A080400 -id:A080400 - 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

A151999 Numbers k such that every prime that divides phi(k) also divides k.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 32, 34, 36, 40, 42, 48, 50, 54, 60, 64, 68, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 114, 120, 126, 128, 136, 144, 150, 156, 160, 162, 168, 170, 180, 192, 200, 204, 210, 216, 220, 222, 228, 234, 240, 250, 252, 256, 270

Offset: 1

J. Luis A. Yebra and J. Jimenez Urroz (yebra(AT)mat.upc.es), Nov 19 2008

Alternative descriptions:
(a) For every prime p|n and every prime q|p-1 we have q|n;
(b) Numbers n such that radical of phi(n) divides radical of n, where phi is Euler's totient function and radical is the squarefree kernel function.
These numbers are "valid bases".
Numbers n such that radical of phi(n) divides n. - Michel Marcus, Nov 06 2017
Pollack and Pomerance call these numbers "phi-deficient numbers". - Amiram Eldar, Jun 02 2020

Paolo P. Lava, Table of n, a(n) for n = 1..10000
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.
J. Jimenez Urroz and J. Luis A. Yebra, On the equation a^x=x mod b^n, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.8.

Cf. A005117, A055744, A073539.
Cf. A007947 (radical of n), A007694 (phi(n) divides n, a subsequence).
Cf. A080400 (radical of phi(n)).
Cf. A152000.

[n: n in [1..300] | forall{d: d in PrimeDivisors(EulerPhi(n)) | IsIntegral(n/d)}]; // Bruno Berselli, Nov 04 2017

A151999 := proc(n)
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            pdvs := numtheory[factorset](a) ;
            aworks := true;
            for p in numtheory[factorset](a) do
                for q in numtheory[factorset](p-1) do
                    if a mod q = 0 then
                        ;
                    else
                        aworks := false;
                    end if;
                end do:
            end do:
            if aworks then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jun 01 2013

Rad[n_]:=Times@@Transpose[FactorInteger[n]][[1]]; Select[1 + Range[300], Mod[Rad[#], Rad[EulerPhi[#]]]==0 &] (* José María Grau Ribas, Jan 09 2012 *)

isok(n) = {fp = factor(n); for (i=1, #fp[, 1], fq = factor(fp[i, 1] - 1); for (j=1, #fq[, 1], if (n % fq[j, 1], return (0)););); return (1);} \\ Michel Marcus, Jun 01 2013

isok(n) = (n % factorback(factor(eulerphi(n))[, 1])) == 0; \\ Michel Marcus, Nov 04 2017

for n in range(1, 271):
    if euler_phi(n)**2 == euler_phi(euler_phi(n) * n): print(n, end=', ') # Torlach Rush, Oct 01 2024

Corrected by Michel Marcus, Jun 01 2013
Edited by N. J. A. Sloane, Jun 02 2013 at the suggestion of Michel Marcus, merging this with A204010
Deleted erroneous comment and added correct b-file by Paolo P. Lava, Nov 06 2017

A187731 Numbers n such that rad(phi(n)) divides n-1.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, 85, 89, 91, 97, 101, 103, 107, 109, 113, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 247, 251, 255, 257, 259

Offset: 1

José María Grau Ribas, Mar 13 2011

nonn

Subsequence of A003277 (cyclic numbers).
Let L(x) = exp(log x log log log x/log log x). McNew shows that there are at most x/L(x)^(1+o(1)) members of this sequence up to x. - Charles R Greathouse IV, Oct 08 2012
Contains all primes A000040 and all Carmichael numbers A002997. - Jeppe Stig Nielsen, Jul 27 2020

			15 is in the sequence because phi(15)=8, rad(8)=2 and 2 divides 15-1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
José María Grau and Antonio M. Oller-Marcén, On k-Lehmer numbers, Integers, 12(2012), #A37
Nathan McNew, Radically weakening the Lehmer and Carmichael conditions (2012)

Cf. A000010, A002997 (Carmichael numbers), A003277 (cyclic numbers), A007947, A080400.

rad[n_]:=Times@@Transpose[FactorInteger[n]][[1]]; Select[Range[1000], Mod[#-1,rad[EulerPhi[#]]]==0&]

rad(n)=n=factor(n);prod(i=1,#n[,1],n[i,1]);
for(n=1,1e4,if((n-1)%rad(eulerphi(n))==0,print1(n", "))) \\ Charles R Greathouse IV, Jul 04 2011

is(n)=my(p=eulerphi(n), g=n); n--; while((g=gcd(p, g))>1, p/=g); p==1 && n \\ Charles R Greathouse IV, Mar 03 2014

A289624 a(n) = A002322(n)/A007947(A034380(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 1, 6, 4, 10, 1, 12, 6, 2, 2, 16, 6, 18, 2, 3, 10, 22, 1, 20, 12, 18, 3, 28, 2, 30, 4, 5, 16, 6, 3, 36, 18, 6, 2, 40, 3, 42, 5, 6, 22, 46, 2, 42, 20, 8, 6, 52, 18, 10, 3, 9, 28, 58, 2, 60, 30, 1, 8, 6, 5, 66, 8, 11, 6, 70, 3, 72, 36, 10, 9, 15, 6, 78, 2, 54, 40, 82, 3, 8, 42, 14, 5, 88, 6, 2, 11, 15, 46

Offset: 1

Antti Karttunen, Jul 17 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A000010, A002322, A007947, A034380, A080400.

with(numtheory):
rad := n -> ilcm(op(factorset(n))):
a := n -> lambda(n)/rad(phi(n)/lambda(n)):
seq(a(n), n=1..94); # Peter Luschny, Jul 17 2017

a007947[n_]:=Last@ Select[Divisors[n], SquareFreeQ[#] &]; Table[Numerator[CarmichaelLambda[n]/a007947[EulerPhi[n]/CarmichaelLambda[n]]], {n, 100}] (* Indranil Ghosh, Jul 17 2017 *)

A002322(n) = lcm(znstar(n)[2]); \\ This function from Charles R Greathouse IV, Aug 04 2012
A007947(n) = factorback(factorint(n)[, 1]); \\ This function from Andrew Lelechenko, May 09 2014
A289624(n) = A002322(n)/A007947(eulerphi(n)/A002322(n));

from sage.crypto.util import carmichael_lambda
def A007947(n): return mul(p for p in prime_divisors(n))
def A000010(n): return euler_phi(n)
def A002322(n): return carmichael_lambda(n)
def A289624(n): return A002322(n)/A007947(A000010(n)/A002322(n))
print([A289624(n) for n in (1..94)]) # Peter Luschny, Jul 17 2017

a(n) = A002322(n) / A007947(A034380(n)) = A002322(n) / A007947(A000010(n) / A002322(n)).

A226384 Numbers k such that rad(phi(k)) = phi(rad(k)).

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 12, 14, 22, 23, 24, 28, 31, 43, 44, 46, 47, 48, 56, 59, 62, 67, 71, 79, 83, 86, 88, 92, 94, 96, 103, 107, 112, 118, 124, 131, 134, 139, 142, 158, 166, 167, 172, 176, 179, 184, 188, 191, 192, 206, 211, 214, 223, 224, 227, 236, 239, 248, 262

Offset: 1

José María Grau Ribas, Jun 05 2013

nonn

Numbers k such that A080400(k) = A173557(k). - Amiram Eldar, Apr 09 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000010, A007947, A080400, A173557.

with(numtheory):
rad:= n-> mul(i, i=factorset(n)):
a:= proc(n) option remember; local k; for k from 1+a(n-1)
      while phi(rad(k))<>rad(phi(k)) do od; k
    end: a(0):=0:
seq(a(n), n=1..80);  # Alois P. Heinz, Jun 07 2013

rad[n_] := Product[fa[n][[i, 1]], {i,
     Length[fa[n]]}]; fa = FactorInteger;
      Select[Range[500], rad[EulerPhi[#]] == EulerPhi[rad[#]] &]

is(n)=my(f=factor(n)); lcm(factor(eulerphi(f))[,1])==prod(i=1,#f~, f[i,1]-1) \\ Charles R Greathouse IV, Nov 13 2013

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

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A151999 Numbers k such that every prime that divides phi(k) also divides k.

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A187731 Numbers n such that rad(phi(n)) divides n-1.

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A289624 a(n) = A002322(n)/A007947(A034380(n)).

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A226384 Numbers k such that rad(phi(k)) = phi(rad(k)).

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