A091155 Numbers m such that m - 2^k is squarefree for all 1 <= 2^k < m.

2, 3, 4, 7, 15, 23, 39, 63, 75, 87, 111, 135, 147, 159, 195, 219, 231, 255, 267, 315, 387, 399, 411, 423, 435, 447, 459, 495, 519, 567, 615, 663, 675, 699, 711, 735, 747, 759, 771, 819, 867, 915, 999, 1011, 1023, 1035, 1047, 1071, 1095, 1119, 1155, 1167, 1263

Offset: 1

Author

T. D. Noe, Dec 23 2003

Keywords

• nonn

Comments

Erdős conjectures that this sequence is infinite. It appears that m == 3 (mod 12) except for m = 2, 4, 7 and 23.

Examples

			39 is on the list because 38, 37, 35, 31, 23 and 7 are all squarefree.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, A19.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• P. Erdős, On integers of the form 2^k + p and some related problems, Summa Brasil. Math. 2 (1950), pp. 113-123.

Crossrefs

Cf. A039669 (m such that m-2^k are all primes).

Programs

Mathematica

a={}; Do[k=1; While[sf=SquareFreeQ[n-k]; sf&&2k
				

PARI

is(n)=for(k=1,log(n+.5)\log(2),if(!issquarefree(n-2^k),return(0))); 1 \\ Charles R Greathouse IV, Apr 13 2014