A173614 -id:A173614 - cp's OEIS frontend

A173751 a(n) = gcd(n, lcm_{p is a prime divisor of n} (p-1)) = gcd(n, A173614(n)).

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 5, 2, 3, 2, 1, 4, 1, 2, 3, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 4, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 3

Offset: 1

José María Grau Ribas, Feb 23 2010

nonn

a(n) is divisor of A126864(n).

			84 = 2^2*3*7; lcm{p-1|p is prime and divisor of 84} = lcm{1,2,6} = 6; gcd(84,6) = 6 ==> a(84)=6.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
A. M. Oller-Marcen, On arithmetic numbers, arXiv preprint arXiv:1206.1823 [math.NT], 2012. From _N. J. A. Sloane_, Nov 25 2012

Cf. A173614.

fa=FactorInteger; lcm[n_] := Module[{aux = 1, lon = Length[fa[n]]}, Do[aux = LCM[aux, (fa[n][[i]][[1]] - 1)], {i, lon}]; aux] a[n_] := GCD[lcm[n], n]; Table[a[n], {n, 1, 300}]

a(n)=gcd(n, lcm(apply(p->p-1, factor(n)[,1]))) \\ Andrew Howroyd, Aug 06 2018

A174824 a(n) = period of the sequence {m^m, m >= 1} modulo n.

Original entry on oeis.org

1, 2, 6, 4, 20, 6, 42, 8, 18, 20, 110, 12, 156, 42, 60, 16, 272, 18, 342, 20, 42, 110, 506, 24, 100, 156, 54, 84, 812, 60, 930, 32, 330, 272, 420, 36, 1332, 342, 156, 40, 1640, 42, 1806, 220, 180, 506, 2162, 48, 294, 100, 816, 156, 2756, 54, 220, 168, 342

Offset: 1

Franklin T. Adams-Watters, Mar 30 2010

nonn

This is a divisibility sequence: if n divides m, a(n) divides a(m).

We have the equality n = a(n) for numbers n in A124240, which is related to Carmichael's function (A002322). The largest values of a(n) occur when n is prime, in which case a(n) = n*(n-1). - T. D. Noe, Feb 21 2014

			For n=3, 1^1 == 1 (mod 3), 2^2 == 1 (mod 3), 3^3 == 0 (mod 3), etc. The sequence of residues 1, 1, 0, 1, 2, 0, 1, 1, 0, ... has period 6, so a(3) = 6. - _Michael B. Porter_, Mar 13 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
José María Grau and Antonio M. Oller-Marcén, On the last digit and the last non-zero digit of n^n in base b, arXiv:1203.4066 [math.NT], 2012. (See page 3)
Index to divisibility sequences

Cf. A000312, A009262, A127699.

Cf. A002322, A124240, A268336.

Table[LCM[n, CarmichaelLambda[n]], {n, 100}] (* T. D. Noe, Feb 20 2014 *)

a(n)=local(ps);ps=factor(n)[,1]~;for(k=1,#ps,n=lcm(n,ps[k]-1));n

a(n) = lcm(n, lcm(znstar(n)[2])); \\ Michel Marcus, Mar 18 2016; corrected by Michel Marcus, Nov 13 2019

apply( {A174824(n)=lcm(lcm([p-1|p<-factor(n)[,1]]),n)}, [1..99]) \\ [...] = znstar(n)[2], but 3x faster. - M. F. Hasler, Nov 13 2019

a(n) = lcm(n, A173614(n)) = lcm(n, A002322(n)) = lcm(n, A011773(n)).
If n and m are relatively prime, a(n*m) = lcm(a(n), a(m)); a(p^k) = (p-1)*p^k for p prime and k > 0.
a(n) = n*A268336(n). - M. F. Hasler, Nov 13 2019

A346467 a(n) is the least common multiple of the divisors d of n-1 such that d+1 is prime; a(1) = 1.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 4, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 20, 1, 22, 1, 12, 1, 2, 1, 28, 1, 30, 1, 16, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 44, 1, 46, 1, 48, 1, 10, 1, 52, 1, 18, 1, 28, 1, 58, 1, 60, 1, 2, 1, 16, 1, 66, 1, 4, 1, 70, 1, 72, 1, 2, 1, 4, 1, 78, 1, 80, 1, 82, 1, 84, 1, 2, 1, 88, 1, 90, 1, 92, 1, 2, 1, 96

Offset: 1

Antti Karttunen and Thomas Ordowski, Jul 22 2021

nonn

Original definition: a(n) is the least common multiple of p-1 computed over all primes p for which p-1 is a divisor of n-1; a(1) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002322, A027642, A173614, A346466, A346468, A346481.

Cf. also A343979.

f:= proc(n)
  if n::even then return 1 fi;
  ilcm(op(select(d -> isprime(d+1), numtheory:-divisors(n-1))));
end proc:
f(1):= 1:
map(f, [$1..200]); # Robert Israel, Aug 30 2021

{1}~Join~Array[CarmichaelLambda@ Denominator@ BernoulliB@ # &, 96] (* Michael De Vlieger, Jul 22 2021 *)

A346467(n) = if(1==n,n,my(m=1); fordiv(n-1,d,if(isprime(1+d),m = lcm(m,d))); (m));

apply( {A346467(n)=if(n>1, lcm([d|d<-divisors(n-1),isprime(d+1)]), 1)}, [1..99]) \\ M. F. Hasler, Nov 23 2021

a(n) = A002322(A027642(n-1)).
a(n) = A346466(n) * A346481(n).
For n > 1, a(n) = (n-1) / A346468(n).
a(n) = LCM { d | n-1; d+1 is prime }, where "|" means "divides". - M. F. Hasler, Nov 23 2021

A346466 The least common multiple of all divisors d of n-1 such that d+1 is a prime divisor of n; a(1) = 1.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 2, 1, 22, 1, 4, 1, 2, 1, 28, 1, 30, 1, 2, 1, 1, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 46, 1, 6, 1, 2, 1, 52, 1, 1, 1, 2, 1, 58, 1, 60, 1, 2, 1, 4, 1, 66, 1, 2, 1, 70, 1, 72, 1, 2, 1, 1, 1, 78, 1, 2, 1, 82, 1, 4, 1, 2, 1, 88, 1, 6, 1, 2, 1, 1, 1, 96, 1, 2, 1, 100, 1, 102, 1, 4

Offset: 1

Antti Karttunen, Jul 19 2021

nonn

			From _M. F. Hasler_, Nov 23 2021: (Start)
For n = 2, the only prime factor of n is p = 2, and p-1 = 1 divides n-1 = 1, therefore a(2) = LCM { 1 } = 1.
For n = 35, the prime factors of n are p = 5 and p = 7; but neither 5-1 = 4 nor 7-1 = 6 divides n-1 = 34, therefore a(35) = LCM {} = 1. (End)

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to lcm's

Cf. also A173614, A346467.

A346466(n) = lcm(apply(p->if((n-1)%(p-1),1,(p-1)), factor(n)[, 1]));

A346466(n) = if(1==n,n,my(m=1); fordiv(n-1,d,if(isprime(1+d)&&!(n%(1+d)),m = lcm(m,d))); (m));

apply( {A346466(n)=lcm([p-1|p<-factor(n)[,1],(n-1)%(p-1)==0])}, [1..99]) \\ M. F. Hasler, Nov 23 2021

a(n) = LCM_{p-1|n-1, p|n, p prime} (p-1).

a(n) = p-1 for prime powers n = p^e, e >= 1; a(n) = 1 for any even n = 2k. - M. F. Hasler, Nov 23 2021

A343979 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.

Original entry on oeis.org

5615659951, 36901698733, 55723044637, 776733036121, 2752403727511, 7725145165297, 14475486778537, 15723055492417, 22824071195485, 29325910221631, 54669159894469, 62086332981241, 125685944708905, 180225455689481, 298620660945331, 335333122310629, 426814989321721

Offset: 1

Amiram Eldar and Thomas Ordowski, May 06 2021

nonn

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.

Cf. A002322, A002997, A027642, A033502, A173614, A174590, A346467, A346468.

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]) &]

A002322(n) = lcm(znstar(n)[2]); \\ From A002322
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));
isA343979(n) = ((n>1) && !isprime(n) && (!((n-1)%A002322(n))) && A173614(n)==A346467(n)); \\ Antti Karttunen, Jul 22 2021

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A173751 a(n) = gcd(n, lcm_{p is a prime divisor of n} (p-1)) = gcd(n, A173614(n)).

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A174824 a(n) = period of the sequence {m^m, m >= 1} modulo n.

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A346467 a(n) is the least common multiple of the divisors d of n-1 such that d+1 is prime; a(1) = 1.

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A346466 The least common multiple of all divisors d of n-1 such that d+1 is a prime divisor of n; a(1) = 1.

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A343979 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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