A289624 -id:A289624 - cp's OEIS frontend

A034380 Ratio of totient to Carmichael's lambda function: a(n) = A000010(n) / A002322(n).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 4, 1, 2, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 4, 1, 1, 6, 2, 4, 2, 1, 2, 2, 2, 1, 4, 1, 1, 2, 2, 2, 2, 1, 8, 1, 1, 1, 4, 4, 1, 2, 4, 1, 2, 6, 2, 2, 1, 2, 4, 1, 1, 2, 2, 1, 2, 1, 4, 4

Offset: 1

Alex Fink

nonn

a(n)=1 if and only if the multiplicative group modulo n is cyclic (that is, if n is either 1, 2, 4, or of the form p^k or 2*p^k where p is an odd prime). In other words: a(n)=1 if n is a term of A033948, otherwise a(n) > 1 (and n is a term of A033949). - Joerg Arndt, Jul 14 2012

T. D. Noe, Table of n, a(n) for n = 1..10000
W. D. Banks and F. Luca, On integers with a special divisibility property, Archivum Mathematicum (BRNO) 42 (2006) pp 31-42.

Cf. A000010, A002322, A033948, A033949, A062373-A062377, A080400, A289624.

a034380 n = a000010 n `div` a002322 n
-- Reinhard Zumkeller, Sep 02 2014

[1] cat [EulerPhi(n) div CarmichaelLambda(n): n in [2..100]]; // Vincenzo Librandi, Jul 18 2017

Maple
```
A034380 := n-> phi(n) / lambda(n);
```

Table[EulerPhi[n]/CarmichaelLambda[n], {n, 1, 200}] (* Geoffrey Critzer, Dec 23 2014 *)

eulerphi(n)/lcm(znstar(n)[2]) \\ Charles R Greathouse IV, Feb 01 2013

a(n) = A000010(n) / A002322(n).
a(A033948(n)) = 1 [Banks & Luca]. - R. J. Mathar, Jul 29 2007
A002322(n)/A007947(a(n)) = A289624(n). - Antti Karttunen, Jul 17 2017

A289626 Restricted growth sequence transform of A289625, related to the structure of multiplicative group of integers modulo n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 5, 4, 3, 6, 5, 7, 4, 8, 8, 9, 4, 10, 8, 11, 6, 12, 13, 14, 7, 10, 11, 15, 8, 16, 17, 18, 9, 19, 11, 20, 10, 19, 21, 22, 11, 23, 18, 19, 12, 24, 21, 23, 14, 25, 19, 26, 10, 27, 28, 29, 15, 30, 21, 31, 16, 32, 25, 33, 18, 34, 25, 35, 19, 36, 28, 37, 20, 27, 29, 38, 19, 39, 40, 41, 22, 42, 28, 43, 23, 44, 45, 46, 19, 47, 35, 38, 24, 48, 49

Offset: 1

Antti Karttunen, Jul 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A289625, A101296.

Cf. A000010, A002322, A034380, A046072, A289624 (some of the matching sequences).

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
write_to_bfile(1,rgs_transform(vector(16384,n,A289625(n))),"b289626_upto16384.txt");

A080400 Largest squarefree number dividing phi(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 6, 2, 6, 2, 10, 2, 6, 6, 2, 2, 2, 6, 6, 2, 6, 10, 22, 2, 10, 6, 6, 6, 14, 2, 30, 2, 10, 2, 6, 6, 6, 6, 6, 2, 10, 6, 42, 10, 6, 22, 46, 2, 42, 10, 2, 6, 26, 6, 10, 6, 6, 14, 58, 2, 30, 30, 6, 2, 6, 10, 66, 2, 22, 6, 70, 6, 6, 6, 10, 6, 30, 6, 78, 2, 6, 10, 82, 6, 2, 42, 14, 10, 22

Offset: 1

Labos Elemer, Mar 19 2003

nonn

Also largest squarefree number dividing Carmichael's lambda, A002322(n). - Antti Karttunen, Jul 17 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A002322, A007947, A034380, A289624.

Table[Max[Select[Divisors[EulerPhi[n]],SquareFreeQ]],{n,90}] (* Harvey P. Dale, Jul 05 2012 *)
a[n_] := Times @@ FactorInteger[EulerPhi[n]][[;; , 1]]; Array[a, 100] (* Amiram Eldar, Jun 18 2022 *)

A080400(n) = factorback(factorint(eulerphi(n))[, 1]); \\ Antti Karttunen & David A. Corneth, Jul 16 2017

a(n) = A007947(A000010(n)).
a(m * n) = A007947(a(m) * a(n)) if gcd(m, n) = 1. - David A. Corneth, Jul 16 2017
a(n) = A007947(A002322(n)). - Antti Karttunen, Jul 17 2017

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A034380 Ratio of totient to Carmichael's lambda function: a(n) = A000010(n) / A002322(n).

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A289626 Restricted growth sequence transform of A289625, related to the structure of multiplicative group of integers modulo n.

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PARI

A080400 Largest squarefree number dividing phi(n).

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Mathematica

PARI

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