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

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, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91

Offset: 1

Author

Reinhard Zumkeller, Oct 12 2003

Keywords

• nonn

Comments

Complement of A088723.

Union of A132895 and A005408, the odd numbers. - Ray Chandler, May 29 2008

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

Examples

			From _Gus Wiseman_, Oct 16 2019: (Start)
The sequence of terms together with their divisors > 1 begins:
   1: {}
   2: {2}
   3: {3}
   4: {2,4}
   5: {5}
   7: {7}
   8: {2,4,8}
   9: {3,9}
  10: {2,5,10}
  11: {11}
  13: {13}
  14: {2,7,14}
  15: {3,5,15}
  16: {2,4,8,16}
  17: {17}
  19: {19}
  21: {3,7,21}
  22: {2,11,22}
  23: {23}
  25: {5,25}
(End)
		

Crossrefs

Positions of 0's and 1's in A129308.

Positions of 0's and 1's in A328457 (also).

Numbers whose divisors (including 1) have no non-singleton runs are A005408.

The number of runs of divisors of n is A137921(n).

The longest run of divisors of n has length A055874(n).

Cf. A000005, A027750, A060680, A088722, A088723 (complement), A088724, A088726, A328166, A328450.

Programs

Mathematica

Select[Range[100],FreeQ[Differences[Rest[Divisors[#]]],1]&] (* Harvey P. Dale, Sep 16 2017 *)

PARI

isok(n) = {my(d=setminus(divisors(n), [1])); #setintersect(d, apply(x->x+1, d)) == 0;} \\ Michel Marcus, Oct 28 2019

Formula

A088722(a(n)) = 0.

Extensions

Extended by Ray Chandler, May 29 2008