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A253597 Least Lucas-Carmichael number divisible by the n-th prime. 3

%I #37 Feb 12 2015 22:15:40

%S 399,935,399,935,2015,935,399,4991,51359,2015,1584599,20705,5719,

%T 18095,2915,46079,162687,22847,46079,16719263,12719,7055,80189,104663,

%U 20705,482143,196559,60059,90287,162687,3441239,13971671

%N Least Lucas-Carmichael number divisible by the n-th prime.

%C Has any odd prime number at least one Lucas-Carmichael multiple?

%H Tim Johannes Ohrtmann and Charles R Greathouse IV, <a href="/A253597/b253597.txt">Table of n, a(n) for n = 2..1000</a> (terms up to 202 from Ohrtmann)

%F a(n) >> n^2 log^2 n. - _Charles R Greathouse IV_, Feb 03 2015

%e a(2) = 399 because this is the least Lucas-Carmichael number which is divisible by 3 (the second prime number).

%t LucasCarmichaelQ[n_] := Block[{fi = FactorInteger@ n}, !PrimeQ@ n && Times @@ (Last@# & /@ fi) == 1 && Plus @@ Mod[n + 1, 1 + First@# & /@ fi] == 0]; f[n_] := Block[{k = p = Prime@ n}, While[ !LucasCarmichaelQ@ k, k += p]; k]; Array[f, 35, 2] (* _Robert G. Wilson v_, Feb 11 2015 *)

%o (PARI) is_A006972(n)=my(f=factor(n)); for(i=1, #f[, 1], if((n+1)%(f[i, 1]+1) || f[i, 2]>1, return(0))); #f[, 1]>1

%o a(n) = pn = prime(n); ln = 1; until (is_A006972(ln) && (ln % pn == 0), ln++); ln;

%o (PARI) is_A006972(n)=my(f=factor(n)); for(i=1, #f~, if((n+1)%(f[i, 1]+1) || f[i, 2]>1, return(0))); #f~>1

%o a(n)=my(p=prime(n), c=p^2+p, t=p); while(!is_A006972(t+=c),); t \\ _Charles R Greathouse IV_, Feb 03 2015

%Y Cf. A006972, A065091, A253598.

%K nonn

%O 2,1

%A _Tim Johannes Ohrtmann_, Jan 05 2015

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Last modified June 24 00:02 EDT 2024. Contains 373661 sequences. (Running on oeis4.)