cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A065403 Primes of the form sigma(m^2) where m is a composite number ordered by values m.

Original entry on oeis.org

31, 127, 1093, 2801, 8191, 19531, 30941, 131071, 88741, 524287, 292561, 797161, 732541, 3500201, 5229043, 12207031, 25646167, 28792661, 39449441, 48037081, 305175781, 262209281, 917087137, 2147483647, 1394714501, 2666986681
Offset: 1

Views

Author

Labos Elemer, Nov 06 2001

Keywords

Comments

There are 46 cases below 10^12.
All Mersenne primes are here: sigma((2^((p-1)/2))^2) = sigma(2^(p-1)) = -1 + 2^p, for suitable p.
m is of the form p^(2*e) for some prime p and e > 1 as sigma is multiplicative and m is composite. Terms are sorted by values of m. The sequence isn't monotonic. - David A. Corneth, Jul 18 2020

Examples

			19531 is in the sequence as for the composite m = 125 we have sigma(m^2) = 19531. - _David A. Corneth_, Jul 18 2020

Links

Crossrefs

Cf. A062700, A000203, A001348, A065404, A065405.

Programs

  • Mathematica
    Do[s=DivisorSigma[1, n]; If[PrimeQ[s]&&!PrimeQ[Sqrt[n]], Print[{n, Sqrt[n], s}]], {n, 1, 20000000}]
  • PARI
    { n=0; for (m=1, 10^9, if (isprime(m), next); x=sigma(m^2); if (isprime(x), write("b065403.txt", n++, " ", x); if (n==100, return)) ) } \\ Harry J. Smith, Oct 18 2009
  • PARI
    upto(n) = {res = List(); forstep(e = 4, logint(n, 2), 2, forprime(p = 2, sqrtnint(n, e), c = (p^(e + 1) - 1)/(p - 1); if(isprime(c), listput(res, [p^e, c]) ) ) ); listsort(res); vector(#res, i, res[i][2]) } \\ David A. Corneth, Jul 18 2020

Extensions

Name corrected by David A. Corneth, Jul 18 2020

A065405 Composite numbers k such that the sum of the divisors of k^2 is a prime.

Original entry on oeis.org

4, 8, 27, 49, 64, 125, 169, 256, 289, 512, 529, 729, 841, 1849, 2197, 3125, 4913, 5329, 6241, 6889, 15625, 16129, 29791, 32768, 37249, 51529, 57121, 69169, 76729, 113569, 117649, 128881, 139129, 157609, 192721, 208849, 226981, 229441, 253009
Offset: 1

Views

Author

Labos Elemer, Nov 06 2001

Keywords

Comments

All these composite numbers k should be prime powers because if k=a*b with gcd(a,b)=1, then sigma(aabb) = sigma(aa)*sigma(bb) cannot be a prime; 46 of the 236 prime powers below 1000000 are here.

Links

Crossrefs

Cf. A062700, A000203, A065403, A065404, A053182, A053183, A028982.

Programs

  • Mathematica
    Select[ Range[3 10^5], ! PrimeQ[ # ] && PrimeQ[ DivisorSigma[1, #^2]] & ]
  • PARI
    isok(k) = { !isprime(k) && isprime(sigma(k^2)) } \\ Harry J. Smith, Oct 18 2009

Formula

sigma(a(n)^2) = sigma(A065404(n)) = A065403(n) is prime.

A065549 a(1) = 1; for n > 1, a(n) = 2^((A000043(n) - 1)/2).

Original entry on oeis.org

2, 4, 8, 64, 256, 512, 32768, 1073741824, 17592186044416, 9007199254740992, 9223372036854775808, 1852673427797059126777135760139006525652319754650249024631321344126610074238976
Offset: 2

Views

Author

Labos Elemer, Nov 13 2001

Keywords

Comments

Proper subset of A065405.
These values also relate to the sequence of perfect numbers. Every even perfect number except 6 can be written as Sum_{k=1..a(n)} (2*k-1)^3. - Derek Orr, Sep 28 2013
Positive real roots of 2n^4 - n^2 - A000396(n) = 0 for A000396(n) > 6. - César Aguilera, Nov 11 2018

Links

Crossrefs

Cf. A000043, A065403, A065404, A065405, A000396.

Programs

  • Mathematica
    Array[2^((MersennePrimeExponent@ # - 1)/2) &, 12, 2] (* Michael De Vlieger, Aug 25 2018 *)
  • PARI
    lista(nn) = {forprime(p=3, nn, if (isprime(2^p-1), print1(2^((p-1)/2), ", ")););} \\ Michel Marcus, Aug 04 2016

Formula

log(n) is approximately log(sqrt(A000668(n)/2)). - César Aguilera, Nov 11 2018

A065772 Nontrivial prime powers k from A025475 such that tau(k^2) is prime but sigma(k^2) is a composite number.

Original entry on oeis.org

9, 25, 32, 121, 243, 343, 361, 961, 1331, 1369, 1681, 2048, 2209, 2809, 3481, 3721, 4489, 5041, 6561, 6859, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16384, 16807, 17161, 18769, 19321, 19683, 22201, 22801, 24389, 24649, 26569
Offset: 1

Views

Author

Labos Elemer, Nov 19 2001

Keywords

Comments

Numbers k = A025475(m) such that A000005(k^2) is prime but A000203(k^2) is composite number.

Examples

			For k = 32: k^2 = 1024, tau(1024) = 11, sigma(1024) = 2047 = 23*89.
For k = 243, k^2 = 59049, tau(59049) = 11, sigma(59049) = 88573 = 23*3851.
Up to 10000000, 453 terms were found.

Links

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A065403, A065404, A065405, A028982, A025475.

Programs

  • Mathematica
    Do[ s=DivisorSigma[ 0, n^2 ]; y=DivisorSigma[ 1, n^2 ]; If[ Equal[ Length[ FactorInteger[ n ] ], 1 ]&&!PrimeQ[ n ] &&PrimeQ[ s ]&&!PrimeQ[ y ], Print[ n ] ], {n, 1, 10000000} ]

A065746 Number of divisors of squares of all true powers of primes: a(n) = A000005(A025475(n+1)^2).

Original entry on oeis.org

5, 7, 5, 9, 5, 7, 11, 5, 13, 9, 5, 7, 15, 5, 11, 17, 5, 7, 5, 19, 5, 9, 13, 5, 5, 21, 7, 5, 5, 5, 23, 15, 7, 5, 9, 5, 11, 5, 5, 25, 5, 7, 5, 5, 5, 17, 7, 5, 5, 27, 5, 5, 5, 5, 5, 7, 5, 9, 13, 5, 29, 11, 5, 5, 5, 19, 5, 5, 7, 5, 5, 5, 9, 7, 5, 5, 5, 31, 5, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 5, 21, 5, 33
Offset: 1

Views

Author

Labos Elemer, Nov 16 2001

Keywords

Examples

			tau(p^(2c)) = 2c+1 is prime if c = (odd prime -1)/2 = 1, 2, 3, 5, 6, 8, ... = A005097.

Crossrefs

Cf. A000005, A025475, A065403, A065404, A065405, A005097.

Programs

  • Mathematica
    DivisorSigma[0, Select[Range[60000], ! PrimeQ[#] && PrimePowerQ[#] &]^2] (* Amiram Eldar, Apr 13 2024 *)

Formula

tau(p^(2c)), where tau is the number of divisors, c > 1 and p is prime.

Extensions

Name corrected by Amiram Eldar, Apr 13 2024
Showing 1-5 of 5 results.

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