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%I #15 Aug 19 2019 03:17:02
%S 676,1352,2704,5408,5476,8788,10816,10952,14884,21316,21632,21904,
%T 29768,35152,42632,43264,43808,59536,70304,85264,86528,95048,114244,
%U 119072,140608,148996,170528,173056,175232,190096,202612,209764,228488,238144,262088,281216
%N Numbers whose sum of squarefree divisors and sum of nonsquarefree divisors are both squarefree numbers.
%C 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.
%C 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.
%C 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.
%H Amiram Eldar, <a href="/A300984/b300984.txt">Table of n, a(n) for n = 1..1000</a>
%e 676 is in the sequence because A048250(676) = 42 = 2*3*7 and A162296(676) = 1239 = 3*7*59 are both squarefree numbers.
%t lst={};Do[If[SquareFreeQ[Total[Select[Divisors[n],SquareFreeQ]]]&& SquareFreeQ[DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]]],AppendTo[lst,n]],{n,300000}];lst
%o (PARI) isok(n) = my(sd = sumdiv(n,d,d*issquarefree(d))); issquarefree(sd) && issquarefree(sigma(n) - sd); \\ _Michel Marcus_, Mar 17 2018
%Y Cf. A013929, A048250, A082539, A162296.
%K nonn
%O 1,1
%A _Michel Lagneau_, Mar 17 2018