A065824 - OEIS

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

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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 *)

PROG

(Python)

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

CROSSREFS

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