A322370 For any n > 4: let p be the n-th prime number; a(n) is the least squarefree p-smooth integer congruent to 4 modulo p.

15, 30, 21, 42, 119, 33, 35, 78, 209, 133, 51, 57, 299, 65, 138, 217, 77, 399, 87, 93, 295, 105, 210, 111, 222, 230, 258, 266, 141, 143, 451, 155, 161, 330, 505, 177, 183, 185, 195, 390, 201, 203, 215, 1342, 231, 462, 237, 721, 1209, 255, 518, 267, 273, 546

Offset: 5

Author

Rémy Sigrist, Dec 05 2018

Keywords

• nonn

Comments

This sequence is well-defined per the work of Booker and Pomerance.

The number 4 in the congruence in the name could be replaced by any value; this number was chosen for being the first integer that is not squarefree.

Examples

			For n = 7:
- the 7th prime is 17,
- the first squarefree 17-smooth integers s, alongside (s-4) mod 17, are:
     s              1   2   3  5  6  7  10  11  13  14  15  17  21
     ------------  --  --  --  -  -  -  --  --  --  --  --  --  --
     (s-4) mod 17  14  15  16  1  2  3   6   7   9  10  11  13   0
- hence a(7) = 21.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 5..10000
• Andrew R. Booker, Carl Pomerance, Squarefree smooth numbers and Euclidean prime generators, arXiv:1607.01557 [math.NT], 2016-2017.
• Andrew R. Booker and Carl Pomerance, Squarefree smooth numbers and Euclidean prime generators, Proceedings of the American Mathematical Society 145 (2017), 5035-5042.
• Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 5..1000000 (where the color is function of (a(n)-4)/A000040(n)).

Crossrefs

Cf. A000040, A005117, A261144.

Programs

Mathematica

a[n_] := Module[{p = Prime[n], k = 4}, While[! SquareFreeQ[k] || FactorInteger[k][[-1, 1]] > p, k += p; Continue[]]; k]; Array[a, 100, 5] (* Amiram Eldar, Dec 08 2018 *)

PARI

a(n) = my (p=prime(n)); forstep (v=4, oo, p, if (issquarefree(v), my (f=factor(v)); if (f[#f~,1] <= p, return (v))))

Formula

a(n) = A261144(n, k) for some k in 1..2^n.