A337377 - OEIS

%I #27 Jul 08 2021 14:22:10

%S 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,

%T 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,

%U 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

%N Primorial deflation (denominator) of Doudna-tree.

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

%C                                       1

%C                                       |

%C                    ...................1...................

%C                   2                                       1

%C         3......../ \........1                   4......../ \........1

%C        / \                 / \                 / \                 / \

%C       /   \               /   \               /   \               /   \

%C      /     \             /     \             /     \             /     \

%C     5       3           2       1           9       2           8       1

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

%C etc.

%C 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.

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

%H Antti Karttunen, <a href="/A337377/b337377.txt">Table of n, a(n) for n = 0..8191</a>

%H Antti Karttunen, <a href="/A337377/a337377.txt">Data supplement: n, a(n) computed for n = 0..65537</a>

%H <a href="/index/Fo#fraction_trees">Index entries for fraction trees</a>

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

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

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

%t 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 *)

%o (PARI)

%o A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };

%o 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)};

%o A319627(n) = (A064989(n) / gcd(n, A064989(n)));

%o A337377(n) = A319627(A005940(1+n));

%Y Cf. A337376 (numerators).

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

%Y Cf. A000041, A025487, A277569.

%Y Positions of 1's: A194602.

%Y Cf. also A329886, A346097.

%K nonn,frac,look

%O 0,3

%A _Antti Karttunen_, _Michael De Vlieger_ and _Peter Munn_, Aug 25 2020