A300984 Numbers whose sum of squarefree divisors and sum of nonsquarefree divisors are both squarefree numbers.

676, 1352, 2704, 5408, 5476, 8788, 10816, 10952, 14884, 21316, 21632, 21904, 29768, 35152, 42632, 43264, 43808, 59536, 70304, 85264, 86528, 95048, 114244, 119072, 140608, 148996, 170528, 173056, 175232, 190096, 202612, 209764, 228488, 238144, 262088, 281216

Offset: 1

Author

Michel Lagneau, Mar 17 2018

Keywords

• nonn

Comments

Conjecture: a(n) is of the form a(n) = 2^i*p^j with i, j integers and p prime. This has been verified for n up to 10^7.

Observation: For n < = 10^7, p belongs to the set E = {13, 37, 61, 73, 109, 157, 181, 193, 229, 277, 313, 373, 397, 409, 421, 433, 457, 541, 601, 613, 661, 673, 709, 733, 757, 769, 829, 853, 877, 997, 1009, 1021, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1297, 1381, 1429, 1453, 1489}. We observe that E minus {181, 433, 601, 769, 853, 1021, 1429} belongs to A082539.

Generalization: For n <= 10^m with m > 7, it is conjectured that a majority of primes p where a(n) = 2^i*p^j are in A082539. For example, with m = 7, 84% of the primes p are in A082539.

Examples

			676 is in the sequence because A048250(676) = 42 = 2*3*7 and A162296(676) = 1239 = 3*7*59 are both squarefree numbers.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A013929, A048250, A082539, A162296.

Programs

Mathematica

lst={};Do[If[SquareFreeQ[Total[Select[Divisors[n],SquareFreeQ]]]&& SquareFreeQ[DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]]],AppendTo[lst,n]],{n,300000}];lst

PARI

isok(n) = my(sd = sumdiv(n,d,d*issquarefree(d))); issquarefree(sd) && issquarefree(sigma(n) - sd); \\ Michel Marcus, Mar 17 2018