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A353955
a(n) = A019565(A353709(n)).
1
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
0,2
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.
LINKS
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.
FORMULA
a(n) = Product p_k where A353709(n) = Sum 2^(k-1).
EXAMPLE
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
...
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, May 12 2022
STATUS
approved