A065405 -id:A065405 - cp's OEIS frontend

A065773 Number of divisors of square of true prime powers arising in A065405.

5, 7, 7, 5, 13, 7, 5, 17, 5, 19, 5, 13, 5, 5, 7, 11, 7, 5, 5, 5, 13, 5, 7, 31, 5, 5, 5, 5, 5, 5, 13, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 5, 7, 7, 5, 5, 5, 5, 5, 11, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 5, 5, 5, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Labos Elemer and Robert G. Wilson v, Nov 19 2001

nonn

			For k = 3125, tau(k^2) = 11, sigma(k^2) = 12207031 = (5^(tau(k^2)) - 1)/4 = A065403(16) is also a prime.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000005, A065405, A000203, A065403, A065771, A065772, A025475.

DivisorSigma[0, Select[Range[10^5], ! PrimeQ[#] && PrimeQ[DivisorSigma[1, #^2]] &]^2] (* Amiram Eldar, Jan 31 2025 *)

{ n=0; for (m=1, 10^9, if (isprime(m), next); x=sigma(m^2); if (isprime(x), a=numdiv(m^2); write("b065773.txt", n++, " ", a); if (n==100, return)) ) } \\ Harry J. Smith, Oct 30 2009

a(n) = A000005(A065405(n)^2).

If A065405(n) = q^c, a prime-power, then sigma(q^(2c)) = A000203(q^(2c)) = (-1 + q^(2c+1))/(q-1) = (-1 + q^A000005(A065405(n)^2))/(q-1) also a prime, from A065403.

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

Labos Elemer, Nov 06 2001

nonn

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

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

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 100 terms from Harry J. Smith, terms 101-500 from Amiram Eldar)

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

Do[s=DivisorSigma[1, n]; If[PrimeQ[s]&&!PrimeQ[Sqrt[n]], Print[{n, Sqrt[n], s}]], {n, 1, 20000000}]

{ 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

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

Name corrected by David A. Corneth, Jul 18 2020

A065404 Squares of composite numbers k such that sigma(k) (sum of divisors of k, A000203) is a prime.

Original entry on oeis.org

16, 64, 729, 2401, 4096, 15625, 28561, 65536, 83521, 262144, 279841, 531441, 707281, 3418801, 4826809, 9765625, 24137569, 28398241, 38950081, 47458321, 244140625, 260144641, 887503681, 1073741824, 1387488001, 2655237841

Offset: 1

Labos Elemer, Nov 06 2001

nonn

			46 cases below 10^12; for M a Mersenne prime, (M+1)/2 is here: M=8191, 4096=(M+1)/2.

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..100 from Harry J. Smith)

Cf. A062700, A000203 (sigma), A065403, A065405, A053182, A053183, A028982.

{ n=0; for (m=1, 10^9, if (isprime(m), next); x=sigma(m^2); if (isprime(x), write("b065404.txt", n++, " ", m^2); if (n==100, return)) ) } \\ Harry J. Smith, Oct 18 2009

sigma(a(n)) = A065403(n).

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

Labos Elemer, Nov 13 2001

nonn

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

Muniru A Asiru, Table of n, a(n) for n = 2..20

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

Array[2^((MersennePrimeExponent@ # - 1)/2) &, 12, 2] (* Michael De Vlieger, Aug 25 2018 *)

lista(nn) = {forprime(p=3, nn, if (isprime(2^p-1), print1(2^((p-1)/2), ", ")););} \\ Michel Marcus, Aug 04 2016

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

A065771 Prime powers n such that both tau(n^2) and sigma(n^2) are composite numbers.

Original entry on oeis.org

16, 81, 128, 625, 1024, 2187, 2401, 4096, 8192, 14641, 28561, 59049, 65536, 78125, 83521, 130321, 131072, 279841, 524288, 531441, 707281, 823543, 923521, 1594323, 1874161, 2825761, 3418801, 4194304, 4879681, 7890481, 9765625, 12117361, 13845841, 16777216

Offset: 1

Labos Elemer, Nov 19 2001

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000005, A000203, A065403-A065405, A028982, A025475.

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} ]
Select[Range[16778000],PrimePowerQ[#]&&AllTrue[{DivisorSigma[ 0,#^2],DivisorSigma[ 1,#^2]},CompositeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 18 2021 *)

A000005(A025475(n)^2) and A000203(A025475(n)^2) are composite numbers.

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

Labos Elemer, Nov 19 2001

nonn

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

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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} ]

A065813 Least m such that (p^(2*m+1)-1)/(p-1) is a prime, where p = prime(n).

Original entry on oeis.org

1, 1, 1, 2, 8, 2, 1, 9, 2, 2, 3, 6, 1, 2, 63, 5, 1, 3, 9, 1, 2, 2, 2, 1, 8, 1, 9, 8, 8, 11, 2, 1, 5, 81, 3, 6, 8, 3, 1, 1, 9, 8, 8, 2, 15, 288, 20, 119, 2, 5, 56, 2, 8, 3, 11, 2

Offset: 1

Vladeta Jovovic and Labos Elemer, Nov 13 2001

			a(5) = 8 because ithprime(5) = 11, (11^(2*m+1)-1)/10 is not a prime for m = 1..7 and (11^17-1)/10 = 50544702849929377 is a prime.

Andy Steward, Prime Generalized Repunits

Cf. A000005, A000203, A000040, A065403-A065405, A000043, A001348, A065854.

Do[p=Prime[w]; s=DivisorSigma[1, (p^r)^2]; z=DivisorSigma[0, (p^r)^2]; If[PrimeQ[s], Print[{p, r, p^r, s, z}]], {w, 1, 100}, {r, 1, 100}] For w=12, this prints out first {37, 6, 2565726409, 6765811783780036261, 13}.
lm[n_]:=Module[{m=1},While[!PrimeQ[(n^(2m+1)-1)/(n-1)],m++];m]; lm/@Prime[ Range[ 56]] (* Harvey P. Dale, Feb 16 2014 *)

{ allocatemem(932245000); for (n=1, 100, x=prime(n); s=x^2; q=x - 1; m=1; while (!isprime(((x*=s) - 1)/q), m++); write("b065813.txt", n, " ", m) ) } \\ Harry J. Smith, Oct 31 2009

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

Labos Elemer, Nov 16 2001

nonn

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

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

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

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

Name corrected by Amiram Eldar, Apr 13 2024

cp's OEIS Frontend

A065773 Number of divisors of square of true prime powers arising in A065405.

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

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A065404 Squares of composite numbers k such that sigma(k) (sum of divisors of k, A000203) is a prime.

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

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A065771 Prime powers n such that both tau(n^2) and sigma(n^2) are composite numbers.

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

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A065813 Least m such that (p^(2*m+1)-1)/(p-1) is a prime, where p = prime(n).

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

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