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A289507
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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.
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14
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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
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = (Sum_j e_j^2)/gcd_j(e_j), where n = Product_j p_{e_j}.
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EXAMPLE
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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.
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MAPLE
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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])):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A056239, where the same encoding for integer multisets ('Heinz encoding') is used.
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KEYWORD
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AUTHOR
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STATUS
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approved
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