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

1,1

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

Cf. A065508, A074268, A206369.

Sequence in context: A367804 A361268 A355038 * A051760 A213519 A352423

Adjacent sequences: A297865 A297866 A297867 * A297869 A297870 A297871

Keyword

nonn

Author

Michel Marcus, Jan 07 2018

Status

approved