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

5,1

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.

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)).

Formula

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

Example

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.

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 *)

Prog

(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))))

Crossrefs

Cf. A000040, A005117, A261144.

Sequence in context: A196184 A201136 A200896 * A040210 A296758 A297285

Adjacent sequences: A322367 A322368 A322369 * A322371 A322372 A322373

Keyword

nonn

Author

Rémy Sigrist, Dec 05 2018

Status

approved