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

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

1,1

Comments

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).

Links

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

Formula

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

Mathematica

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 *)

Crossrefs

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

See also A005097, A047845, A014076, A001223.

Sequence in context: A268693 A068832 A046472 * A191546 A069463 A288212

Adjacent sequences: A065821 A065822 A065823 * A065825 A065826 A065827

Keyword

nice,nonn

Author

Labos Elemer, Nov 23 2001

Status

approved