A380756 a(n) is the smallest number not yet in the sequence which is coprime to n and has the same number of 0's in its binary expansion as n.

1, 5, 7, 9, 2, 11, 3, 17, 4, 19, 6, 25, 14, 13, 31, 33, 8, 35, 10, 37, 22, 21, 27, 41, 12, 43, 23, 39, 30, 29, 15, 65, 16, 67, 18, 73, 20, 49, 28, 69, 24, 71, 26, 75, 46, 45, 55, 97, 38, 77, 53, 83, 51, 79, 47, 85, 58, 57, 61, 91, 59, 95, 127, 129, 32, 131, 34, 133

Offset

1,2

Comments

a(n) is the smallest novel number coprime to n with A023416(a(n)) = A023416(n). Sequence is self inverse and conjectured to be a permutation of the positive integers.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Michael De Vlieger, Log log scatterplot of a(n), n = 2^15..2^16-1, enlarging a portion of the seemingly repetitive details of the above graph.

Formula

a(a(n)) = n for all n; a(2^k) = 2^(k+1) + 1, (k >= 1); a(2^k + 1) = 2^(k-1), (k > 1).

Example

a(1) = 1, the smallest novel number coprime to 1, and A023416(1) = A023416(1) = 0.
a(2) = 5, the smallest novel number coprime to 2, and A023416(2) = A023416(5) = 1.
a(3) = 7, the smallest novel number prime to 3, and A023416(3) = A023416(7) = 0.

Mathematica

nn = 120; c[_] := False; u = 1;
f[x_] := f[x] = DigitCount[x, 2, 0];
Reap[Do[w = f[n]; k = u;
  While[Or[c[k], ! CoprimeQ[k, n], w != f[k]], k++];
  Sow[k]; c[k] = True;
If[k == u, While[c[u], u++]], {n, nn}] ][[-1, 1]] (* Michael De Vlieger, Feb 02 2025 *)

Prog

(PARI) first(n) = {
	my(res = vector(n), c = 0);
	for(i = 1, n,
		qzeros = A023416(i);
		for(j = 1, oo,
			if(bitand(c, 1<<j) == 0 && gcd(j, i) == 1 && A023416(j) == qzeros,
				res[i] = j;
				c = bitor(c, 1<<j);
				next(2)
			);
		);
	); res
}
A023416(n) = if(n == 0, 1, 1+logint(n, 2) - hammingweight(n)) \\ David A. Corneth, Feb 02 2025, A023416 from Gheorghe Coserea, Sep 01 2015

Crossrefs

Cf. A000120, A023416.

Sequence in context: A388801 A296490 A290151 * A117031 A394838 A068456

Adjacent sequences: A380753 A380754 A380755 * A380757 A380758 A380759

Keyword

nonn

Author

David James Sycamore, Feb 01 2025

Extensions

More terms from David A. Corneth, Feb 02 2025

Status

approved