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A299148 a(n) is the smallest number k such that sigma(k) and sigma(k^n) are both primes. 1
2, 2, 4, 2, 25, 2, 262144, 4, 4, 64, 734449, 2, 3100870943041, 9066121, 4, 2, 729, 2, 214355670008317962105386619478205641151753401, 5041, 64, 16, 25, 10651330026288961, 16610312161, 2607021481, 38950081, 1817762776525603445521, 5331481, 2, 2160067977820518171249529658520145004718584607049, 21203610154988994565561 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Sequence b(n) of the smallest numbers m such that sigma(m^k) are all primes for k = 1..n: 2, 2, 4, ... (if fourth term exists, it must be bigger than 10^16).

a(n) is of the form p^e where p, e+1 and e*n+1 are primes. e=1 is possible only in the case p=2. - Robert Israel, Feb 06 2018

LINKS

Robert Israel, Table of n, a(n) for n = 1..79

FORMULA

a(n) >= A279094(n).

EXAMPLE

For n = 3; a(3) = 4 because 4 is the smallest number such that sigma(4) = 7 and sigma(4^3) = 127 are both primes.

MAPLE

f:= proc(n, Nmin, Nmax) local p, e, M, Res;

M:= Nmax;

Res:= -1;

e:= 0;

do

e:= nextprime(e+1)-1;

if 2^e > M then return Res fi;

if not isprime(e*n+1) then next fi;

p:= floor(Nmin^(1/e));

do

p:= nextprime(p);

if p^e > M then break fi;

if e = 1 and p > 2 then break fi;

if isprime((p^(e+1)-1)/(p-1)) and isprime((p^(e*n+1)-1)/(p-1)) then

Res:= p^e;

M:= p^e;

break

fi

od

od;

end proc:

g:= proc(n) local Nmin, Nmax, v;

Nmax:= 1;

do

Nmin:= Nmax;

Nmax:= Nmax*10^3;

v:= f(n, Nmin, Nmax);

if v > 0 then return v fi;

od;

end proc:

seq(g(n), n=1..50); # Robert Israel, Feb 06 2018

MATHEMATICA

Array[Block[{k = 2}, While[! AllTrue[DivisorSigma[1, #] & /@ {k, k^#}, PrimeQ], k++]; k] &, 10] (* Michael De Vlieger, Feb 05 2018 *)

PROG

(Magma) [Min([n: n in[1..10000000] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n^k))]): k in [2..12]]

(PARI) a(n) = {my(k=1); while (!(isprime(sigma(k)) && isprime(sigma(k^n))), k++); k; } \\ Michel Marcus, Feb 05 2018

CROSSREFS

Cf. A000203, A279094.

Sequence in context: A333595 A227509 A279094 * A129243 A013551 A252040

Adjacent sequences: A299145 A299146 A299147 * A299149 A299150 A299151

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Feb 03 2018

EXTENSIONS

a(13) to a(32) from Robert Israel, Feb 06 2018

STATUS

approved

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Last modified December 6 04:57 EST 2022. Contains 358594 sequences. (Running on oeis4.)