

A319927


Numbers k such that the sum of the squares of the odd nonunitary divisors of k divides the sum of the squares of the nonunitary divisors of k.


2



9, 18, 25, 27, 45, 49, 50, 54, 63, 75, 81, 90, 98, 99, 117, 121, 125, 126, 135, 147, 150, 153, 162, 169, 171, 175, 189, 198, 207, 225, 234, 242, 243, 245, 250, 261, 270, 275, 279, 289, 294, 297, 306, 315, 325, 333, 338, 342, 343, 350, 351, 361, 363, 369, 375, 378, 387, 405
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OFFSET

1,1


COMMENTS

Conjecture: For any nonnegative integer power p the sum of the pth powers of the odd nonunitary divisors of a(n) divides the sum of the pth powers of the nonunitary 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 nonunitary 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

Cf. A048105 (number, i.e., sum of the 0th powers, of nonunitary divisors of n), A034444 (number of unitary divisors of n).
Sequence in context: A038837 A325359 A307579 * A034046 A069562 A072502
Adjacent sequences: A319924 A319925 A319926 * A319928 A319929 A319930


KEYWORD

nonn


AUTHOR

Ivan N. Ianakiev, Oct 02 2018


STATUS

approved



