A297868 Prime powers p^e with odd exponent e such that rho(p^(e+1)) is prime, where rho is A206369.

8, 27, 32, 125, 243, 512, 1331, 2048, 32768, 50653, 79507, 103823, 131072, 161051, 177147, 357911, 1419857, 2097152, 2248091, 3869893, 11089567, 15813251, 16974593, 20511149, 28934443, 69343957, 115501303, 147008443, 263374721, 536870912, 844596301, 1284365503, 1305751357

Offset: 1

Author

Michel Marcus, Jan 07 2018

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

Cf. A065508, A074268, A206369.

Programs

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 *)

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, ", ");););}