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A343979
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Composite numbers m such that lambda(m) = lambda(D_{m-1}), where lambda(n) is the Carmichael function of n (A002322) and D_k is the denominator (A027642) of Bernoulli number B_k.
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1
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5615659951, 36901698733, 55723044637, 776733036121, 2752403727511, 7725145165297, 14475486778537, 15723055492417, 22824071195485, 29325910221631, 54669159894469, 62086332981241, 125685944708905, 180225455689481, 298620660945331, 335333122310629, 426814989321721
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OFFSET
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1,1
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COMMENTS
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Squarefree composites m such that LCM_{prime p|m} (p-1) = LCM_{prime p, p-1|m-1} (p-1).
Carmichael numbers m such that LCM_{prime p|m} (p-1) = LCM_{prime p, p-1|m-1} (p-1), i.e., with A173614(m) = A346467(m).
Carmichael numbers m such that their index (m-1)/lambda(m) = A346468(m), cf. A174590.
Carl Pomerance noted that, for k = 40826, Chernick's Carmichael number (6k+1)*(12k+1)*(18k+1) = 88189878776579929 satisfies this condition.
Theorem: lambda(m) | lambda(D_{m-1}) if and only if m | D_{m-1}.
Composites m such that lambda(m) | lambda(D_{m-1}) are all Carmichael numbers, defined as composites m such that lambda(m) | m-1, while lambda(D_{m-1}) | m-1 for every m.
Note that if p is prime, then lambda(p) = lambda(D_{p-1}) = p-1.
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LINKS
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MATHEMATICA
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c = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {_, _}][[;; , 2]]; q[d_] := If[PrimeQ[d + 1], d, 1]; Select[c, LCM @@ (FactorInteger[#][[;; , 1]] - 1) == LCM @@ (q /@ Divisors[# - 1]) &]
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PROG
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(PARI)
A173614(n) = lcm(apply(p->p-1, factor(n)[, 1]));
A346467(n) = if(1==n, n, my(m=1); fordiv(n-1, d, if(isprime(1+d), m = lcm(m, d))); (m));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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