A332815 a(n) = A108548(A005940(1+n)).

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 27, 16, 13, 14, 21, 20, 35, 30, 45, 24, 49, 50, 75, 36, 125, 54, 81, 32, 11, 26, 39, 28, 65, 42, 63, 40, 91, 70, 105, 60, 175, 90, 135, 48, 169, 98, 147, 100, 245, 150, 225, 72, 343, 250, 375, 108, 625, 162, 243, 64, 17, 22, 33, 52, 55, 78, 117, 56, 77, 130, 195, 84

Offset: 0

Author

Antti Karttunen, Feb 28 2020

Keywords

• nonn
• tabf

Comments

This is variant of Doudna-sequence, A005940 and thus can be represented as a binary tree. Each child to the left is obtained by applying A332818 to the parent, and each child to the right is obtained by doubling the parent:

1

|

...................2...................

3 4

5......../ \........6 9......../ \........8

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

7 10 15 12 25 18 27 16

13 14 21 20 35 30 45 24 49 50 75 36 125 54 81 32

etc.

Note the indexing: the sequence starts with a(0)=1, as is natural for sequences based on maps from base-2 expansion to prime factorization. This is

in contrast to A005940, which for historical reasons starts from offset 1.

For any n > 1, A332893(n) gives the value of the parent node. For any n >= 1, A332894(n) gives the distance to 1, and A332899(n) gives the number of odd numbers that occur (inclusively) on the path from 1 to n.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8191
• Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A332816 (inverse permutation).

Cf. A005940, A108548, A332818, A332893, A332894, A332897, A332898, A332899.

Cf. A108546 (the left edge of the tree from 2 downward).

Programs

PARI

up_to = 26927;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n];
A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); };
A332815(n) = A108548(A005940(1+n));

Formula

a(n) = A108548(A005940(1+n)).