A073539 -id:A073539 - cp's OEIS frontend

A055744 Numbers k such that k and phi(k) have the same prime factors.

1, 4, 8, 16, 18, 32, 36, 50, 54, 64, 72, 100, 108, 128, 144, 162, 200, 216, 250, 256, 288, 294, 324, 400, 432, 450, 486, 500, 512, 576, 578, 588, 648, 800, 864, 882, 900, 972, 1000, 1014, 1024, 1152, 1156, 1176, 1210, 1250, 1296, 1350, 1458, 1600, 1728, 1764

Offset: 1

Labos Elemer, Jul 11 2000

nonn

Contains products of suitable powers of 2 and Fermat primes. For x = 2^u*3^w, phi(x) = 2^u*3^(w-1) with suitable exponents. Analogous constructions are possible with {2,3,7} prime divisors, etc.
From Ivan Neretin, Mar 19 2015: (Start)
Also, numbers k that meet the following criteria for every prime p dividing k:
  1. All prime divisors of p-1 must also divide k;
  2. If k has no prime divisors of the form m*p+1, and k is divisible by p, then k must be divisible by p^2.
Also, numbers k for which {k, phi(k), phi(phi(k))} is a geometric progression.
(End)
All terms > 1 are even. An even number k is in the sequence iff 2*k is in the sequence. - Robert Israel, Mar 19 2015
For n > 1, the largest prime factor of a(n) has multiplicity >= 2. For all prime factors more than half of the largest prime factor of a(n), the multiplicity differs from 1.
If k = p1^a1 * p2^a2 * ... * pm^am is in the sequence, then so is p1^b1 * p2^b2 * ... * pm^bm for 1 <= i <= m and prime pi and bi >= ai.
If m * p^2 is not in the sequence, for a prime p and some m > 0, then neither is m * p^3. - David A. Corneth, Mar 22 2015
A027748(a(n),j) = A027748(A000010(a(n)),j) for j=1..A001221(a(n)); also numbers k such that k and phi(k) have the same squarefree kernel: A007947(a(n)) = A007947(A000010(a(n))). - Reinhard Zumkeller, Jun 01 2015
Pollack and Pomerance call these numbers "phi-perfect numbers". - Amiram Eldar, Jun 02 2020

			k = 578 = 2*17*17, phi(578) = 272 = 2*2*2*2*17 with 2 and 17 prime factors, so 578 is a term.
k = 588 = 2*2*3*7*7, phi(588) = 168 = 2*2*2*3*7, so 588 is a term.
k = 264196 = 2*2*257*257, phi(264196) = 512*257 = 131584, so 264196 is a term.

David A. Corneth, Table of n, a(n) for n = 1..117561 (terms <= 10^11; first 10000 terms from T. D. Noe)
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.

Intersection of A073539 and A151999. - Amiram Eldar, Jun 02 2020
Cf. A001221, A000010, A110751, A110819, A027598, A081377.
Cf. A007947, A027748, A055742, A173557, A256248, subsequence of A124240.

a055744 n = a055744_list !! (n-1)
a055744_list = 1 : filter f [2..] where
   f x = all ((== 0) . mod x) (concatMap (a027748_row . subtract 1) ps) &&
         all ((== 0) . mod (a173557 x))
             (map fst $ filter ((== 1) . snd) $ zip ps $ a124010_row x)
         where ps = a027748_row x
-- Reinhard Zumkeller, Jun 01 2015

select(numtheory:-factorset = numtheory:-factorset @ numtheory:-phi,
[1, 2*i $ i=1..2000]); # Robert Israel, Mar 19 2015
isA055744 := proc(n)
    nfs := numtheory[factorset](n) ;
    phinfs := numtheory[factorset](numtheory[phi](n)) ;
    if nfs = phinfs then
        true;
    else
        false;
    end if;
end proc:
A055744 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA055744(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 23 2016

Select[Range@ 1800,
First /@ FactorInteger@ # == First /@ FactorInteger@ EulerPhi@ # &] (* Michael De Vlieger, Mar 21 2015 *)

is(n)=factor(n)[,1]==factor(eulerphi(n))[,1] \\ Charles R Greathouse IV, Oct 31 2011

is(n)=my(f=factor(n)); f[,1]==factor(eulerphi(f))[,1] \\ Charles R Greathouse IV, May 26 2015

Corrected and extended by James Sellers, Jul 11 2000

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

A090779 Numbers k that divide phi(k)^2.

Original entry on oeis.org

1, 4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 81, 98, 100, 108, 121, 125, 128, 144, 147, 162, 169, 196, 200, 216, 225, 242, 243, 250, 256, 288, 289, 294, 324, 338, 343, 361, 392, 400, 432, 441, 450, 484, 486, 500, 507, 512, 529, 576, 578, 588, 605, 625

Offset: 1

Benoit Cloitre, Feb 08 2004

nonn

Coincides with A073539 in the first 850000 terms and possibly more. - R. J. Mathar, Sep 08 2008

Vaclav Kotesovec, Table of n, a(n) for n = 1..20000
Vaclav Kotesovec, Plot of a(n)/n^2 and a(n)/(n^2/log(n)) for n = 1..20000

Select[Range[1000], Divisible[EulerPhi[#]^2,#]&] (* Vaclav Kotesovec, Feb 16 2019 *)

a(n) seems to be asymptotic to c*n^2 with c = 0.1....

A230400 Numbers n such that n = abc = 2(ab+ac+bc) for some positive integers a,b,c.

Original entry on oeis.org

216, 250, 256, 288, 400, 432, 450, 486, 576, 882

Offset: 1

M. F. Hasler, Oct 18 2013

Otherwise said: Volumes of integer-sided cubes equal to their surface area (assuming dimensionless unit of length).

The sequence is a finite subsequence of A055744, A069167, A073539, A090779 and A137845.

			The triples (a,b,c) ordered by largest member(s) are (6,6,6), (8,8,4), (10,5,5), (12,6,4), (12,12,3), (15,10,3), (18,9,3), (20,5,4), (24,8,3), (42,7,3).

"Mathematically possible", Volume = Surface Area?, on facebook.com.

Cf. A229941.

L=[];for(a=1,99,for(b=1,a,for(c=1,b,a*b*c==2*(a*b+b*c+a*c)&&!printf("(%d,%d,%d), ",a,b,c)&&L=concat(L,a*b*c))));vecsort(L)

A280936 Numbers k such that phi(k) = rad(k) * sopfr(k), where phi(k) is the Euler totient function of k, rad(k) the squarefree kernel of k and sopfr(k) the integer log of k.

Original entry on oeis.org

288, 1225, 4116, 35378, 54450, 1693776, 5969418, 9396618, 24509696310, 246465324525, 5876919827760, 71516027973936, 89547553939440, 370544528449590, 4014732589250736, 565869696542012100

Offset: 1

Paolo P. Lava, Jan 11 2017

nonn

If p is the largest prime divisor of a term k, then p^2 divides k. - Max Alekseyev, Feb 03 2024

			Prime factors of 288 are 2, 2, 2, 2, 2, 3, 3. Then phi(288) = 96, rad(288) = 2 * 3 = 6, sopfr(288) = 2 + 2 + 2 + 2 + 2 + 3 + 3 = 16 and 6 * 16 = 96.

Subsequence of A073539.

Cf. A000010, A001414, A007947.

with(numtheory): P:=proc(q) local a,k,n; for n from 1 to q do a:=ifactors(n)[2];
if phi(n)=mul(a[k][1],k=1..nops(a))*add(a[k][1]*a[k][2],k=1..nops(a)) then print(n);
fi; od; end: P(10^9);

a(9)-a(16) from Max Alekseyev, Feb 03 2024

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A055744 Numbers k such that k and phi(k) have the same prime factors.

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

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A090779 Numbers k that divide phi(k)^2.

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A230400 Numbers n such that n = abc = 2(ab+ac+bc) for some positive integers a,b,c.

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A280936 Numbers k such that phi(k) = rad(k) * sopfr(k), where phi(k) is the Euler totient function of k, rad(k) the squarefree kernel of k and sopfr(k) the integer log of k.

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