A353955 a(n) = A019565(A353709(n)).

1, 2, 3, 5, 7, 6, 11, 35, 13, 22, 15, 91, 17, 10, 21, 143, 34, 105, 19, 26, 33, 85, 14, 39, 55, 119, 78, 95, 77, 51, 65, 154, 57, 221, 70, 209, 663, 23, 110, 273, 323, 46, 165, 1547, 38, 69, 385, 442, 437, 231, 130, 391, 133, 30, 187, 247, 42, 935, 299, 114, 595

Offset: 0

Author

Michael De Vlieger, May 12 2022

Keywords

• nonn

Comments

Interpretation of A353709(n) written in binary instead as if written in "multiplicity notation", that is, as if we write 1 if divisible by prime(k+1), otherwise 0 in the k-th place. Example, decimal 12 is written in binary as 1100 = 2^2 + 2^3, and take exponents 2 and 3 and instead construe them as prime(2+1) * prime(3+1) = 5*7 = 35.

If A353709 is a permutation of nonnegative numbers, then this sequence is a permutation of squarefree numbers A005117.

Examples

			Table showing n, A353709(n), and b(n), the binary expansion of A353709(n) writing "." for zeros for clarity. a(n) interprets 1's in the k-th place of b(n) as prime(k+1) and thereafter takes the product. We find a(n) = A005117(j).
   n A353709(n) b(n)  a(n)   j
  ----------------------------
   1    0         .     1    1
   2    1         1     2    2
   3    2        1.     3    3
   4    4       1..     5    4
   5    8      1...     7    6
   6    3        11     6    5
   7   16     1....    11    8
   8   12      11..    35   23
   9   32    1.....    13    9
  10   17     1...1    22   15
  11    6       11.    15   11
  12   40    1.1...    91   57
  13   64   1......    17   12
  14    5       1.1    10    7
  15   10      1.1.    21   14
  16   48    11....   143   89
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^14, with records in red and local minima in blue, highlighting primes in green and fixed points in gold.

Crossrefs

Cf. A005117, A019565, A353709.

Programs

Mathematica

nn = 2^7; c[_] = -1; c[0] = i = 0; a[0] = c[1] = j = 1; a[1] = u = 2; Do[k = u; While[Nand[c[k] == -1, BitAnd[i, k] == 0, BitAnd[j, k] == 0], k++]; If[k == u, While[c[u] > -1, u++]]; i = j; j = k; Set[{a[n], c[k]}, {Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ IntegerDigits[k, 2]], n}], {n, 2, nn}]; Array[a, nn + 1, 0]

Formula

a(n) = Product p_k where A353709(n) = Sum 2^(k-1).