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A297868 Prime powers p^e with odd exponent e such that rho(p^(e+1)) is prime, where rho is A206369. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A097054 A304291 A056824 * A051760 A213519 A066215

Adjacent sequences:  A297865 A297866 A297867 * A297869 A297870 A297871

KEYWORD

nonn

AUTHOR

Michel Marcus, Jan 07 2018

STATUS

approved

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Last modified June 19 02:43 EDT 2021. Contains 345125 sequences. (Running on oeis4.)