A075926 -id:A075926 - cp's OEIS frontend

A335440 Lexicographically earliest sequence of distinct positive terms such that two distinct terms differ by at least 3 prime factors.

1, 8, 18, 50, 60, 64, 81, 98, 105, 144, 225, 242, 308, 338, 400, 429, 441, 480, 512, 546, 578, 625, 648, 722, 756, 784, 884, 935, 969, 1058, 1089, 1122, 1152, 1190, 1225, 1235, 1428, 1430, 1458, 1463, 1485, 1521, 1547, 1682, 1748, 1800, 1820, 1922, 1936, 2001

Offset: 1

Rémy Sigrist, Jun 10 2020

nonn

In other words, for any distinct m and n, let a(m)/a(n) = u/v in reduced form, then bigomega(u) + bigomega(v) >= 3 (where bigomega corresponds to A001222(n), the number of distinct prime factors of n with multiplicity).
The variant where distinct terms differ by at least 1 prime factor simply corresponds to the positive numbers.
The variant where distinct terms differ by at least 2 prime factors corresponds to A028260.
No term is prime nor the square of a prime.
This sequence has similarities with A075926 and A333568; here we consider prime factors, there digits.

			The first terms, alongside their p-adic valuations for p = 2..11 (with dots instead of zeros), are:
  n   a(n)  v2  v3  v5  v7  v11
  --  ----  --  --  --  --  ---
   1     1   .   .   .   .    .
   2     8   3   .   .   .    .
   3    18   1   2   .   .    .
   4    50   1   .   2   .    .
   5    60   2   1   1   .    .
   6    64   6   .   .   .    .
   7    81   .   4   .   .    .
   8    98   1   .   .   2    .
   9   105   .   1   1   1    .
  10   144   4   2   .   .    .
  11   225   .   2   2   .    .
  12   242   1   .   .   .    2
  13   308   2   .   .   1    1

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A335440

Cf. A001222, A028260, A075926, A333568.

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A341273 If A261283(n) = 0, then a(n) = n, otherwise a(n) is obtained by flipping the A261283(n)-th rightmost bit in the binary expansion of n.

Original entry on oeis.org

0, 0, 0, 7, 0, 7, 7, 7, 0, 25, 42, 75, 76, 45, 30, 7, 0, 25, 82, 51, 52, 85, 30, 7, 25, 25, 30, 25, 30, 25, 30, 30, 0, 97, 42, 51, 52, 45, 102, 7, 42, 45, 42, 42, 45, 45, 42, 45, 52, 51, 51, 51, 52, 52, 52, 51, 120, 25, 42, 51, 52, 45, 30, 127, 0, 97, 82, 75

Offset: 0

Rémy Sigrist, Feb 08 2021

All terms belong to A075926.

			For n = 7:
- A261283(7) = 0,
- so a(7) = 7.
For n = 43:
- A261283(43) = 1,
- so a(43) is obtained by flipping the rightmost binary digit in 43,
- a(43) = 42.

Rémy Sigrist, Table of n, a(n) for n = 0..32768

Cf. A075926, A261283.

a(n) = { my (m=n, x=0); while (m, my (v=valuation(m, 2)); x=bitxor(x, v+1); m-=2^v); if (x, bitxor(n, 2^(x-1)), n) }

A261283(a(n)) = 0.
a(a(n)) = a(n).
a(n) = n iff n belongs to A075926.
a(2^k) = 0 for any k >= 0.

A341242 Numbers whose binary representation encodes a subset S of the natural numbers such that the XOR of the binary representations of all s in S gives 0.

Original entry on oeis.org

0, 1, 14, 15, 50, 51, 60, 61, 84, 85, 90, 91, 102, 103, 104, 105, 150, 151, 152, 153, 164, 165, 170, 171, 194, 195, 204, 205, 240, 241, 254, 255, 770, 771, 780, 781, 816, 817, 830, 831, 854, 855, 856, 857, 868, 869, 874, 875, 916, 917, 922, 923, 934, 935, 936

Offset: 1

Marc A. A. van Leeuwen, Feb 07 2021

The numbers for which the set S of positions of bits 1 in the binary representation, interpreted as a set of distinct-sized Nim heaps (including a possible uninteresting size 0 heap for the least significant bit) is losing for the player to move.
Viewing the list as a set of valid code words, every natural number N can be "corrected" to a valid code word by changing exactly one bit, in exactly one way. The position of that bit is found by computing for N the XOR of its raised-bit positions of the title (if the result is 0, then N is already valid but flipping the irrelevant bit 0 makes it valid again).
The "error correcting" interpretation, applied to 64-bit numbers interpreted as orientation of 64 coins, corresponds to a solution of the "coins on a chessboard" puzzle described in the Nick Berry's blog, and also mentioned at A253315.
Numbers 2*n and 2*n+1 for n = A075926(m).
Numbers m such that A253315(m) = 0. - Rémy Sigrist, Feb 09 2021

Nick Berry, Impossible Escape?, DataGenetics blog, December 2014.

Cf. A075926, A253315.

def ok(n):
  xor, b = 0, (bin(n)[2:])[::-1]
  for i, c in enumerate(b):
    if c == '1': xor ^= i
  return xor == 0
print([m for m in range(937) if ok(m)]) # Michael S. Branicky, Feb 07 2021

a(2*n+1) = 2*A075926(n), a(2*n+2) = 2*A075926(n) + 1 for any n >= 0. - Rémy Sigrist, Feb 09 2021

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A335440 Lexicographically earliest sequence of distinct positive terms such that two distinct terms differ by at least 3 prime factors.

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A341273 If A261283(n) = 0, then a(n) = n, otherwise a(n) is obtained by flipping the A261283(n)-th rightmost bit in the binary expansion of n.

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A341242 Numbers whose binary representation encodes a subset S of the natural numbers such that the XOR of the binary representations of all s in S gives 0.

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