A337377 Primorial deflation (denominator) of Doudna-tree.

1, 1, 2, 1, 3, 1, 4, 1, 5, 3, 2, 1, 9, 2, 8, 1, 7, 5, 10, 3, 3, 1, 4, 1, 25, 9, 6, 1, 27, 4, 16, 1, 11, 7, 14, 5, 21, 5, 20, 3, 5, 3, 2, 1, 9, 2, 8, 1, 49, 25, 50, 9, 15, 3, 4, 1, 125, 27, 18, 2, 81, 8, 32, 1, 13, 11, 22, 7, 33, 7, 28, 5, 55, 21, 14, 5, 63, 10, 40, 3, 7, 5, 10, 3, 3, 1, 4, 1, 25, 9, 6, 1, 27, 4, 16, 1, 121

Offset: 0

Author

Antti Karttunen, Michael De Vlieger and Peter Munn, Aug 25 2020

Keywords

• nonn
• frac
• look

Comments

Like A005940, also this irregular table can be represented as a binary tree:

1

|

...................1...................

2 1

3......../ \........1 4......../ \........1

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

5 3 2 1 9 2 8 1

7 5 10 3 3 1 4 1 25 9 6 1 27 4 16 1

etc.

A194602 gives the positions of nodes that have value 1. They correspond to terms of A005940 that are products of primorials (A025487). The first 2^k nodes contain A000041(k+1) 1's.

a(n) is even if and only if A005940(1+n) occurs in A277569.

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 fraction trees

Crossrefs

Cf. A337376 (numerators).

A003961, A005940, A006519, A026741, A064989, A319627 are used in a formula defining this sequence.

Cf. A000041, A025487, A277569.

Positions of 1's: A194602.

Cf. also A329886, A346097.

Programs

Mathematica

Array[#2/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &@ Function[p, Times @@ Flatten@ Table[Prime[Count[Flatten[#], 0] + 1]^#[[1, 1]] &@ Take[p, -i], {i, Length[p]}]]@ Partition[Split[Join[IntegerDigits[# - 1, 2], {2}]], 2] &[# + 1] &, 96] (* Michael De Vlieger, Aug 27 2020 *)

PARI

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A337377(n) = A319627(A005940(1+n));

Formula

a(n) = A319627(A005940(1+n)).

For n >= 1, a(2*n) = A003961(a(n)) * A006519(n+1).

a(2*n+1) = A026741(a(n)).