

Cf. (with base 2 and b): A062845 (b=3), A062846 (b=4), A062847 (b=6), A062848 (b=7), A062849 (b=8), A062850 (b=9), A032533 (b=10).

[0] cat [k:k in [1..12000]|Seqint(Intseq(Seqint(Intseq(k, 2))), 5) mod k eq 0]; // Marius A. Burtea, Jan 29 2020

Prepend[Select[Range[12000], Divisible[FromDigits[IntegerDigits[#, 2], 5], #] &], 0] (* Amiram Eldar, Jan 29 2020 *)

isok(n) = (n == 0) || (fromdigits(digits(n, 2), 5) % n) == 0; \\ Michel Marcus, Jan 29 2020

A357111 For n >= 1, a(n) = n / A076775(n).

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 9, 1, 11, 3, 13, 7, 15, 1, 17, 9, 19, 1, 1, 11, 23, 3, 25, 13, 27, 7, 29, 3, 31, 1, 3, 17, 35, 9, 37, 19, 39, 1, 41, 1, 43, 11, 45, 23, 47, 3, 49, 5, 51, 13, 53, 27, 55, 7, 57, 29, 59, 3, 61, 31, 3, 1, 65, 3, 67, 17, 23, 7, 71, 9, 73, 37, 75

Offset: 1

Ctibor O. Zizka, Sep 11 2022

			a(12) = 12 / A076775(12) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000265, A032533, A076775, A354837.

f:= proc(n) local L,i;
  L:= convert(n,base,2);
  n/igcd(n,add(10^(i-1)*L[i],i=1..nops(L)));
end proc:
map(f, [$1..100]); # Robert Israel, Sep 11 2022

a[n_] := n / GCD[n, FromDigits@IntegerDigits[n, 2]]; Array[a, 100] (* Amiram Eldar, Sep 11 2022 *)

from math import gcd
def A357111(n): return n//gcd(n,int(bin(n)[2:])) # Chai Wah Wu, Sep 12 2022

a(n) = 1 for n = A076775(A032533(k)).
a(n) != n for n = A354837(k), a(n) = n for all other odd n.
a(n) != A000265(n) for n = 10*k and for n = 2^r * A354837(k), r >= 0, k >= 1.

A032535 Odd numbers that, when expressed in base 2 and then interpreted in base 10, yield a multiple of the original number.

Original entry on oeis.org

1, 21, 273, 2231, 10101, 28261, 611123, 1200341, 3427673, 8108919, 38636301, 51484647, 2202416417, 11102657671, 42822560781

Offset: 1

Patrick De Geest, Apr 15 1998

			273 is a term: 273_10 = 100010001_2; 100010001 = 273*366337. - _Jon E. Schoenfield_, Oct 25 2019

= odd(A032533). See also A032532 for explanation.

Select[Range[1, 10^6, 2], Mod[ FromDigits@ IntegerDigits[#, 2], #] == 0 &] (* Giovanni Resta, Jul 12 2016 *)

a(13)-a(15) from Giovanni Resta, Jul 12 2016

A174415 Numbers such that when expressed in base 2 and then interpreted in base 10, is not a multiple of the original number.

Original entry on oeis.org

3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60

Offset: 1

Jaroslav Krizek, Mar 19 2010

Complement of A032533.

			For n=5; a(5) = 9, 9 in base 2 is 1001, which interpreted in base 10, 1001 is not multiple of 9.

Select[Range[100],!Divisible[FromDigits[IntegerDigits[#,2]],#]&] (* Harvey P. Dale, Jul 10 2014 *)

A355297 a(n) = A007088(n) mod n.

Original entry on oeis.org

0, 0, 2, 0, 1, 2, 6, 0, 2, 0, 10, 8, 9, 4, 1, 0, 5, 2, 17, 0, 0, 12, 14, 8, 1, 12, 22, 12, 23, 10, 13, 0, 11, 16, 16, 20, 16, 18, 37, 0, 18, 0, 4, 32, 31, 2, 14, 32, 45, 10, 4, 16, 20, 4, 1, 8, 22, 56, 32, 40, 20, 6, 42, 0, 41, 44, 36, 24, 15, 20, 5, 56, 25, 12, 61, 28, 24, 58, 23, 0

Offset: 1

Ctibor O. Zizka, Jun 27 2022

a(n) = 0 see A032533, a(n) = 1 see A339567.

			n = 7; a(7) = A007088(7) mod 7 = 111 mod 7 = 6.

Cf. A007088, A032532, A032533, A339567.

a[n_] := Mod[FromDigits[IntegerDigits[n, 2]], n]; Array[a, 100] (* Amiram Eldar, Jun 27 2022 *)

a(n) = A007088(n) mod n. a(n) = A007088(n) - n * floor(A007088(n)/n).

A355300 a(0) = 0; for n >= 1, a(n) = a(A007088(n) mod n) + 1.

Original entry on oeis.org

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

Offset: 0

Ctibor O. Zizka, Jun 27 2022

Number of steps needed to reach zero when starting from k = n and repeatedly applying the map that replaces k by A007088(k) mod k. For a(n) = 1 see A032533.

a(95980631) = 26. - Charles R Greathouse IV, Jun 30 2022

			n = 12, a(12) = a(1100 mod 12) + 1 = a(8) + 1 = a(1000 mod 8) + 2 = a(0) + 2 = 2.

Cf. A007088, A032533.

a[n_] := a[n] = a[Mod[FromDigits[IntegerDigits[n, 2]], n]] + 1; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jun 27 2022 *)

f(n) = fromdigits(binary(n), 10); \\ A007088
a(n) = if (n, a(f(n) % n)+1, 0); \\ Michel Marcus, Jun 27 2022

cp's OEIS Frontend

A032534 Quotient of 'base-2' division described in A032533.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A032532 Integer part of decimal 'base-2 looking' numbers divided by their actual base-2 values (denominator of a(n) is n, numerator is n written in binary but read in decimal).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A062845 When expressed in base 2 and then interpreted in base 3, is a multiple of the original number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A062847 When expressed in base 2 and then interpreted in base 6, is a multiple of the original number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

PARI

A331841 When expressed in base 2 and then interpreted in base 5, is a multiple of the original number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A357111 For n >= 1, a(n) = n / A076775(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A032535 Odd numbers that, when expressed in base 2 and then interpreted in base 10, yield a multiple of the original number.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A174415 Numbers such that when expressed in base 2 and then interpreted in base 10, is not a multiple of the original number.

Views

Author

Keywords

Comments