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

1,1

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.

Links

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

Example

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

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

Prog

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

Crossrefs

Cf. A013929, A048250, A082539, A162296.

Sequence in context: A258963 A309787 A064963 * A185794 A253330 A255076

Adjacent sequences: A300981 A300982 A300983 * A300985 A300986 A300987

Keyword

nonn

Author

Michel Lagneau, Mar 17 2018

Status

approved