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A356803
a(n) = product of prohibited prime factors of A354790(n).
2
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, 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
Sequence in context: A190339 A078328 A038111 * A375522 A376051 A261726
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Sep 06 2022
STATUS
approved