A355374 -id:A355374 - cp's OEIS frontend

A355482 a(1) = 2; for n > 1, a(n) is the smallest positive number that has not yet appeared such that the number of 1-bits in the binary expansion of a(n) equals the number of proper divisors of a(n-1).

2, 4, 3, 8, 7, 16, 15, 11, 32, 31, 64, 63, 47, 128, 127, 256, 255, 191, 512, 511, 13, 1024, 1023, 223, 2048, 2047, 14, 19, 4096, 4095, 8388607, 21, 22, 25, 5, 8192, 8191, 16384, 16383, 239, 32768, 32767, 247, 26, 28, 55, 35, 37, 65536, 65535, 49151, 38, 41, 131072, 131071, 262144, 262143

Offset: 1

Scott R. Shannon, Jul 03 2022

This sequence is similar to A355374 but the rules for determining a(n) are reversed. The only fixed point in the first 145 terms is a(3) = 3. It is unknown if all numbers eventually appear. The last known term is a(145) which is a 154 digit number whose complete factorization is unknown.

			a(7) = 15 = 1111_2 as a(6) = 16 which has four proper divisors, and 15 is the smallest unused number that has four 1-bits in its binary expansion.

Scott R. Shannon, Table of n, a(n) for n = 1..145

Cf. A355483 (all divisors), A355374, A000120, A032741, A005179, A027751.

A355483 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared such that the number of 1-bits in the binary expansion of a(n) equals the number of divisors of a(n-1).

Original entry on oeis.org

1, 2, 3, 5, 6, 15, 23, 9, 7, 10, 27, 29, 12, 63, 95, 30, 255, 383, 17, 18, 111, 39, 43, 20, 119, 45, 123, 46, 51, 53, 24, 447, 54, 479, 33, 57, 58, 60, 4095, 16777215, 79228162514264337593543950335

Offset: 1

Scott R. Shannon, Jul 03 2022

This sequence is similar to A355482 except that here all divisors of a(n-1) are counted.
The fixed points in the first 41 terms are 1,2,3,10.
It is unknown if all numbers eventually appear.
Since a(41) has 6144 divisors, a(42) = 2^6144 - 1  is a 1850-digit number.

			a(7) = 23 = 10111_2 as a(6) = 15 which has four divisors, and 23 is the smallest unused number that has four 1-bits in its binary expansion.

Cf. A355482 (proper divisors), A355374, A000120, A032741, A005179, A027751.

A355715 a(0) = 0; for n > 0, a(n) is the total number of binary bits that n has in common with all previous terms.

Original entry on oeis.org

0, 0, 2, 1, 3, 2, 7, 8, 8, 9, 16, 15, 17, 17, 18, 19, 32, 35, 39, 42, 33, 36, 40, 40, 50, 50, 57, 57, 50, 49, 53, 54, 92, 91, 94, 93, 85, 87, 89, 90, 101, 105, 106, 113, 103, 109, 108, 116, 143, 146, 144, 149, 145, 151, 146, 153, 161, 169, 161, 170, 159, 169, 158, 170, 184, 192, 187, 194, 181

Offset: 0

Scott R. Shannon, Jul 15 2022

Scott R. Shannon, Table of n, a(n) for n = 0..10000

cf. A030190, A342303, A351753, A355374.

a(1) = 0 as a(0) = 0, and 0 shares no bits in common with 1.
a(2) = 2 as a(0) = 0, a(1) = 0, and 2 = 10_2 has the 0-bit in common with both previous terms.
a(3) = 1 as a(2) = 2 = 10_2 and 3 = 11_2 shares a 1-bit in common with 2.
a(6) = 7 as a(0) = 0, a(1) = 0, a(2) = a(5) = 2 = 10_2, a(4) = 3 = 11_2 and 6 = 110_2 shares four 0-bits and three 1-bits, seven bits in all, with these previous terms.

cp's OEIS Frontend

A355482 a(1) = 2; for n > 1, a(n) is the smallest positive number that has not yet appeared such that the number of 1-bits in the binary expansion of a(n) equals the number of proper divisors of a(n-1).

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A355483 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared such that the number of 1-bits in the binary expansion of a(n) equals the number of divisors of a(n-1).

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A355715 a(0) = 0; for n > 0, a(n) is the total number of binary bits that n has in common with all previous terms.

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