A289507 The sum of squares of the elements of a finite multiset of positive integers divided by their gcd, the multiset {s_j} being indexed by n = Product_j p_{s_j}, where p_{s_j} is the s_j-th prime.

0, 1, 2, 2, 3, 5, 4, 3, 4, 10, 5, 6, 6, 17, 13, 4, 7, 9, 8, 11, 10, 26, 9, 7, 6, 37, 6, 18, 10, 14, 11, 5, 29, 50, 25, 10, 12, 65, 20, 12, 13, 21, 14, 27, 17, 82, 15, 8, 8, 19, 53, 38, 16, 13, 34, 19, 34, 101, 17, 15, 18, 122, 12, 6, 15, 30, 19, 51

Offset: 1

Author

Christopher J. Smyth, Jul 07 2017

Keywords

• nonn
• look

Comments

Given an integer linear equation Sum_{j=1..k} e_j x_j = 0, a(n) is also the modulus of the determinant whose first row is e_1, e_2, ..., e_k and whose other k-1 rows form an integral basis for the integer solution space of the equation. Here n = Product_j p_{e_j}, where p_{e_j} is the e_j-th prime.

For the proof, see Links.

Also a(n) = A289506(n) when gcd_j e_j = 1, which occurs for the numbers n in A289509.

Examples

			For n = 63 = 3^2*7 = p_2*p_2*p_4, the corresponding multiset is {2,2,4}, and a(63) = (2^2 + 2^2 + 4^2)/2 = 12. Also the relevant determinant is Det([[2,2,4],[-1,1,0],[-2,0,1]]) = 12.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• Christopher J. Smyth, A determinant associated to an integer linear equation

Crossrefs

Cf. A289506, A289509.

Cf. A056239, where the same encoding for integer multisets ('Heinz encoding') is used.

Programs

Maple

p:=1: for ind to 1000 do p:=nextprime(p); primeindex[p]:=ind; od:
# so primeindex[p]:=k if p is the k-th prime
out:=[0]: for n from 2 to 100 do f:=ifactors(n)[2];
m:=[];g:=0; for k to nops(f) do pow:=f[k]; ind:=primeindex[pow[1]];g:=gcd(g,ind); for e to pow[2] do
m:=[op(m), ind]; od; od; out:=[op(out), sum(m[jj]^2, jj=1..nops(m))/g];
od:print(out);
# second Maple program:
with(numtheory):
a:= n-> (l-> add(i[1]^2*i[2], i=l)/`if`(n=1, 1, igcd(seq(i[1],
         i=l))))(map(i-> [pi(i[1]), i[2]], ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 05 2017

Mathematica

a[n_] := Module[{m}, m = Table[{p, e} = pe; Table[PrimePi[p], {e}], {pe, FactorInteger[n]}] // Flatten; (m.m)/GCD @@ m]; a[1] = 0; Array[a, 80] (* Jean-François Alcover, May 05 2019 *)

PARI

a(n) = if (n==1, 0, my(f=factor(n)); sum(k=1, #f~, f[k,2]*primepi(f[k,1])^2) /gcd(apply(x->primepi(x), f[,1]))); \\ Michel Marcus, Jul 19 2017

Formula

a(n) = (Sum_j e_j^2)/gcd_j(e_j), where n = Product_j p_{e_j}.