A134708 -id:A134708 - cp's OEIS frontend

A014635 a(n) = 2n(4*n - 1).

0, 6, 28, 66, 120, 190, 276, 378, 496, 630, 780, 946, 1128, 1326, 1540, 1770, 2016, 2278, 2556, 2850, 3160, 3486, 3828, 4186, 4560, 4950, 5356, 5778, 6216, 6670, 7140, 7626, 8128, 8646, 9180, 9730, 10296, 10878, 11476, 12090, 12720, 13366, 14028, 14706

Offset: 0

Mohammad K. Azarian

Even hexagonal numbers.
Number of edges in the join of two complete graphs of order 3n and n, K_3n * K_n - Roberto E. Martinez II, Jan 07 2002
Bisection of A000384. Also, this sequence arises from reading the line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the triangular numbers A000217. Perfect numbers are members of this sequence because a(A134708(n)) = A000396(n). Also, positive members are a bisection of A139596. - Omar E. Pol, May 07 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..880
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
Leo Tavares, Illustration: Diamond Cut Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000384, A000396, A134708, A139596.

Cf. A154105, A016754.

[2*n*(4*n-1): n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

[seq(binomial(4*n,2),n=0..43)]; # Zerinvary Lajos, Jan 02 2007

s=0;lst={s};Do[s+=n++ +6;AppendTo[lst, s], {n, 0, 7!, 16}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
Table[2*n*(4*n - 1), {n,0,50}] (* G. C. Greubel, Jul 18 2017 *)
PolygonalNumber[6,Range[0,90,2]] (* or *) LinearRecurrence[{3,-3,1},{0,6,28},50] (* Harvey P. Dale, Jan 21 2023 *)

a(n)=2*n*(4*n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = C(4*n,2), n>=0. - Zerinvary Lajos, Jan 02 2007
O.g.f.: 2*x*(3+5*x)/(1-x)^3. - R. J. Mathar, May 06 2008
a(n) = 8*n^2 - 2*n. - Omar E. Pol, May 07 2008
a(n) = a(n-1) + 16*n - 10 (with a(0)=0). - Vincenzo Librandi, Nov 20 2010
E.g.f.: (8*x^2 + 6*x)*exp(x). - G. C. Greubel, Jul 18 2017
From Vaclav Kotesovec, Aug 18 2018: (Start)
Sum_{n>=1} 1/a(n) = 3*log(2)/2 - Pi/4.
Sum_{n>=1} (-1)^n / a(n) = log(2)/2 + log(1+sqrt(2))/sqrt(2) - Pi / 2^(3/2). (End)
a(n) = A154105(n-1) - A016754(n-1). - Leo Tavares, May 02 2023

More terms from Erich Friedman

A133028 Even perfect numbers divided by 2.

Original entry on oeis.org

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

A139045 Largest proper divisor of the Fibonacci numbers > 1.

Original entry on oeis.org

1, 1, 1, 4, 1, 7, 17, 11, 1, 72, 1, 29, 305, 329, 1, 1292, 113, 2255, 5473, 199, 1, 23184, 15005, 521, 98209, 105937, 1, 416020, 2417, 726103, 1762289, 3571, 1845493, 7465176, 330929, 1056437, 31622993, 34111385, 59369, 133957148, 1, 233802911, 567451585

Offset: 3

Omar E. Pol, Apr 23 2008

nonn

See the list of divisors of positive Fibonacci numbers in the triangle A133021.
See the largest proper divisor of n in A032742.
Fibonacci(1)=Fibonacci(2)=1 do not have proper divisors. - Emeric Deutsch, May 18 2008

			a(9) = 17 because the 9th Fibonacci number is 34 and the divisors of 34 are 1, 2, 17, 34, then the largest proper divisor of 34 is 17.

Amiram Eldar, Table of n, a(n) for n = 3..1000

Cf. A000045, A032742, A060383, A133021, A133028, A134708.

with(combinat): with(numtheory): a:=proc(n) options operator, arrow: op(tau(fibonacci(n))-1, divisors(fibonacci(n))) end proc: seq(a(n),n=3..40); # Emeric Deutsch, May 18 2008
# second Maple program:
a:= n-> (f-> f/min(numtheory[factorset](f)))((<<0|1>, <1|1>>^n)[1, 2]):
seq(a(n), n=3..47);  # Alois P. Heinz, Sep 03 2019

lpd[n_]:=Divisors[n][[-2]]; lpd/@(Fibonacci[Range[3,40]]) (* Harvey P. Dale, Mar 29 2015 *)

a(n) = A032742(A000045(n)).

a(n) = A000045(n)/A060383(n). - Alois P. Heinz, Sep 03 2019

More terms from Emeric Deutsch, May 18 2008

A153798 a(n) = A000043(n)-2.

Original entry on oeis.org

0, 1, 3, 5, 11, 15, 17, 29, 59, 87, 105, 125, 519, 605, 1277, 2201, 2279, 3215, 4251, 4421, 9687, 9939, 11211, 19935, 21699, 23207, 44495, 86241, 110501, 132047, 216089, 756837, 859431, 1257785, 1398267, 2976219, 3021375, 6972591, 13466915, 20996009, 24036581

Offset: 1

Omar E. Pol, Jan 20 2009

nonn

Base 2 logarithm of A134708(n).

Amiram Eldar, Table of n, a(n) for n = 1..48

Cf. A000043, A014635, A134708.

MersennePrimeExponent[Range[48]] - 2 (* Amiram Eldar, Oct 17 2024 *)

a(40)-a(41) from Amiram Eldar, Oct 17 2024

A138814 Divisors of 4064 (half the 4th perfect number).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 127, 254, 508, 1016, 2032, 4064

Offset: 1

Omar E. Pol, Mar 31 2008

The n-th perfect number divided by 2 (A133028(n)) has 2*A090748(n) divisors, then this sequence has 12 members. First 6 members are the first 6 powers of 2 A000079. Last 6 members are multiples of 4th Mersenne prime A000668(4)=127. a(n) written in base 2 has n digit. See A138824 for the structure of this sequence.

Index entries for sequences related to divisors of numbers

Perfect number divided by 2: A133028.

Cf. A000043, A000079, A000396, A000668, A090748, A134708, A135654, A138824.

Divisors[4064] (* Paolo Xausa, Jul 22 2024 *)

divisors(4064) \\ Charles R Greathouse IV, Jun 21 2017

A138815 Divisors of 16775168 (half the 5th perfect number).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168

Offset: 1

Omar E. Pol, Mar 31 2008

The n-th perfect number divided by 2 (A133028(n)) has 2*A090748(n) divisors, then this sequence has 24 members. First 12 members are the first 12 powers of 2 A000079. Last 12 members are multiples of 5th Mersenne prime A000668(5)=8191. a(n) written in base 2 has n digits. See A138825 for the structure of this sequence.

Index entries for sequences related to divisors of numbers

Perfect number divided by 2: A133028. Cf. A000043, A000079, A000396, A000668, A090748, A134708, A135655, A138825.

Divisors[16775168] (* Wesley Ivan Hurt, Mar 20 2022 *)

divisors(16775168) \\ Charles R Greathouse IV, Jun 22 2017

A138824 Divisors of 4064 (the 4th perfect number divided by 2), written in base 2.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 31 2008

a(n) has n digits. See A138814 for more information.

			The structure of divisors of 4064 (see A138814)
.................................................................
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) A134708(4) = .. 32 = 2^5 ........... 100000
7) A000668(4) = . 127 = 2^7 - 2^0 ..... 1111111
8) .............. 254 = 2^8 - 2^1 ..... 11111110
9) .............. 508 = 2^9 - 2^2 ..... 111111100
10) ............ 1016 = 2^10- 2^3 ..... 1111111000
11) ............ 2032 = 2^11- 2^4 ..... 11111110000
12) A133028(4) = 4064 = 2^12- 2^5 ..... 111111100000

Index entries for sequences related to divisors of numbers

Perfect number divided by 2: A133028. Cf. A000043, A000396, A000668, A090748, A134708, A135654, A138814.

FromDigits/@(IntegerDigits[#,2]&/@Divisors[4064]) (* Harvey P. Dale, Oct 12 2016 *)

A138825 Divisors of 16775168 (the 5th perfect number divided by 2), written in base 2.

Original entry on oeis.org

1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1111111111111, 11111111111110, 111111111111100, 1111111111111000, 11111111111110000, 111111111111100000

Offset: 1

Omar E. Pol, Mar 31 2008

a(n) has n digits. See A138815 for more information.

			The structure of divisors of 16775168 (see A138815)
.....................................................................
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
8) .................. 128 = 2^7 ........... 10000000
9) .................. 256 = 2^8 ........... 100000000
10) ................. 512 = 2^9 ........... 1000000000
11) ................ 1024 = 2^10 .......... 10000000000
12) A134708(5) = ... 2048 = 2^11 .......... 100000000000
13) A000668(5) = ... 8191 = 2^13 - 2^0 .... 1111111111111
14) ............... 16382 = 2^14 - 2^1 .... 11111111111110
15) ............... 32764 = 2^15 - 2^2 .... 111111111111100
16) ............... 65528 = 2^16 - 2^3 .... 1111111111111000
17) .............. 131056 = 2^17 - 2^4 .... 11111111111110000
18) .............. 262112 = 2^18 - 2^5 .... 111111111111100000
19) .............. 524224 = 2^19 - 2^6 .... 1111111111111000000
20) ............. 1048448 = 2^20 - 2^7 .... 11111111111110000000
21) ............. 2096896 = 2^21 - 2^8 .... 111111111111100000000
22) ............. 4193792 = 2^22 - 2^9 .... 1111111111111000000000
23) ............. 8387584 = 2^23 - 2^10 ... 11111111111110000000000
24) A133028(5) = 16775168 = 2^24 - 2^11 ... 111111111111100000000000

Index entries for sequences related to divisors of numbers

Perfect number divided by 2: A133028. Cf. A000043, A000396, A000668, A090748, A134708, A135654, A138815.

FromDigits[IntegerDigits[#,2]]&/@Divisors[16775168] (* Harvey P. Dale, May 26 2015 *)

A135612 Even superperfect numbers divided by 2, written in base 2.

Original entry on oeis.org

1, 10, 1000, 100000, 100000000000, 1000000000000000, 100000000000000000, 100000000000000000000000000000, 100000000000000000000000000000000000000000000000000000000000

Offset: 1

Omar E. Pol, Mar 01 2008

Also, concatenation of "1" and A000043(n)-2 digits "0".

The number of divisors of a(n) is equal to the number of its digits. This number is equal to A090748(n)=A000043(n)-1.

			a(3)=1000 because A134708(n)=8 and 8 written in base 2 is 1000.

Even superperfect numbers divided by 2: A134708. Cf. A000043, A019279, A090748, A135651, A135656.

a(n)=A134708(n) written in base 2.

A138823 Divisors of 248 (the 3rd perfect number divided by 2), written in base 2.

Original entry on oeis.org

1, 10, 100, 1000, 11111, 111110, 1111100, 11111000

Offset: 1

Omar E. Pol, Mar 31 2008, corrected Apr 03 2008

248 is the number of dimensions of E_8.

a(n) has n digits.

			The structure of divisors of 248 (see A018355)
..................................................................
n ............ Divisor . Formula ....... Divisor written in base 2
..................................................................
1) ................. 1 = 2^0 ........... 1
2) ................. 2 = 2^1 ........... 10
3) ................. 4 = 2^2 ........... 100
4) A134708(3) = .... 8 = 2^3 ........... 1000
5) A000668(3) = ... 31 = 2^5 - 2^0 ..... 11111
6) ................ 62 = 2^6 - 2^1 ..... 111110
7) ............... 124 = 2^7 - 2^2 ..... 1111100
8) A133028(3) = .. 248 = 2^8 - 2^3 ..... 11111000

Index entries for sequences related to divisors of numbers

Perfect number divided by 2: A133028. Cf. A000043, A000396, A000668, A018355, A090748, A134708, A135653.

FromDigits[IntegerDigits[#,2]]&/@Divisors[248] (* Harvey P. Dale, Apr 01 2017 *)

divisors(248) \\ Charles R Greathouse IV, Sep 06 2016

cp's OEIS Frontend

A014635 a(n) = 2*n*(4*n - 1).

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A133028 Even perfect numbers divided by 2.

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A139045 Largest proper divisor of the Fibonacci numbers > 1.

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A153798 a(n) = A000043(n)-2.

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A138814 Divisors of 4064 (half the 4th perfect number).

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A138815 Divisors of 16775168 (half the 5th perfect number).

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A138824 Divisors of 4064 (the 4th perfect number divided by 2), written in base 2.

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A014635 a(n) = 2n(4*n - 1).