A133024 -id:A133024 - cp's OEIS frontend

A133028 Even perfect numbers divided by 2.

3, 14, 248, 4064, 16775168, 4294934528, 68719345664, 1152921504069976064, 1329227995784915872327346307976921088, 95780971304118053647396689042151819065498660774084608, 6582018229284824168619876730229361455111736159193471558891864064, 7237005577332262213973186563042994240786838745737417944533177174565599576064

Offset: 1

Omar E. Pol, Oct 20 2007, Apr 23 2008, Apr 28 2009

nonn

a(13) has 314 digits and is too large to include. - R. J. Mathar, Oct 23 2007
Largest proper divisor of n-th even perfect number.
Also numbers k such that A000203(k) is divisible 24. - Ctibor O. Zizka, Jun 29 2009

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

Cf. A028334, A000396 (perfect numbers).
Cf. A000043, A000668, A000203.
Cf. A018254, A018487, A032742, A133024, A133025, A135652, A135653, A135654, A135655.
Cf. A134708, A135650. - Omar E. Pol, Jul 07 2009

a:=proc(n) if isprime(2^n-1)=true then 2^(n-2)*(2^n-1) else end if end proc: seq(a(n),n=1..120); # Emeric Deutsch, Oct 24 2007

p = Select[2^Range[400] - 1, PrimeQ]; p*(p+1)/4 (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
Map[2^(#-2) * (2^# - 1) &, MersennePrimeExponent[Range[12]]] (* Amiram Eldar, Oct 21 2024 *)

a(n) = A000396(n)/2. - R. J. Mathar, Oct 23 2007 [Assuming there are no odd perfect numbers. - Jianing Song, Sep 17 2022]
a(n) = 2^(A000043(n) - 2) * A000668(n). - Omar E. Pol, Mar 01 2008
a(n) = A032742(A000396(n)), assuming there are no odd perfect numbers.

More terms from R. J. Mathar and Emeric Deutsch, Oct 23 2007

A135654 Divisors of 8128 (the 4th perfect number), written in base 2.

Original entry on oeis.org

1, 10, 100, 1000, 10000, 100000, 1000000, 1111111, 11111110, 111111100, 1111111000, 11111110000, 111111100000, 1111111000000

Offset: 1

Omar E. Pol, Feb 23 2008, Mar 03 2008

The number of divisors of the 4th perfect number is equal to 2*A000043(4)=A061645(4)=14.

			The structure of divisors of 8128 (see A133024)
-------------------------------------------------------------------------
n ... Divisor . Formula ....... Divisor written in base 2 ...............
-------------------------------------------------------------------------
1)......... 1 = 2^0 ........... 1
2)......... 2 = 2^1 ........... 10
3)......... 4 = 2^2 ........... 100
4)......... 8 = 2^3 ........... 1000
5)........ 16 = 2^4 ........... 10000
6)........ 32 = 2^5 ........... 100000
7)........ 64 = 2^6 ........... 1000000 ... (The 4th superperfect number)
8)....... 127 = 2^7 - 2^0 ..... 1111111 ... (The 4th Mersenne prime)
9)....... 254 = 2^8 - 2^1 ..... 11111110
10)...... 508 = 2^9 - 2^2 ..... 111111100
11)..... 1016 = 2^10- 2^3 ..... 1111111000
12)..... 2032 = 2^11- 2^4 ..... 11111110000
13)..... 4064 = 2^12- 2^5 ..... 111111100000
14)..... 8128 = 2^13- 2^6 ..... 1111111000000 ... (The 4th perfect number)

Index entries for sequences related to divisors of numbers

For more information see A133024 (Divisors of 8128). Cf. A000043, A000079, A000396, A000668, A019279, A061645, A061652.

FromDigits[IntegerDigits[#,2]]&/@Divisors[8128] (* Harvey P. Dale, Jan 08 2014 *)

a(n)=A133024(n), written in base 2. Also, for n=1 .. 14: If n<=(A000043(4)=7) then a(n) is the concatenation of the digit "1" and n-1 digits "0" else a(n) is the concatenation of A000043(4)=7 digits "1" and (n-1-A000043(4)) digits "0".

A139247 Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 6, 7, 14, 28, 31, 62, 124, 248, 496, 127, 254, 508, 1016, 2032, 4064, 8128, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168, 33550336, 131071, 262142, 524284, 1048568, 2097136, 4193792

Offset: 1

Omar E. Pol, Apr 22 2008

Also, row n list the divisors of n-th perfect number that are not powers of 2.

First term of row n is the n-th Mersenne prime A000668(n). Last term of row n is the n-th perfect number A000396(n). Row n has A000043(n) terms. The sum of row n is equal to A133049(n), the square of n-th Mersenne prime A000668(n).

			Triangle begins:
  3, 6,
  7, 14, 28
  31, 62, 124, 248, 496
  127, 254, 508, 1016, 2032, 4064, 8128
  ...
==========================================================
Row .... First term ..... Last term ....... Row sum ......
n ..... (A000668(n)) ... (A000396(n)) ... (A000668(n)^2) .
==========================================================
1 ............ 3 .............. 6 ......... 3^2 = 9
2 ............ 7 ............. 28 ......... 7^2 = 49
3 ........... 31 ............ 496 ........ 31^2 = 961
4 .......... 127 ........... 8128 ....... 127^2 = 16129
5 ......... 8191 ....... 33550336 ...... 8191^2 = 67092481

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

Cf. A000043, A000396, A000668, A018254, A018487, A133024, A133025, A133031, A133049, A135652, A135653, A135654, A135655.

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 *)

A139248 Triangle read by rows: row n lists the proper divisors of n-th even superperfect number A061652(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 8, 1, 2, 4, 8, 16, 32, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 1, 2, 4, 8, 16

Offset: 1

Omar E. Pol, Apr 22 2008

Also, row n list the proper divisors of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Row n has A000043(n) - 1 = A090748(n) terms.

			Triangle begins:
  1
  1, 2
  1, 2, 4, 8
  1, 2, 4, 8, 16, 32
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768
  ...

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

Cf. A000043, A000668, A018254, A018487, A019279, A061652, A090748, A133024, A133025, A133031, A133049, A135652, A135653, A135654, A135655, A139246.

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.

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A133028 Even perfect numbers divided by 2.

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A135654 Divisors of 8128 (the 4th perfect number), written in base 2.

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A139247 Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime 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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A139248 Triangle read by rows: row n lists the proper divisors of n-th even superperfect number A061652(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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