A274445 a(n) is the smallest composite squarefree number k such that (p+n) | (k-1) for every prime p dividing k.
385, 91, 65, 451, 33, 170171, 145, 1261, 161, 78409, 469, 294061, 649, 13051, 1921, 5251, 721, 8453501, 145, 300243, 1121, 47611, 3601, 1915801, 1057, 41311, 545, 5671, 1261, 19723133, 4321, 37759, 6913, 451, 4033, 102821, 1513, 40891, 11521, 1259497, 721, 364781, 145
Offset: 1
Keywords
Examples
For n=1, prime factors of 385 are 5, 7 and 11. (385 - 1)/(5 + 1) = 384/6 = 64, (385 - 1)/(7 + 1) = 384/8 = 48 and (385 - 1)/(11 + 1) = 384/12 = 32. For n=2, prime factors of 91 are 7 and 13. (91 - 1)/(7 + 2) = 90/9 = 10 and (91 - 1)/(13 + 2) = 90/15 = 6.
Links
- Jinyuan Wang, Table of n, a(n) for n = 1..100
Crossrefs
Programs
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Maple
with(numtheory); P:=proc(q) local d,k,n,ok,p; for k from 1 to q do for n from 2 to q do if not isprime(n) and issqrfree(n) then p:=ifactors(n)[2]; ok:=1; for d from 1 to nops(p) do if not type((n-1)/(p[d][1]+k),integer) then ok:=0; break; fi; od; if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);
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Mathematica
t = Select[Range[10^6], SquareFreeQ@ # && CompositeQ@ # &]; Table[ SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k, Divisible[k - 1, # + n] &]]], {n, 17}] (* Michael De Vlieger, Jun 24 2016, Version 10 *) -
PARI
isok(k,n)=if (! issquarefree(k), return (0)); vp = factor(k) [,1]; if (#vp == 1, return (0)); for (i=1, #vp, if ((k-1) % (n+vp[i]), return (0));); 1; a(n) = my(k=2); while (! isok(k,n), k++); k; \\ Michel Marcus, Jun 28 2016
Extensions
a(18), a(24), a(30) added by Giovanni Resta, Jun 23 2016
More terms from Michel Marcus, Jun 28 2016