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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 27 15:36 EST 2021. Contains 349394 sequences. (Running on oeis4.)