A065819 -id:A065819 - cp's OEIS frontend

A065818 Numbers k such that 3phi(k) = 2sigma(k).

5, 119, 3553, 4147, 8323, 198679, 218569, 248501, 298129, 468809, 547261, 1098349, 1398061, 1947379, 1990417, 3076549, 3453289, 3994753, 6969529, 7690249, 8790439, 11905457, 13097327, 14346577, 14732011, 14988967, 15286973, 16145269, 20851493, 21622867, 23083081, 24924599, 26218777, 26326867

Offset: 1

Labos Elemer, Nov 23 2001

nonn

			n = 3553, 3*phi(3553) = 8640 = 2*4320 = 2*sigma(3553).

Jud McCranie, Table of n, a(n) for n = 1..10000 (first 75 terms from Harry J. Smith)

Cf. A000010, A000203, A062699, A065819.

Select[Range[20000000], 3EulerPhi[#] == 2DivisorSigma[1, #] &]  (* Harvey P. Dale, Apr 18 2011 *)

isok(k) = { 3*eulerphi(k) == 2*sigma(k) } \\ Harry J. Smith, Oct 31 2009

a(22)-a(28) from Harry J. Smith, Oct 31 2009

A065822 Numbers k such that 5phi(k) = 4sigma(k).

Original entry on oeis.org

323, 377, 22591, 42619, 49751, 106711, 119647, 180947, 2782057, 2980823, 2981233, 3794737, 5112427, 5285743, 5732179, 5964229, 6073267, 6669797, 6769927, 7049407, 8025547, 8350633, 8954023, 9373213, 10039471, 10140517, 10842901

Offset: 1

Labos Elemer, Nov 23 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..335 (terms 1..60 from Harry J. Smith)

Cf. A000010, A000203, A065818, A065819, A062699.

Select[Range[1085*10^4],5EulerPhi[#]==4DivisorSigma[1,#]&] (* Harvey P. Dale, Aug 11 2019 *)

{ n=0; for (m=1, 10^9, if (5*eulerphi(m) == 4*sigma(m), write("b065822.txt", n++, " ", m); if (n==60, return)) ) } \\ Harry J. Smith, Nov 01 2009

is(n) = {my(f=factor(n)); 5*eulerphi(f) == 4*sigma(f);} \\ Amiram Eldar, Jun 26 2024

a(9)-a(27) from Harry J. Smith, Nov 01 2009

A065824 Smallest solution m to (n+1)phi(m) = nsigma(m), or -1 if no solution exists.

Original entry on oeis.org

3, 5, 7, 323, 11, 13, 899, 17, 19, 1763, 23, 5249, 3239, 29, 31, 979801, 5459, 37, 10763, 41, 43, 9179, 47, 9701, 10403, 53, 12319, 5646547, 59, 61, 24569, 19109, 67, 19043, 71, 73, 22499, 50819, 79, 41309, 83, 32639, 46979, 89, 34579, 39059, 125969

Offset: 1

Labos Elemer, Nov 23 2001

If p = a(n) is a prime solution, then (n+1)*(p-1) = n*(p+1) and p = 2n+1, so position for p if it is in fact a minimal solution is at n = (p-1)/2. E.g. 29 appears at 14th position shown by A005097. On the other hand large and (seemingly always composite) solutions arise at indices shown essentially by A047845. Also, differences between the sites of two consecutive small prime solutions appears to be d/2, half the difference between consecutive primes (A001223).

Donovan Johnson, Table of n, a(n) for n = 1..456

Cf. A000010, A000203, A062699, A065818, A065819, A065822, A065823.

See also A005097, A047845, A014076, A001223.

max = 10^7; a[n_] := For[m = 3, m <= max, m++, If[(n+1)*EulerPhi[m] == n*DivisorSigma[1, m], Print[m]; Return[m]]] /. Null -> -1; Array[a, 50] (* Jean-François Alcover, Oct 08 2016 *)

from itertools import count
from math import prod
from sympy import factorint
def A065824(n):
    for m in count(1):
        f = factorint(m)
        if (n+1)*m*prod((p-1)**2 for p in f)==n*prod(p**(e+2)-p for p,e in f.items()):
            return m # Chai Wah Wu, Aug 12 2024

(n+1)*A000010(a(n)) = n*A000203(a(n)), smallest x=a(n) solutions.

A065823 Numbers k such that 6phi(k) = 5sigma(k).

Original entry on oeis.org

11, 527, 923, 36859, 40549, 55309, 88519, 120139, 138301, 280579, 293501, 313807, 529789, 719927, 2458859, 4864117, 6191413, 6811243, 7297877, 8402663, 8624107, 9487477, 10475821, 12356441, 12940957, 13624717, 13971229, 14869033, 15293137

Offset: 1

Labos Elemer, Nov 23 2001

nonn

Not all terms are squarefree: a(74) = 137640191 = 13^2 * 89 * 9151. - Charles R Greathouse IV, Nov 13 2015

Apart from the first term, no terms are divisible by 2, 3, 5, 7, or 11. - Charles R Greathouse IV, Nov 13 2015

Harry J. Smith and Charles R Greathouse IV, Table of n, a(n) for n = 1..651 (first 70 terms from Smith)

Subsequence of A008364.

Cf. A000010, A000203, A065818, A065819, A062699, A065822, A274535(5*sigma(n)).

n=0; for (m=1, 10^9, if (6*eulerphi(m) == 5*sigma(m), write("b065823.txt", n++, " ", m); if (n==70, return))) \\ Harry J. Smith, Nov 01 2009

is(n)=my(f=factor(n)); 6*eulerphi(f)==5*sigma(f) \\ Charles R Greathouse IV, Nov 13 2015

Terms a(16)-a(29) from Harry J. Smith, Nov 01 2009

cp's OEIS Frontend

A065818 Numbers k such that 3*phi(k) = 2*sigma(k).

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A065822 Numbers k such that 5*phi(k) = 4*sigma(k).

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A065824 Smallest solution m to (n+1)*phi(m) = n*sigma(m), or -1 if no solution exists.

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A065823 Numbers k such that 6*phi(k) = 5*sigma(k).

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A065818 Numbers k such that 3phi(k) = 2sigma(k).

A065822 Numbers k such that 5phi(k) = 4sigma(k).

A065824 Smallest solution m to (n+1)phi(m) = nsigma(m), or -1 if no solution exists.

A065823 Numbers k such that 6phi(k) = 5sigma(k).