A065880 -id:A065880 - cp's OEIS frontend

A272760 a(n) = A065880(n)/n.

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

Offset: 1

Peter Kagey, May 05 2016

This sequence is bounded above by A272756.

a(n) is the greatest value i such that A000120(n * i) = i or 0 if no such value exists.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A065880, A272756, A272759.

Table[SelectFirst[Reverse@ Range@ #, First@ DigitCount[#, 2] == #/n &] &[n SelectFirst[Range[2^12], # > IntegerLength[n #, 2] &]]/n, {n, 120}] /. k_ /; MissingQ@ k -> 0 (* Michael De Vlieger, May 05 2016, Version 10.2 *)

A272761 List of indices i such that A065879(i) != A065880(i).

Original entry on oeis.org

23, 27, 37, 41, 46, 53, 54, 57, 69, 71, 74, 77, 79, 81, 82, 83, 85, 89, 91, 92, 101, 105, 106, 107, 108, 113, 114, 121, 125, 133, 135, 138, 139, 141, 142, 143, 147, 148, 153, 154, 158, 161, 162, 163, 164, 166, 167, 169, 170, 173, 177, 178, 181, 182, 184, 197

Offset: 1

Peter Kagey, May 06 2016

Equivalently a list of indices i such that A272759(i) != A272760(i).

Also, a list of indices i such that A065413(i) > 1.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A065879, A065880, A065413, A272759, A272760.

A065878 Numbers which are not an integer multiple of their number of binary 1's.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 65, 67, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 97, 98, 99, 100

Offset: 1

Henry Bottomley, Nov 26 2001

			5 is in the sequence since 5 = 101_2 and 5 is not a multiple of 1 + 0 + 1 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Complement of A049445.
The base-10 equivalent is A065877.
Cf. A000120, A058898, A065413, A065879, A065880.

Select[Range[100],!IntegerQ[#/Total[IntegerDigits[#,2]]]&]  (* Harvey P. Dale, Apr 20 2011 *)

isok(k) = k % hammingweight(k); \\ Amiram Eldar, Aug 04 2025

A065879 a(n) is the smallest positive number that is n times the number of 1's in its binary expansion, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 6, 4, 10, 12, 21, 8, 18, 20, 55, 24, 0, 42, 60, 16, 34, 36, 0, 40, 126, 110, 69, 48, 0, 0, 81, 84, 116, 120, 155, 32, 66, 68, 0, 72, 185, 0, 156, 80, 205, 252, 172, 220, 180, 138, 0, 96, 0, 0, 204, 0, 212, 162, 0, 168, 228, 232, 295, 240, 366, 310, 378, 64, 130

Offset: 1

Henry Bottomley, Nov 26 2001

a(n) is bounded above by n*A272756(n), so a program only has to check values up to that point to see if a(n) is zero. - Peter Kagey, May 05 2016

			a(23) is 69 since 69 is written in binary as 1000101, 69/(1+0+0+0+1+0+1)=23 and there is no smaller possibility (neither 23 nor 46 are divisible by their number of binary 1's).

Peter Kagey, Table of n, a(n) for n = 1..10000

A003634 is the base-10 equivalent.

Cf. A000120, A049445, A058898, A065413, A065878, A065880, A272756.

Table[SelectFirst[Range[2^12], # == n First@ DigitCount[#, 2] &] /. k_ /; MissingQ@ k -> 0, {n, 80}] (* Michael De Vlieger, May 05 2016, Version 10.2 *)

A277249 Largest number that is n times sum of its digits when written in base 3.

Original entry on oeis.org

0, 2, 8, 6, 16, 25, 24, 35, 32, 18, 80, 77, 48, 78, 56, 75, 64, 0, 72, 152, 160, 105, 132, 115, 96, 150, 156, 54, 224, 232, 240, 186, 0, 231, 204, 210, 144, 296, 228, 234, 320, 287, 168, 172, 0, 225, 368, 376, 192, 392, 400, 0, 312, 371, 216, 440, 448, 456

Offset: 0

Daniel Mondot, Oct 06 2016

Daniel Mondot, Table of n, a(n) for n = 0..10000

Cf. A058899, A065880, A277250, A277291, A052489.

a(n) = 0 for n in A058899. - Michel Marcus, Oct 09 2016

A277250 Largest number that is n times sum of its digits when written in base 4.

Original entry on oeis.org

0, 3, 6, 9, 12, 30, 18, 63, 24, 54, 60, 33, 36, 91, 126, 90, 48, 102, 108, 190, 120, 189, 154, 207, 72, 250, 234, 243, 252, 0, 180, 248, 96, 198, 204, 315, 216, 148, 228, 351, 240, 0, 378, 430, 132, 270, 506, 423, 144, 490, 300, 459, 364, 477, 486, 440, 504

Offset: 0

Daniel Mondot, Oct 06 2016

Daniel Mondot, Table of n, a(n) for n = 0..10000

Cf. A058900, A065880, A277249, A277291, A052489.

a(n) = 0 for n in A058900. - Michel Marcus, Oct 09 2016

A277291 Largest number that is n times sum of its digits when written in base 5.

Original entry on oeis.org

0, 4, 8, 24, 16, 20, 48, 42, 64, 99, 40, 88, 96, 117, 112, 120, 0, 102, 144, 76, 80, 189, 0, 184, 192, 100, 208, 324, 0, 348, 240, 372, 128, 363, 272, 210, 288, 444, 304, 624, 320, 574, 336, 258, 352, 495, 0, 564, 384, 588, 200, 612, 416, 583, 432, 440, 0, 570

Offset: 0

Daniel Mondot, Oct 09 2016

Daniel Mondot, Table of n, a(n) for n = 0..37057

Cf. A058901, A065880, A277249, A277250, A052489.

a(n) = 0 for n in A058901. - Michel Marcus, Oct 09 2016

cp's OEIS Frontend

A272760 a(n) = A065880(n)/n.

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A272761 List of indices i such that A065879(i) != A065880(i).

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A065878 Numbers which are not an integer multiple of their number of binary 1's.

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PARI

A065879 a(n) is the smallest positive number that is n times the number of 1's in its binary expansion, or 0 if no such number exists.

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A277249 Largest number that is n times sum of its digits when written in base 3.

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A277250 Largest number that is n times sum of its digits when written in base 4.

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A277291 Largest number that is n times sum of its digits when written in base 5.

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