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A270562 a(n) is the largest number m satisfying lambda(m)=n, or zero if there is no solution, where lambda(m) is Carmichael's lambda function A002322(m). 4

%I #37 Dec 01 2023 03:25:39

%S 2,24,0,240,0,504,0,480,0,264,0,65520,0,0,0,16320,0,28728,0,13200,0,

%T 552,0,131040,0,0,0,6960,0,171864,0,32640,0,0,0,138181680,0,0,0,

%U 1082400,0,151704,0,5520,0,1128,0,4455360,0,0,0,12720,0,86184,0,13920,0,1416,0,6814407600,0,0,0,65280

%N a(n) is the largest number m satisfying lambda(m)=n, or zero if there is no solution, where lambda(m) is Carmichael's lambda function A002322(m).

%C a(n) is the largest modulus m such that the largest order of any element in the multiplicative group modulo m is n; a(n) is zero if there is no such group with largest order n.

%C Omitting the zeros gives A143407.

%C a(n) = 0 if n is not a term of A002174.

%H Gheorghe Coserea, <a href="/A270562/b270562.txt">Table of n, a(n) for n = 1..50005</a>

%H R. D. Carmichael, <a href="https://doi.org/10.1090/S0002-9904-1910-01892-9">Note on a new number theory function</a>, Bull. Amer. Math. Soc. 16 (1910), 232-238.

%t a[n_] := Module[{f, fsz, g = 1, h = 1, p, e}, Which[n <= 0, Return[0], n == 1, Return[2], OddQ[n], Return[0]]; f = FactorInteger[n][[All, 1]]; fsz = Length[f]; For[k = 1, k <= fsz, k++, p = f[[k]]; e = 1; While[Mod[n, CarmichaelLambda[p^e]] == 0, e++]; g *= p^(e-1)]; Do[If[PrimeQ[d+1] && Mod[g, d+1] != 0, h *= (d+1)], {d, Divisors[n]}]; g *= h; If[CarmichaelLambda[g] != n, 0, g]];

%t a /@ Range[100] (* _Jean-François Alcover_, Oct 18 2019, after _Gheorghe Coserea_ *)

%o (PARI)

%o lambda(n) = { \\ A002322

%o my(f=factor(n), fsz=matsize(f)[1]);

%o lcm(vector(fsz, k, my(p=f[k,1], e=f[k,2]);

%o if (p != 2, p^(e-1)*(p-1), e > 2, 2^(e-2), 2^(e-1))));

%o };

%o a(n) = {

%o if (n <= 0, return(0), n==1, return(2), n%2, return(0));

%o my(f=factor(n), fsz=matsize(f)[1], g=1, h=1);

%o for (k=1, fsz, my(p=f[k,1], e=1);

%o while (n % lambda(p^e) == 0, e++); g *= p^(e-1));

%o fordiv(n, d, if (isprime(d+1) && g % (d+1) != 0, h *= (d+1)));

%o g *= h; if (lambda(g) != n, 0, g);

%o };

%o vector(64, n, a(n)) \\ _Gheorghe Coserea_, Feb 21 2019

%Y Cf. A002322, A002174, A051222, A143407, A270564, A111725.

%Y See also A321713 (number of solutions).

%K nonn

%O 1,1

%A _Joerg Arndt_, Mar 19 2016

%E Corrected and extended by _Gheorghe Coserea_, Feb 21 2019

%E Entry revised by _N. J. A. Sloane_, May 03 2019

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)