A337376 Primorial deflation (numerator) of Doudna-tree.

1, 2, 3, 4, 5, 3, 9, 8, 7, 10, 5, 6, 25, 9, 27, 16, 11, 14, 21, 20, 7, 5, 15, 12, 49, 50, 25, 9, 125, 27, 81, 32, 13, 22, 33, 28, 55, 21, 63, 40, 11, 14, 7, 10, 35, 15, 45, 24, 121, 98, 147, 100, 49, 25, 25, 18, 343, 250, 125, 27, 625, 81, 243, 64, 17, 26, 39, 44, 65, 33, 99, 56, 91, 110, 55, 42, 275, 63, 189, 80, 13, 22

Offset: 0

Author

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

Keywords

• nonn
• frac
• tabf
• look

Comments

Tree with both numerators (this sequence) and denominators (A337377) shown starts as:

1/1

|

2

-

1

3 / \ 4

- ................. ................. -

2 1

5 / \ 3 9 / \ 8

- ....... ....... - - ....... ....... -

3 1 4 1

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

7 10 5 6 25 9 27 16

- -- - - -- - -- --

5 3 2 1 9 2 8 1

/ \ / \ / \ / \ / \ / \ / \ / \

11 14 21 20 7 5 15 12 49 50 25 9 125 27 81 32

-- -- -- -- - - -- -- -- -- -- - --- -- -- --

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

etc.

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

A005940, A319626, A337375 are used in a formula defining this sequence.

Cf. A064989.

Cf. A337377 (denominators).

A000265, A001222, A003961, A007814, A337821 are used to express relationship between terms of this sequence.

Cf. also A329886, A346096.

Programs

Mathematica

Array[#1/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] &, 82] (* 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)};
A319626(n) = (n / gcd(n, A064989(n)));
A337376(n) = A319626(A005940(1+n));

Formula

a(n) = A319626(A005940(1+n)).

a(n) = A005940(1+n) / A337375(n).

a(2*n) = A003961(a(n)).

If A007814(n+1) < A337821(n+1) then a(2*n+1) = a(n), otherwise a(2*n+1) = 2 * a(n).

If A337377(n) mod 2 = 0 then a(2*n+1) = a(n), otherwise a(2*n+1) = 2 * a(n).

A000265(a(2*n+1)) = A000265(a(n)).

A001222(a(2*n)) = A001222(A337377(2*n)) = A001222(a(n)).

A001222(a(2*n+1)) - A001222(A337377(2*n+1)) = 1 + A001222(a(n)) - A001222(A337377(n)).