A097945 -id:A097945 - cp's OEIS frontend

A349694 Dirichlet convolution of the squarefree kernel function (A007947) with itself.

1, 4, 6, 8, 10, 24, 14, 12, 15, 40, 22, 48, 26, 56, 60, 16, 34, 60, 38, 80, 84, 88, 46, 72, 35, 104, 24, 112, 58, 240, 62, 20, 132, 136, 140, 120, 74, 152, 156, 120, 82, 336, 86, 176, 150, 184, 94, 96, 63, 140, 204, 208, 106, 96, 220, 168, 228, 232, 118, 480

Offset: 1

Ilya Gutkovskiy, Nov 25 2021

Cf. A007947, A097945, A176345, A191750.

Table[Sum[Last[Select[Divisors[d], SquareFreeQ]] Last[Select[Divisors[n/d], SquareFreeQ]], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := (e - 1)*p^2 + 2*p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Nov 25 2021 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A349694(n) = sumdiv(n,d,A007947(n/d)*A007947(d)); \\ Antti Karttunen, Nov 25 2021

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + p^(1-s) - p^(-s))^2.
a(n) = Sum_{d|n} A007947(d) * A007947(n/d).
a(n) = Sum_{d|n} abs(A097945(d)) * A191750(n/d).
Multiplicative with a(p^e) = (e-1)*p^2 + 2*p. - Amiram Eldar, Nov 25 2021
From Vaclav Kotesovec, Nov 26 2021: (Start)
Dirichlet g.f.: zeta(s-1)^2 * zeta(s)^2 * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s))^2.
Let f(s) = Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)), then
Sum_{k=1..n} a(k) ~ Pi^2 * f(2)^2 * n^2 / 144 * (Pi^2 * (2*log(n) + 4*gamma - 1 + 4*f'(2)/f(2)) + 24*zeta'(2)), where f(2) = Product_{primes p} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298513355936144..., f'(2) = f(2) * Sum_{primes p} log(p) * (3*p - 2) / (p^3 - 2*p + 1) = 0.6293283828324697510445630056425352981207558777167836747744750359407300858..., zeta'(2) = -A073002 and gamma is the Euler-Mascheroni constant A001620. (End)

A097927 If a(n-1) is divisible by 2 then a(n) = a(n-1)/2. If a(n-1) is not divisible by 2 then a(n) = 47-(a(n-1)+1)/2; a(0)=1.

Original entry on oeis.org

1, 46, 23, 35, 29, 32, 16, 8, 4, 2, 1, 46, 23, 35, 29, 32, 16, 8, 4, 2, 1, 46, 23, 35, 29, 32, 16, 8, 4, 2, 1, 46, 23, 35, 29, 32, 16, 8, 4, 2, 1, 46, 23, 35, 29, 32, 16, 8, 4, 2, 1, 46, 23, 35, 29, 32, 16, 8, 4, 2, 1, 46, 23, 35, 29, 32, 16, 8, 4, 2, 1, 46, 23, 35, 29, 32, 16, 8, 4, 2, 1, 46, 23

Offset: 0

N. J. A. Sloane, Sep 25 2008, based on email from Zak Seidov, May 12 2008

nonn

The old entry with this sequence number was a duplicate of A097945.

Periodic, with period = 10. - Harvey P. Dale, Sep 30 2016

a:=proc(n) option remember; local k; k:=47;
if n=0 then RETURN(1); fi;
if a(n-1) mod 2 = 0 then RETURN(a(n-1)/2); fi;
RETURN(k-(a(n-1)+1)/2); end;

NestList[If[EvenQ[#],#/2,47-(#+1)/2]&,1,90] (* or *) PadRight[{},90,{1,46,23,35,29,32,16,8,4,2}] (* Harvey P. Dale, Sep 30 2016 *)

Periodic with period of length 10.

Definition corrected by Harvey P. Dale, Sep 30 2016

A127475 Triangle T(n,k) read by rows: T(n,k) = mu(n)*phi(k) if k|n, else T(n,k)=0.

Original entry on oeis.org

1, -1, -1, -1, 0, -2, 0, 0, 0, 0, -1, 0, 0, 0, -4, 1, 1, 2, 0, 0, 2, -1, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 4, 0, 0, 0, 0, 4, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12

Offset: 1

Gary W. Adamson, Jan 15 2007

			The triangle starts in row n=1 as:
1;
-1, -1;
-1, 0, -2;
0, 0, 0, 0;
-1, 0, 0, 0, -4;
1, 1, 2, 0, 0, 2;
-1, 0, 0, 0, 0, 0, -6;
...

Cf. A008683, A054522, A055615 (row sums), A097945, A023900.

T(n,k) = A054522(n,k)*A008683(n).
T(n,1) = A008683(n) = mu(n).
T(n,n) = A097945(n).

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A349694 Dirichlet convolution of the squarefree kernel function (A007947) with itself.

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A097927 If a(n-1) is divisible by 2 then a(n) = a(n-1)/2. If a(n-1) is not divisible by 2 then a(n) = 47-(a(n-1)+1)/2; a(0)=1.

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A127475 Triangle T(n,k) read by rows: T(n,k) = mu(n)*phi(k) if k|n, else T(n,k)=0.

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