A139247 -id:A139247 - cp's OEIS frontend

A133049 Squares of Mersenne primes A000668(n).

9, 49, 961, 16129, 67092481, 17179607041, 274876858369, 4611686014132420609, 5316911983139663487003542222693990401, 383123885216472214589586755549637256619304505646776321

Offset: 1

Omar E. Pol, Oct 30 2007, Apr 23 2008

nonn

Sum of last A000043(n) divisors of the n-th even perfect number. In other words; sum of divisors that are not powers of 2 of the n-th even perfect number, or sum of divisors that are multiples of the n-th Mersenne prime A000668(n) of the n-th even perfect number. See A139247 for more information.

See the structure of the divisors of perfect numbers in A135652, A135653, A135654 and A135655.

			a(3)=961 because the 3rd Mersenne prime is 31 and 31^2=961.

G. C. Greubel, Table of n, a(n) for n = 1..15
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000290, A001248. Mersenne primes: A000668.

Cf. A000043, A000396, A135652, A135653, A135654, A135655, A138247.

Select[2^Range[1000] - 1, PrimeQ]^2 (* G. C. Greubel, Oct 03 2017 *)

forprime(p=2, 1000, if(ispseudoprime(2^p-1), print1((2^p-1)^2", "))) \\ G. C. Greubel, Oct 03 2017

a(n) = A000668(n)^2

More terms from Olaf Voß, Feb 13 2008

A139246 Triangle read by rows: row n lists the proper divisors of n-th perfect number A000396(n).

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 7, 14, 1, 2, 4, 8, 16, 31, 62, 124, 248, 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168, 1

Offset: 1

Omar E. Pol, Apr 22 2008, corrected Apr 25 2008

Rows n has A133033(n) terms.

The n-th row sum is the n-th perfect number A000396(n).

			Triangle begins:
  1, 2, 3
  1, 2, 4, 7, 14
  1, 2, 4, 8, 16, 31, 62, 124, 248
  1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064
  ...

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A018254, A018487, A133024, A133025, A133031, A133033, A135652, A135653, A135654, A135655, A139247.

Table[Most[Divisors[PerfectNumber[n]]],{n,6}]//Flatten (* Harvey P. Dale, Jul 08 2024 *)

A233757 Triangle read by rows: T(n,k) = (2^n-1)*2^(k-1), for n >= 1 and 1<=k<=n.

Original entry on oeis.org

1, 3, 6, 7, 14, 28, 15, 30, 60, 120, 31, 62, 124, 248, 496, 63, 126, 252, 504, 1008, 2016, 127, 254, 508, 1016, 2032, 4064, 8128, 255, 510, 1020, 2040, 4080, 8160, 16320, 32640, 511, 1022, 2044, 4088, 8176, 16352, 32704, 65408, 130816, 1023, 2046, 4092

Offset: 1

Omar E. Pol, Jan 12 2014

Column 1 gives the positive terms of A000225.
Leading diagonal gives the positive terms of A006516.
The sum of row n is T(n,1)^2 = A000225(n)^2, hence row sums give A060867.
If n = A000043(m) then T(n,1) = A000668(m) and row n lists last n divisors of m-th even perfect number, which are also the divisors that are multiples of m-th Mersenne prime, for m >= 1.
If n = A000043(m) then T(n,n) = A000396(m), assuming there are no odd perfect numbers, for m >= 1.

			Triangle begins:
1;
3, 6;
7, 14, 28;
15, 30, 60, 120;
31, 62, 124, 248, 496;
63, 126, 252, 504, 1008, 2016;
127, 254, 508, 1016, 2032, 4064, 8128;
255, 510, 1020, 2040, 4080, 8160, 16320, 32640;
511, 1022, 2044, 4088, 8176, 16352, 32704, 65408, 130816;
...

Harvey P. Dale, Table of n, a(n) for n = 1..5000

Cf. A000043, A000079, A000225, A000396, A000668, A006516, A018254, A018487, A059268, A060867, A133024, A133025, A135652-A135655, A139247.

Table[(2^n-1)2^(k-1),{n,10},{k,n}]//Flatten (* Harvey P. Dale, Oct 10 2018 *)

T(n,k) = A000225(n)*A000079(k-1), n >= 1, 1<=k<=n.

cp's OEIS Frontend

A133049 Squares of Mersenne primes A000668(n).

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A139246 Triangle read by rows: row n lists the proper divisors of n-th perfect number A000396(n).

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A233757 Triangle read by rows: T(n,k) = (2^n-1)*2^(k-1), for n >= 1 and 1<=k<=n.

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Keywords

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