OFFSET
1,1
COMMENTS
Conjecture: For any nonnegative integer power p the sum of the p-th powers of the odd non-unitary divisors of a(n) divides the sum of the p-th powers of the non-unitary divisors of a(n).
The start of this sequence, including the 58 terms currently shown in the data section, is consistent with a definition "nonsquarefree numbers not divisible by 4", but some larger terms are divisible by 4: for example, a(1305) = 9216 = 2^10 * 3^2. - Peter Munn, Sep 21 2020
It seems that, for every n and odd prime p, p*a(n) is a term. - Ivan N. Ianakiev, Sep 11 2022
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor
Wikipedia, Unitary divisor
EXAMPLE
The non-unitary divisors of 18 are 3 and 6, of which only 3 is odd. 3^2=9 divides 3^3 + 6^2 = 45 and therefore 18 is in the sequence.
MATHEMATICA
sigmaNU[n_, p_]:=Total[Select[Divisors[n], GCD[#, n/#]>1&]^p];
sigmaNUOdd[n_, p_]:=Total[Select[Divisors[n], OddQ[#]&&GCD[#, n/#]>1&]^p];
p=2; (*if you want to check the conjecture for the power of 99, replace 2 with 99*)
Select[Range[1000], IntegerQ[sigmaNU[#, p]/sigmaNUOdd[#, p]]&]//Quiet
PROG
(PARI) su2(n) = sumdiv(n, d, if(gcd(d, n/d)!=1, d^2));
suo2(n) = sumdiv(n, d, if ((d%2) && (gcd(d, n/d)!=1), d^2));
isok(n) = my(suo = suo2(n)); if (suo, (su2(n) % suo) == 0); \\ Michel Marcus, Oct 28 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Ivan N. Ianakiev, Oct 02 2018
STATUS
approved