A293928 -id:A293928 - cp's OEIS frontend

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

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.

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

Original entry on oeis.org

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=', ')

A068997 Numbers k such that Sum_{d|k} d*mu(d) divides k.

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

Offset: 1

Benoit Cloitre, Apr 07 2002

Numbers k such that A023900(k) divides k.
The only squarefree terms so far are a(1), a(2), and a(4). - Torlach Rush, Dec 04 2017
There are no more squarefree terms. The squarefree terms are also the squarefree terms of A007694 since A023900(n) = A008683(n) * A000010(n) for squarefree numbers n, and A007694 contains only 3-smooth numbers (A003586). - Amiram Eldar, Apr 19 2025
There is a surjective mapping from all even numbers not in this sequence to terms of the sequence. The first such is 10 to a(9). The next is 14, 28, 42 to a(19). All even numbers not in the sequence are divisors of some term in the sequence. - Torlach Rush, Dec 08 2017

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A000010, A003586, A007694, A008683, A023900, A173557, A293928.

a068997 n = a068997_list !! (n - 1)
a068997_list = filter (\x -> mod x (a173557 x) == 0) [1..]
-- Reinhard Zumkeller, Jun 01 2015

with(numtheory): A068997 := i->`if`(i mod phi(mul(j,j=factorset(i)))=0,i,NULL): seq(A068997(i),i=1..650); # Peter Luschny, Nov 02 2010

Select[Range[650], Divisible[#, DivisorSum[#, # MoebiusMu[#] &]] &] (* Michael De Vlieger, Nov 20 2017 *)
q[1] =True; q[n_] := Divisible[n, Times @@ ((First[#] - 1) & /@ FactorInteger[n])]; Select[Range[650], q] (* Amiram Eldar, Apr 19 2025 *)

for(n=1,1000,if(n%sumdiv(n,d,moebius(d)*d)==0,print1(n,",")))

isok(k) = !(k % vecprod(apply(x -> 1-x, factor(k)[, 1]))); \\ Amiram Eldar, Apr 19 2025

cp's OEIS Frontend

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

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

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A068997 Numbers k such that Sum_{d|k} d*mu(d) divides k.

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