A304721 Numbers m with A304720(m) = 1.

2, 3, 5, 7, 9, 10, 11, 12, 13, 19, 21, 23, 26, 28, 30, 39, 41, 46, 50, 51, 53, 55, 57, 59, 77, 89, 93, 101, 113, 129, 149, 151, 153, 161, 165, 178, 185, 189, 201, 221, 237, 245, 246, 297, 364, 377, 489, 553, 581, 639

Offset: 1

Author

Zhi-Wei Sun, May 17 2018

Keywords

• nonn

Comments

Conjecture: The sequence only has 112 terms as listed in the b-file.

We have verified that there is no new term below 2*10^9.

Examples

			a(9) = 13 since 13 - (4^1 - 1) = 2*5 is squarefree,  13 - (4^0 - 0) = 2^2*3 is not squarefree, and 13 - (4^k -k ) < 0 for any integer k > 1.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..112
• Zhi-Wei Sun, Mixed sums of primes and other terms, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.
• Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Crossrefs

Cf. A000302, A005117, A024037, A304034, A304081, A304331, A304333, A304522, A304523, A304689, A304720.

Programs

Mathematica

f[n_]:=f[n]=4^n-n;
tab={};Do[r=0;k=0;Label[bb];If[f[k]>=m,Goto[aa]];If[SquareFreeQ[m-f[k]],r=r+1];If[r>1,Goto[cc]];k=k+1;Goto[bb];Label[aa];If[r==1,tab=Append[tab,m]];Label[cc],{m,1,640}];Print[tab]