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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.)