A151999 -id:A151999 - cp's OEIS frontend

A376639 Terms of A151999 which are not a term of A293928.

10, 30, 34, 42, 50, 60, 68, 78, 90, 102, 110, 114, 126, 136, 150, 156, 170, 180, 204, 210, 220, 222, 228, 234, 250, 270, 294, 300, 306, 330, 340, 342, 378, 390, 408, 410, 420, 438, 444, 450, 456, 468, 510, 514, 540, 546, 550, 570, 578, 582, 612, 630, 654, 660, 666

Offset: 1

Torlach Rush, Sep 30 2024

nonn

Conjecture: For each a(n) there is no a(n) = A000010(a(k)), k > n.

Conjecture: Every term of A293928 exists in A151999.

			10 is a term because 2 divides 4 and 10 and 10 is not a term of A293928.
666 is a term because 666 is a term of A151999 and 666 is not a term of A293928 as it has no totient inverses.

Cf. A000010, A151999, A293928.

terms = []
for n in range(1, 10000): # Equivalent of A151999/b151999.txt
    if euler_phi(n)**2 == euler_phi(euler_phi(n) * n): terms.append(n)
displayTerms = []
for n in range(0,10000):
    searchTerms = terms[n+1::]
    found = False
    for k in range(0, len(searchTerms)):
        if terms[n] == euler_phi(searchTerms[k]):
            found = True
            break
    if False == found and n < len(terms):
        displayTerms.append(terms[n])
for n in range(0, 55):
    print(displayTerms[n], end=', ')

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

Original entry on oeis.org

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

A073539 Numbers k such that if p is a prime dividing k then p divides phi(k).

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

Offset: 1

Benoit Cloitre, Aug 27 2002

Converse does not necessarily hold: phi(k) may have prime factors not dividing k.
Numbers k for which phi(k)*lambda(k) == 0 (mod k), where lambda(k) = A002322(k) is the Carmichael function. - Michel Lagneau, Nov 18 2012
Pollack and Pomerance call these numbers "phi-abundant numbers". Numbers k such that rad(k) | phi(k), where rad(k) is the squarefree kernel of k (A007947). - Amiram Eldar, Jun 02 2020
If p is the largest prime divisor of a term k, then p^2 divides k. - Max Alekseyev, Aug 27 2024

			98 = 2*7^2 and phi(98)=2*3*7 so if p divides 98 then p divides phi(98), hence 98 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.

Cf. A090779, A000010, A002322, A007947, A055744, A151999, A280936.

[n: n in [1..620] | IsZero(EulerPhi(n)^NumberOfDivisors(n) mod n)]; // Bruno Berselli, Jul 27 2012

Select[Range[700],And@@Divisible[EulerPhi[#],Transpose[FactorInteger[#]] [[1]]]&] (* Harvey P. Dale, Nov 02 2011 *)

A293928 Totients phi(m) having one or more solutions m to phi(m)^2 = phi(phi(m)*m).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 64, 72, 80, 84, 96, 100, 108, 120, 128, 144, 160, 162, 168, 192, 200, 216, 240, 252, 256, 272, 288, 312, 320, 324, 336, 360, 384, 400, 432, 440, 480, 486, 500, 504, 512, 544, 576, 588, 600, 624, 640, 648, 672, 684

Offset: 1

Torlach Rush, Oct 19 2017

nonn

"Totients" are terms of A000010. - N. J. A. Sloane, Oct 22 2017
The smallest totient absent from the list is 10. This is because the totient inverses of 10, 11 and 22 are not solutions to phi(m)^2 = phi(phi(m)*m).
The formula is recursive. For example, taking a(22) we get the following: 11664 = phi(108*324), 1259712 = phi(11664*324), 136048896 = phi(1259712*324), ...
Where (if ever) does this first differ from A068997? - R. J. Mathar, Oct 30 2017
Apparently the set of the m is A151999. - R. J. Mathar, Mar 25 2024
If m satisfies phi(m)^2 = phi(phi(m)*m), then it satisfies phi(m)^(k+1) = phi(phi(m)^k*m) for all k >= 1. - Max Alekseyev, Dec 03 2024

			96 is a term since 96^2 = phi(96*288), with m=288 where phi(288) = 96.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A006511, A032447, A007366.

Subsequence of A002202.

isok(n) = {my(iv = invphi(n)); if (#iv, for (m = 1, #iv, if (n^2 == eulerphi(n*iv[m]), return (1)););); return (0);} \\ using the invphi script by Max Alekseyev; Michel Marcus, Nov 01 2017

More terms from Michel Marcus, Oct 24 2017

Definition simplified by Max Alekseyev, Dec 03 2024

A152000 a(n) is squarefree and such that for every prime p|a(n) and every prime q|p-1 then q|a(n) holds.

Original entry on oeis.org

2, 6, 10, 30, 34, 42, 78, 102, 110, 114, 170, 210, 222, 330, 390, 410, 438, 510, 514, 546, 570, 582, 654, 714, 798, 930, 978, 1010, 1110, 1158, 1218, 1230, 1326, 1482, 1542, 1554, 1806, 1830, 1870, 1938, 2190, 2310, 2510, 2530

Offset: 1

J. Luis, A. Yebra and J. Jimenez Urroz, Nov 19 2008

a(n) is a squarefree valid base.

Numbers m > 1 such that m equals A007947(A002618(m)). - Jon Maiga, Aug 08 2019

Jinyuan Wang, Table of n, a(n) for n = 1..10000
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. A002618, A007947, A151999.

A152000 := proc(n)
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if issqrfree(a) then
                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 if;
        end do:
    end if;
end proc: # R. J. Mathar, Jun 01 2013

rad[n_] := Times @@ (First@# & /@ FactorInteger@n);
Select[Range[2, 2530], # == rad[#*EulerPhi[#]] &] (* Jon Maiga, Aug 08 2019 *)

is(m) = factorback(factorint(m*eulerphi(m))[, 1]) == m && m > 1; \\ Jinyuan Wang, Aug 08 2019

Corrected and extended by Michel Marcus, Jun 01 2013

A294618 a(n) is the number of solutions of x^2 = eulerphi(x * m) where x is A293928(n).

Original entry on oeis.org

2, 2, 3, 1, 4, 2, 5, 1, 1, 4, 6, 3, 3, 5, 1, 7, 6, 4, 1, 7, 1, 3, 1, 8, 10, 5, 1, 1, 9, 3, 8, 4, 1, 9, 1, 13, 1, 7, 4, 3, 1, 12, 5, 14, 1, 7, 1, 1, 2, 10, 2, 18, 1, 1, 1, 9, 9, 3, 1, 5, 1, 14, 7, 22, 3, 1

Offset: 1

Torlach Rush, Nov 05 2017

The valid values of m in the equation are the terms of the sequence A151999 in order.
m is a solution if all squarefree divisors of x also divide m.
The formula is recursive. For example, taking A151999(68) we get the following: 11664=phi(108*324), 1259712=phi(11664*324), 136048896=phi(1259712*324), ...
If a solution exists then x^(k+1) = phi(x^k * m) for a fixed m, and the smallest value of k must be 1. This follows from a|b implies phi(a)|phi(b), and for k >= 1 a^(k-1)|a^k.
The smallest solution where solutions exist are the terms of the sequence A055744 not in order.
The values of phi(m) are the terms of the sequence A068997 not in order.

			The first 1 is a term since there is only 1 solution when phi(m)=6. The solution is m=18.
The first 5 is a term since there are 5 solutions when phi(m)=16. These are 32, 34, 40, 48, and 60.
From _Michel Marcus_, Nov 08 2017: (Start)
Illustration of first few terms:
   1: [1, 2],
   2: [4, 6],
   4: [8, 10, 12],
   6: [18],
   8: [16, 20, 24, 30],
  12: [36, 42],
  16: [32, 34, 40, 48, 60],
  18: [54],
  20: [50],
  24: [72, 78, 84, 90],
  32: [64, 68, 80, 96, 102, 120],
  ... (End)

Max Alekseyev, PARI scripts for various problems

Cf. A000010, A006511, A032447, A151999, A055744, A068997, A293928.

isok(n) = {iv = invphi(n); if (#iv, return (sum(m=1, #iv, n^2 == eulerphi(n*iv[m])))); return (0);}
lista(nn) = {for (n=1, nn, if (v = isok(n), print1(v, ", ")););} \\ \\ using the invphi script by Max Alekseyev; Michel Marcus, Nov 07 2017

0 < (phi(m)^(k+1) = phi(phi(m)^k*m)), k >= 1, m >= 1.

cp's OEIS Frontend

A376639 Terms of A151999 which are not a term of A293928.

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

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A073539 Numbers k such that if p is a prime dividing k then p divides phi(k).

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A293928 Totients phi(m) having one or more solutions m to phi(m)^2 = phi(phi(m)*m).

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A152000 a(n) is squarefree and such that for every prime p|a(n) and every prime q|p-1 then q|a(n) holds.

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A294618 a(n) is the number of solutions of x^2 = eulerphi(x * m) where x is A293928(n).

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