OFFSET
1,1
COMMENTS
Along with A065508, these are the integers mentioned at the bottom of page 4 of the Iannucci link. Let x = p^e, and q = rho(p^(e+1)), then x/rho(x) = (x*p*q)/rho(x*p*q). An example with A065508 is 3, for which rho(3) is 7, so 3 and 3*3*7 have the same x/rho(x) ratio, 3/2.
Note that there are other "rho-friendly pairs" that have a different, yet simple, form like for instance 7^5 and 7^8*117307.
Number of terms < 10^k: 1, 3, 6, 8, 11, 16, 20, 26, 31, 46, 73, 110, 198, 327, 611, 1157, 2135, 4107, 7724, 14771, 28610, etc. - Robert G. Wilson v, Jan 07 2018
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..28610
Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
EXAMPLE
8=2^3 is a term because rho(2*8)=11 is prime, so 8 and 8*2*11 have the same x/rho(x) ratio, 8/5.
MATHEMATICA
rho[n_] := n*DivisorSum[n, LiouvilleLambda[#]/# &]; fQ[n_] := Block[{p = FactorInteger[n][[1, 1]]}, PrimeQ[ rho[p n]]]; mx = 10^9; lst = Sort@ Flatten@ Table[ Prime[n]^e, {n, PrimePi[mx^(1/3)]}, {e, 3, Floor@ Log[ Prime@ n, mx], 2}]; Select[lst, fQ] (* Robert G. Wilson v, Jan 07 2018 *)
PROG
(PARI) rhope(p, e) = my(s=1); for(i=1, e, s=s*p + (-1)^i); s;
lista(nn) = {for (n=1, nn, if ((e = isprimepower(n, &p)) && (e > 1) && (e % 2) && isprime(rhope(p, e+1)), print1(n, ", "); ); ); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Jan 07 2018
STATUS
approved