

A300984


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


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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



