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

Author

Michael De Vlieger, Sep 06 2022

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A354790, A355057.

Programs

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]] ]

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.