A322607 -id:A322607 - cp's OEIS frontend

A322608 Values of k such that (product of squarefree numbers <= k) / (sum of squarefree numbers <= k) is an integer.

1, 3, 11, 14, 17, 21, 23, 33, 34, 37, 46, 47, 55, 58, 59, 61, 62, 67, 69, 73, 82, 83, 87, 94, 95, 97, 101, 106, 107, 109, 114, 115, 119, 123, 127, 133, 134, 141, 146, 151, 157, 158, 159, 161, 165, 166, 173, 181, 187, 197, 202, 203, 210, 218, 219, 223, 226, 230

Offset: 1

Paolo P. Lava, Dec 20 2018

			3 is in the sequence because (1*2*3)/(1+2+3) = 1.
11 is in the sequence because (1*2*3*5*6*7*10*11)/(1+2+3+5+6+7+10+11) = 138600/45 = 3080.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A051838, A111059, A116536, A141092, A173143, A322607 (values of the quotient), A347690 (numbers of terms in the numerators).

with(numtheory): P:=proc(q) local a,b,c,n; a:=1; b:=0; c:=[];
for n from 1 to q do if issqrfree(n) then a:=a*n; b:=b+n;
if frac(a/b)=0 then c:=[op(c),n];
fi; fi; od; op(c); end: P(60);

seq = {}; sum = 0; prod = 1; Do[If[SquareFreeQ[n], sum += n; prod *= n; If[Divisible[prod, sum], AppendTo[seq, n]]], {n, 1, 230}]; seq (* Amiram Eldar, Mar 05 2021 *)

Definition corrected by N. J. A. Sloane, Sep 19 2021 at the suggestion of Harvey P. Dale.

A347690 Values of k such that (product of first k squarefree numbers) / (sum of first k squarefree numbers) is an integer.

Original entry on oeis.org

1, 3, 8, 10, 12, 14, 16, 21, 22, 24, 30, 31, 34, 36, 37, 38, 39, 42, 43, 46, 51, 52, 55, 59, 60, 61, 62, 66, 67, 68, 72, 73, 75, 77, 78, 82, 83, 87, 91, 93, 96, 97, 98, 99, 101, 102, 105, 110, 115, 121, 124, 125, 129, 135, 136, 139, 140, 143, 145, 147, 151

Offset: 1

N. J. A. Sloane, Sep 19 2021

nonn

a(n) is the number of terms in the numerator (or denominator) when A322609(n) is found.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005117, A111059, A173143, A322607, A322608.

Module[{nn=4000,sf},sf=Select[Range[nn],SquareFreeQ];Table[ If[ IntegerQ[ Times@@Take[sf,n]/Total[Take[sf,n]]],n,Nothing],{n,Length[sf]}]] (* Harvey P. Dale, Sep 19 2021 *)

More terms from Harvey P. Dale, Sep 19 2021

cp's OEIS Frontend

A322608 Values of k such that (product of squarefree numbers <= k) / (sum of squarefree numbers <= k) is an integer.

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