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A088725 Numbers having no divisors d>1 such that also d+1 is a divisor.

%I A088725 #24 Jun 14 2022 02:27:48
%S A088725 1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,28,29,31,
%T A088725 32,33,34,35,37,38,39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,
%U A088725 62,63,64,65,67,68,69,70,71,73,74,75,76,77,79,81,82,83,85,86,87,88,89,91
%N A088725 Numbers having no divisors d>1 such that also d+1 is a divisor.
%C A088725 Complement of A088723.
%C A088725 Union of A132895 and A005408, the odd numbers. - _Ray Chandler_, May 29 2008
%C A088725 The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 9, 79, 778, 7782, 77813, 778055, 7780548, 77805234, 778052138, 7780519314, ... . Apparently, the asymptotic density of this sequence exists and equals 0.77805... . - _Amiram Eldar_, Jun 14 2022
%F A088725 A088722(a(n)) = 0.
%e A088725 From _Gus Wiseman_, Oct 16 2019: (Start)
%e A088725 The sequence of terms together with their divisors > 1 begins:
%e A088725    1: {}
%e A088725    2: {2}
%e A088725    3: {3}
%e A088725    4: {2,4}
%e A088725    5: {5}
%e A088725    7: {7}
%e A088725    8: {2,4,8}
%e A088725    9: {3,9}
%e A088725   10: {2,5,10}
%e A088725   11: {11}
%e A088725   13: {13}
%e A088725   14: {2,7,14}
%e A088725   15: {3,5,15}
%e A088725   16: {2,4,8,16}
%e A088725   17: {17}
%e A088725   19: {19}
%e A088725   21: {3,7,21}
%e A088725   22: {2,11,22}
%e A088725   23: {23}
%e A088725   25: {5,25}
%e A088725 (End)
%t A088725 Select[Range[100],FreeQ[Differences[Rest[Divisors[#]]],1]&] (* _Harvey P. Dale_, Sep 16 2017 *)
%o A088725 (PARI) isok(n) = {my(d=setminus(divisors(n), [1])); #setintersect(d, apply(x->x+1, d)) == 0;} \\ _Michel Marcus_, Oct 28 2019
%Y A088725 Positions of 0's and 1's in A129308.
%Y A088725 Positions of 0's and 1's in A328457 (also).
%Y A088725 Numbers whose divisors (including 1) have no non-singleton runs are A005408.
%Y A088725 The number of runs of divisors of n is A137921(n).
%Y A088725 The longest run of divisors of n has length A055874(n).
%Y A088725 Cf. A000005, A027750, A060680, A088722, A088723 (complement), A088724, A088726, A328166, A328450.
%K A088725 nonn
%O A088725 1,2
%A A088725 _Reinhard Zumkeller_, Oct 12 2003
%E A088725 Extended by _Ray Chandler_, May 29 2008