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%I #17 Mar 04 2020 18:08:24
%S 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,
%T 75,36,125,54,81,32,11,26,39,28,65,42,63,40,91,70,105,60,175,90,135,
%U 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
%N a(n) = A108548(A005940(1+n)).
%C 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:
%C 1
%C |
%C ...................2...................
%C 3 4
%C 5......../ \........6 9......../ \........8
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C 7 10 15 12 25 18 27 16
%C 13 14 21 20 35 30 45 24 49 50 75 36 125 54 81 32
%C etc.
%C 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
%C in contrast to A005940, which for historical reasons starts from offset 1.
%C 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.
%H Antti Karttunen, <a href="/A332815/b332815.txt">Table of n, a(n) for n = 0..8191</a>
%H Antti Karttunen, <a href="/A332815/a332815.txt">Data supplement: n, a(n) computed for n = 0..65537</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(n) = A108548(A005940(1+n)).
%o (PARI)
%o up_to = 26927;
%o A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
%o 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); };
%o v108546 = A108546list(up_to);
%o A108546(n) = v108546[n];
%o A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); };
%o A332815(n) = A108548(A005940(1+n));
%Y Cf. A332816 (inverse permutation).
%Y Cf. A005940, A108548, A332818, A332893, A332894, A332897, A332898, A332899.
%Y Cf. A108546 (the left edge of the tree from 2 downward).
%K nonn,tabf
%O 0,2
%A _Antti Karttunen_, Feb 28 2020
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