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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.

%I A322370 #27 Feb 04 2019 07:35:55
%S A322370 15,30,21,42,119,33,35,78,209,133,51,57,299,65,138,217,77,399,87,93,
%T A322370 295,105,210,111,222,230,258,266,141,143,451,155,161,330,505,177,183,
%U A322370 185,195,390,201,203,215,1342,231,462,237,721,1209,255,518,267,273,546
%N 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.
%C A322370 This sequence is well-defined per the work of Booker and Pomerance.
%C A322370 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.
%H A322370 Rémy Sigrist, <a href="/A322370/b322370.txt">Table of n, a(n) for n = 5..10000</a>
%H A322370 Andrew R. Booker, Carl Pomerance, <a href="https://arxiv.org/abs/1607.01557">Squarefree smooth numbers and Euclidean prime generators</a>, arXiv:1607.01557 [math.NT], 2016-2017.
%H A322370 Andrew R. Booker and Carl Pomerance, <a href="https://doi.org/10.1090/proc/13576">Squarefree smooth numbers and Euclidean prime generators</a>, Proceedings of the American Mathematical Society 145 (2017), 5035-5042.
%H A322370 Rémy Sigrist, <a href="/A322370/a322370.png">Colored scatterplot of (n, a(n)) for n = 5..1000000</a> (where the color is function of (a(n)-4)/A000040(n)).
%F A322370 a(n) = A261144(n, k) for some k in 1..2^n.
%e A322370 For n = 7:
%e A322370 - the 7th prime is 17,
%e A322370 - the first squarefree 17-smooth integers s, alongside (s-4) mod 17, are:
%e A322370      s              1   2   3  5  6  7  10  11  13  14  15  17  21
%e A322370      ------------  --  --  --  -  -  -  --  --  --  --  --  --  --
%e A322370      (s-4) mod 17  14  15  16  1  2  3   6   7   9  10  11  13   0
%e A322370 - hence a(7) = 21.
%t A322370 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 *)
%o A322370 (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))))
%Y A322370 Cf. A000040, A005117, A261144.
%K A322370 nonn
%O A322370 5,1
%A A322370 _Rémy Sigrist_, Dec 05 2018