A092965
Greatest prime arising as the product of numbers chosen from among the first n numbers + 1.
Original entry on oeis.org
2, 3, 7, 13, 61, 241, 2521, 20161, 72577, 604801, 39916801, 59875201, 3113510401, 17435658241, 186810624001, 10461394944001, 118562476032001, 246245142528001, 24329020081766401, 304112751022080001
Offset: 1
a(5) = 61 = 3*4*5 + 1. 5! + 1, 4!+ 1, are composite and 2*4*5 + 1 = 41 <61, etc.
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Do[l = Map[Times @@ #&, Subsets[Range[n]]]; Print[Max[Select[Map[ #+1&, l], PrimeQ]]], {n, 20}] (* Ryan Propper, Aug 13 2005 *)
f[n_] := Max@ Select[ Union[ Times @@@ Subsets@ Range@ n] + 1, PrimeQ]; Array[f, 20] (* Robert G. Wilson v, Nov 13 2014 *)
A272981
Least prime k>1 such that the sum of divisors of powers k^e, 1 <= e <= n, are divisible by the number their divisors, d(k^e).
Original entry on oeis.org
3, 7, 7, 31, 31, 211, 211, 211, 211, 2311, 2311, 120121, 120121, 120121, 120121, 4084081, 4084081, 106696591, 106696591, 106696591, 106696591, 892371481, 892371481, 892371481, 892371481, 892371481, 892371481, 71166625531, 71166625531, 200560490131, 200560490131
Offset: 1
sigma(3) / d(3) = 4 / 2 = 2 but sigma(3^2) / d(3^2) = 13 / 3;
sigma(7) / d(7) = 8 / 2 = 4, sigma(7^2) / d(7^2) = 57 / 3 = 19, sigma(7^3) / d(7^3) = 400 / 4 = 100 but sigma(7^4) / d(7^4) = 2801 / 5.
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with(numtheory): P:= proc(q) local a,j,k,ok,p; global n; a:=2;
for k from 1 to q do for n from a to q do ok:=1;
for j from 1 to k do if not type(sigma(n^j)/tau(n^j),integer) then ok:=0; break; fi; od;
if ok=1 then a:=n; print(n); break; fi; od; od; end: P(10^9);
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Table[SelectFirst[Range[2, 10^6], AllTrue[#^Range@ n, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &] &], {n, 15}] (* Michael De Vlieger, May 12 2016, Version 10 *)
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