

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_jth prime.


13



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
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OFFSET

1,3


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 k1 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_jth 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.


LINKS

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


FORMULA

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


EXAMPLE

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.


MAPLE

p:=1: for ind to 1000 do p:=nextprime(p); primeindex[p]:=ind; od:
# so primeindex[p]:=k if p is the kth 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] (* JeanFrançois Alcover, May 05 2019 *)


PROG

(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


CROSSREFS

Cf. A289506, A289509.
Cf. A056239, where the same encoding for integer multisets ('Heinz encoding') is used.
Sequence in context: A299995 A113167 A036014 * A076228 A317050 A243970
Adjacent sequences: A289504 A289505 A289506 * A289508 A289509 A289510


KEYWORD

nonn,look


AUTHOR

Christopher J. Smyth, Jul 07 2017


STATUS

approved



