A356803 a(n) = product of prohibited prime factors of A354790(n).

1, 1, 2, 6, 15, 105, 385, 2310, 6006, 102102, 277134, 6374082, 16804398, 520936338, 3038795305, 66853496710, 190275336790, 7420738134810, 17897074325130, 769574195980590, 1903683537425670, 100895227483560510, 258818192240437830, 15787909726666707630, 36475515575402393490

Offset

1,3

Comments

Let s(n) = A354790(n), a squarefree number by definition. Prime p | s(n) implies p does not divide s(n+j), 1 <= j <= n. Therefore a(n) is the product of primes p that cannot divide s(n). a(n) = product of distinct primes that divide a(j) for floor((n+1)/2) <= j <= n-1. (After N. J. A. Sloane in A355057.)

Analogous to A355057.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..585

Michael De Vlieger, Annotated plot of prime p | m at (n, pi(p)) for m = a(n) in blue and A354790(n) in red and n = 1..80.

Michael De Vlieger, Plot of prime p | m at (n, pi(p)) for m = a(n) in cyan and A354790(n) in red and n = 1..1024.

Formula

a(n) = a(n-1) * s(n-1) / s((n-1)/2), where the last operation is only carried out iff (n-1)/2 is an integer.

Example

a(1) = 1;
a(2) = 1 since s(1) = 1, and (2-1)/2 is not an integer;
a(3) = a(2) * s(2) / s((3-1)/2) = 1 * 2 / 1 = 2;
a(4) = a(3) * s(3) = 2 * 3 = 6;
a(5) = a(4) * s(4) / s((5-1)/2) = 6 * 5 / 2 = 15;
a(6) = a(5) * s(5) = 15 * 7 = 105;
a(7) = a(6) * s(6) / s((7-1)/2) = 105 * 11 / 3 = 385; etc.

Mathematica

Block[{s = Import["https://oeis.org/A354790/b354790.txt", "Data"][[1 ;; 26, -1]], ww, m = 1, t, w = 3, k = 3}, Reap[Do[m *= Times @@ FactorInteger[s[[If[# == 0, 1, #] &[i - 1]]]][[All, 1]]; If[IntegerQ[#] && # > 0, m /= Times @@ FactorInteger[s[[#]]][[All, 1]]] &[(i - 1)/2]; Sow[m], {i, Length[s] - 1}]][[-1, -1]] ]

Crossrefs

Cf. A354790, A355057.

Sequence in context: A190339 A078328 A038111 * A261726 A302775 A181993

Adjacent sequences: A356800 A356801 A356802 * A356804 A356805 A356806

Keyword

nonn

Author

Michael De Vlieger, Sep 06 2022

Status

approved