A279096
Numbers k such that sigma(k^3) is prime.
Original entry on oeis.org
4, 9, 16, 25, 64, 81, 169, 289, 625, 961, 1024, 2401, 3721, 5329, 7921, 22201, 26569, 63001, 121801, 124609, 212521, 273529, 358801, 418609, 744769, 885481, 896809, 1048576, 1181569, 1247689, 1510441, 1630729, 1666681, 1682209, 1771561, 1874161, 1985281
Offset: 1
4 is in the sequence because sigma(4^3) = sigma(2^6) = 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127, which is prime.
16 is in the sequence because sigma(16^3) = sigma(2^12) = Sum_{m=0..12} 2^m = (2^13 - 1)/(2 - 1) = 8191, which is prime.
36 is not in the sequence because sigma(36^3) = sigma(2^6*3^6) = ((2^7 - 1)/(2 - 1))*((3^7 - 1)/(3 - 1)) = 127*1093, which is not prime. (36 is not of the form p^j where p is prime.)
361 is not in the sequence (even though 361 = 19^2 is of the form p^j where both p and 3*j + 1 are prime) because sigma(361^3) = sigma(19^6) = (19^7 - 1)/(19 - 1) = 49659541 = 701 * 70841.
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mx = 10^7; ee = Select[Range@ Log2@ mx, PrimeQ[3 # + 1] &]; Union@ Reap[ Do[ Do[ If[(v = p^e) <= mx, If[ PrimeQ[(p v^3 - 1)/ (p-1)], Sow@ v], Break[]], {e, ee}], {p, Prime@ Range@ PrimePi@ Sqrt@ mx}]][[2, 1]] (* Giovanni Resta, Mar 12 2017 *)
Select[Range[2*10^6],PrimeQ[DivisorSigma[1,#^3]]&] (* Harvey P. Dale, Jan 10 2024 *)
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isok(n) = isprime(sigma(n^3)); \\ Michel Marcus, Mar 12 2017
A299147
Numbers k such that sigma(k), sigma(k^2) and sigma(k^3) are primes.
Original entry on oeis.org
4, 64, 289, 253541929, 499477801, 1260747049, 14450203681, 25391466409, 256221229489, 333456586849, 341122579249, 459926756041, 911087431081, 928731181849, 1142288550841, 2880002461249, 2923070670601, 3000305515321, 4103999343889, 4123226708329, 4258977385441
Offset: 1
4 is in the sequence because all sigma(4) = 7, sigma(4^2) = 31 and sigma(4^3) = 127 are primes.
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[n: n in[1..10000000] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n^2)) and IsPrime(SumOfDivisors(n^3))];
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N:= 10^14: # to get all terms <= N
Res:= NULL:
p:= 1:
do
p:= nextprime(p);
if p^2 > N then break fi;
for k from 2 by 2 while p^k <= N do
if isprime(k+1) and isprime(2*k+1) and isprime(3*k+1) then
q1:= (p^(k+1)-1)/(p-1);
q2:= (p^(2*k+1)-1)/(p-1);
q3:= (p^(3*k+1)-1)/(p-1);
if isprime(q1) and isprime(q2) and isprime(q3) then
Res:= Res, p^k;
fi
fi
od
od:
sort([Res]); # Robert Israel, Feb 22 2018
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k = 1; A299147 = {}; While[k < 4260000000000, If[Union@ PrimeQ@ DivisorSigma[1, {k, k^2, k^3}] == {True}, AppendTo[A299147, k]]; k++]; A299147 (* Robert G. Wilson v, Feb 10 2018 *)
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isok(n) = isprime(sigma(n)) && isprime(sigma(n^2)) && isprime(sigma(n^3)); \\ Michel Marcus, Feb 05 2018
A299153
Numbers k such that sigma(k) and sigma(k^3) are both primes.
Original entry on oeis.org
4, 9, 16, 25, 64, 289, 2401, 7921, 3418801, 19439281, 24730729, 40819321, 52258441, 67848169, 75151561, 76405081, 142396489, 175006441, 185313769, 198443569, 253541929, 352425529, 369062521, 386554921, 414896161, 499477801, 526105969, 684921241, 775678201
Offset: 1
4 is in the sequence because sigma(4) = 7 and sigma(4^2) = 31 are both primes.
Cf.
A000203 (sigma(n)),
A055638 (sigma(n^2) is prime),
A232444 (sigma(n) and sigma(n^2) are primes),
A279094 (the smallest k such that sigma(k^n) is prime),
A279096 (sigma(n^3) is prime),
A299147 (sigma(n), sigma(n^2) and sigma(n^3) are primes).
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[n: n in[1..10000000] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n^3))];
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Select[Range[10^4], AllTrue[DivisorSigma[1, #] & /@ {#, #^3}, PrimeQ] &] (* Michael De Vlieger, Feb 05 2018 *)
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isok(n) = isprime(sigma(n)) && isprime(sigma(n^3)); \\ Michel Marcus, Feb 05 2018
A299148
a(n) is the smallest number k such that sigma(k) and sigma(k^n) are both primes.
Original entry on oeis.org
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
Offset: 1
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.
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[Min([n: n in[1..10000000] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n^k))]): k in [2..12]];
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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
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Array[Block[{k = 2}, While[! AllTrue[DivisorSigma[1, #] & /@ {k, k^#}, PrimeQ], k++]; k] &, 10] (* Michael De Vlieger, Feb 05 2018 *)
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a(n) = {my(k=1); while (!(isprime(sigma(k)) && isprime(sigma(k^n))), k++); k;} \\ Michel Marcus, Feb 05 2018
Showing 1-4 of 4 results.
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