A232444 -id:A232444 - cp's OEIS frontend

A299147 Numbers k such that sigma(k), sigma(k^2) and sigma(k^3) are primes.

4, 64, 289, 253541929, 499477801, 1260747049, 14450203681, 25391466409, 256221229489, 333456586849, 341122579249, 459926756041, 911087431081, 928731181849, 1142288550841, 2880002461249, 2923070670601, 3000305515321, 4103999343889, 4123226708329, 4258977385441

Offset: 1

Jaroslav Krizek, Feb 03 2018

nonn

All terms are squares (proof in A023194).

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

			4 is in the sequence because all sigma(4) = 7, sigma(4^2) = 31 and sigma(4^3) = 127 are primes.

Chai Wah Wu, Table of n, a(n) for n = 1..12775 (n = 1..997 from Robert G. Wilson v)

Subsequence of A232444.

Cf. A000203, A055638, A279094, A279096, A299153.

[n: n in[1..10000000] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n^2)) and IsPrime(SumOfDivisors(n^3))];

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

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 *)

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

Jaroslav Krizek, Feb 03 2018

nonn

Intersection of A023194 and A279096.

All terms are squares.

			4 is in the sequence because sigma(4) = 7 and sigma(4^2) = 31 are both primes.

Chai Wah Wu, Table of n, a(n) for n = 1..10203

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).

[n: n in[1..10000000] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n^3))];

Select[Range[10^4], AllTrue[DivisorSigma[1, #] & /@ {#, #^3}, PrimeQ] &] (* Michael De Vlieger, Feb 05 2018 *)

isok(n) = isprime(sigma(n)) && isprime(sigma(n^3)); \\ Michel Marcus, Feb 05 2018

A232445 Numbers n such that sigma(n) and sigma(n^2) are squares.

Original entry on oeis.org

1, 11177320, 182937066, 159839399818, 166474679436

Offset: 1

Alex Ratushnyak, Nov 24 2013

Intersection of A006532 and A008847.

sigma(a(6)) >= 10^12. - Hiroaki Yamanouchi, Sep 26 2014

			sigma(a(2)) = (64*9*5*2)^2 and sigma(a(2)^2) = (3*7*13*19*31*127)^2.
sigma(a(3)) = (64*9*5*7)^2 and sigma(a(3)^2) = (3*7*13*37*61*499)^2.
sigma(a(4)) = (256*9*5*47)^2 and sigma(a(4)^2) = (3*49*13*19*37*43*61*67)^2.
sigma(a(5)) = (16*3*7*121*17)^2 and sigma(a(5)^2) = (3*49*13*31*61*109*757)^2.

Cf. A000203 (sigma: sum of divisors of n), A006532, A008847, A232444.

isok(n) = issquare(sigma(n)) && issquare(sigma(n^2)); \\ Michel Marcus, Sep 24 2014

a(4) from Hiroaki Yamanouchi, Sep 24 2014

a(5) from Hiroaki Yamanouchi, Sep 26 2014

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A299147 Numbers k such that sigma(k), sigma(k^2) and sigma(k^3) are primes.

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A299153 Numbers k such that sigma(k) and sigma(k^3) are both primes.

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A232445 Numbers n such that sigma(n) and sigma(n^2) are squares.

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