A319927 Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k.

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

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

Cf. A048105 (number, i.e., sum of the 0th powers, of non-unitary divisors of n), A034444 (number of unitary divisors of n).

Sequence in context: A038837 A325359 A307579 * A034046 A376216 A069562

Adjacent sequences: A319924 A319925 A319926 * A319928 A319929 A319930

Keyword

nonn

Author

Ivan N. Ianakiev, Oct 02 2018

Status

approved