A019279 -id:A019279 - cp's OEIS frontend

A135655 Divisors of 33550336 (the 5th perfect number), written in base 2.

1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 1111111111111, 11111111111110, 111111111111100, 1111111111111000, 11111111111110000

Offset: 1

Omar E. Pol, Feb 23 2008, Mar 01 2008, Mar 03 2008

The number of divisors of the 5th perfect number is equal to 2*A000043(5)=A061645(5)=26.

			The structure of divisors of 33550336 (see A133025)
------------------------------------------------------------------------
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)........ 2048 = 2^11 .......... 100000000000
13) ....... 4096 = 2^12 .......... 1000000000000 ... (The 5th superperfect number)
14) ....... 8191 = 2^13 - 2^0 .... 1111111111111 ... (The 5th Mersenne prime)
15) ...... 16382 = 2^14 - 2^1 .... 11111111111110
16) ...... 32764 = 2^15 - 2^2 .... 111111111111100
17) ...... 65528 = 2^16 - 2^3 .... 1111111111111000
18) ..... 131056 = 2^17 - 2^4 .... 11111111111110000
19) ..... 262112 = 2^18 - 2^5 .... 111111111111100000
20) ..... 524224 = 2^19 - 2^6 .... 1111111111111000000
21) .... 1048448 = 2^20 - 2^7 .... 11111111111110000000
22) .... 2096896 = 2^21 - 2^8 .... 111111111111100000000
23) .... 4193792 = 2^22 - 2^9 .... 1111111111111000000000
24) .... 8387584 = 2^23 - 2^10 ... 11111111111110000000000
25) ... 16775168 = 2^24 - 2^11 ... 111111111111100000000000
26) ... 33550336 = 2^25 - 2^12 ... 1111111111111000000000000 ... (The 5th perfect number)

Omar E. Pol, Table of n, a(n) for n = 1..26 (shows all terms).
Index entries for sequences related to divisors of numbers

For more information see A133025 (Divisors of 33550336). Cf. A000043, A000079, A000396, A000668, A019279, A061645, A061652.

a(n)=A133025(n), written in base 2. Also, for n=1 .. 26: If n<=(A000043(5)=13) then a(n) is the concatenation of the digit "1" and n-1 digits "0" else a(n) is the concatenation of A000043(5)=13 digits "1" and (n-1-A000043(5)) digits "0".

A139237 Second differences of even superperfect numbers A061652, divided by 2.

Original entry on oeis.org

5, 18, 1992, 28704, 67584, 536641536, 576460751229812736, 154742503757751030292414464, 40564509722294095963578081214464, 42535214735633635831150802674328272896

Offset: 1

Omar E. Pol, Apr 18 2008

nonn

Second differences of Mersenne primes A000668, divided by 4 (see A139232).

Also, second differences of superperfect numbers A019279, divided by 2, if there are no odd superperfect numbers.

Paolo Xausa, Table of n, a(n) for n = 1..16

Cf. A000668, A019279, A061652, A139228, A139229, A139230, A139231, A139232, A139233, A139234, A139235, A139236.

Differences[2^MersennePrimeExponent[Range[14]]-1,2]/4 (* Paolo Xausa, Oct 20 2023 *)

a(n) = A139236(n)/2.

More terms from Michel Marcus, Jul 09 2017

A019281 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,3)-perfect numbers.

Original entry on oeis.org

8, 21, 512

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No further term < 10^9 [see Table 1].
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
a(4) > 4*10^12, if it exists. - Giovanni Resta, Feb 26 2020

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Experimental Mathematics, Home Page.

Cf. A019278, A019279, A019282, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019290, A019291.

A019282 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,4)-perfect numbers.

Original entry on oeis.org

15, 1023, 29127, 355744082763

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
a(5) > 4*10^12, if it exists. - Giovanni Resta, Feb 26 2020

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A019278, A019279, A019281, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019290, A019291.

Select[Range[100000], DivisorSigma[1, DivisorSigma[1, #]]/# == 4 &] (* Robert Price, Apr 07 2019 *)

isok(n) = sigma(sigma(n))/n  == 4; \\ Michel Marcus, May 12 2016

a(4) from Jud McCranie, Feb 08 2012

A134708 Even superperfect numbers divided by 2.

Original entry on oeis.org

1, 2, 8, 32, 2048, 32768, 131072, 536870912, 576460752303423488, 154742504910672534362390528, 40564819207303340847894502572032, 42535295865117307932921825928971026432

Offset: 1

Omar E. Pol, Nov 07 2007, Apr 23 2008

nonn

a(13) and a(14) have 157 and 183 digits respectively. - R. J. Mathar, Jan 07 2008
Largest proper divisor of n-th even superperfect number A061652(n). Also, largest proper divisor of n-th superperfect number A019279(n), if there are no odd superperfect numbers.
Indices of even hexagonal numbers (A014635) that are also even perfect numbers. - Omar E. Pol, Jan 11 2009

			a(5) = 2048 because the 5th even superperfect number is 4096 and 4096/2 = 2048.

Amiram Eldar, Table of n, a(n) for n = 1..18
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos [From _Omar E. Pol_, Jan 11 2009]

Cf. A019279, A061652, A133028.
Cf. A000043.
Cf. A032742, A138882, A139248.
Cf. A000217, A000396, A014635. - Omar E. Pol, Jan 11 2009

A000043 := proc(n) op(n,[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213]) ; end: A061652 := proc(n) 2^(A000043(n)-1) ; end: A134708 := proc(n) A061652(n)/2 ; end: seq(A134708(n),n=1..14) ; # R. J. Mathar, Jan 07 2008

With[{max = 12}, 2^(MersennePrimeExponent[Range[max]] - 2)] (* Amiram Eldar, Oct 21 2024 *)

a(n) = A061652(n)/2.
a(n) = 2^(A000043(n)-2). - Omar E. Pol, Mar 01 2008
a(n) = A032742(A061652(n)). Also, a(n) = A032742(A019279(n)), if there are no odd superperfect numbers.
a(n) = Sum_{x=1..n-th superperfect number} x*(-1)^x. - Juri-Stepan Gerasimov, Jul 21 2009

More terms from R. J. Mathar, Jan 07 2008

A135652 Divisors of 28 (the 2nd perfect number), written in base 2.

Original entry on oeis.org

1, 10, 100, 111, 1110, 11100

Offset: 1

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

The number of divisors of the second perfect number is equal to 2*A000043(2)=A061645(2)=6.

			The structure of divisors of 28 (see A018254)
----------------------------------------------------------------------
n ... Divisor . Formula ....... Divisor written in base 2 ............
----------------------------------------------------------------------
1)......... 1 = 2^0 ........... 1
2)......... 2 = 2^1 ........... 10
3)......... 4 = 2^2 ........... 100 .... (The 2nd superperfect number)
4)......... 7 = 2^3 - 2^0 ..... 111 .... (The 2nd Mersenne prime)
5)........ 14 = 2^4 - 2^1 ..... 1110
6)........ 28 = 2^5 - 2^2 ..... 11100... (The 2nd perfect number)

Index entries for sequences related to divisors of numbers

For more information see A018254 (Divisors of 28). Cf. A000043, A000079, A000396, A000668, A019279, A061645, A061652.

FromDigits[IntegerDigits[#,2]]&/@Divisors[28] (* Harvey P. Dale, Nov 14 2020 *)

apply(n->fromdigits(binary(n)), divisors(28)) \\ Charles R Greathouse IV, Jun 21 2017

a(n)=A018254(n), written in base 2. Also, for n=1 .. 6: If n<=(A000043(2)=3) then a(n) is the concatenation of the digit "1" and n-1 digits "0" else a(n) is the concatenation of A000043(2)=3 digits "1" and (n-1-A000043(2)) digits "0".

A139232 Second differences of Mersenne primes A000668.

Original entry on oeis.org

20, 72, 7968, 114816, 270336, 2146566144, 2305843004919250944, 618970015031004121169657856, 162258038889176383854312324857856, 170140858942534543324603210697313091584

Offset: 1

Omar E. Pol, Apr 19 2008

nonn

Second differences of even superperfect numbers, multiplied by 2 (see A139236).

G. C. Greubel, Table of n, a(n) for n = 1..16

Cf. A000668, A019279, A061652, A139228, A139229, A139230, A139231, A139233, A139234, A139235, A139236, A139237.

A000668 := Select[2^Range[1000] - 1, PrimeQ]; Table[A000668[[n + 2]] - 2*A000668[[n + 1]] + A000668[[n]], {n, 1, 6}] (* G. C. Greubel, Oct 03 2017 *)
Differences[2^MersennePrimeExponent[Range[14]]-1,2] (* Paolo Xausa, Oct 20 2023 *)

a(n) = A139236(n)*2.

Terms a(7) - a(10) added by G. C. Greubel, Oct 03 2017

A139235 First differences of even superperfect numbers A061652, divided by 2.

Original entry on oeis.org

1, 6, 24, 2016, 30720, 98304, 536739840, 576460751766552576, 154742504334211782058967040, 40564664464798430175360140181504, 42535255300298100629580978034468454400

Offset: 1

Omar E. Pol, Apr 18 2008

nonn

First differences of Mersenne primes A000668, divided by 4 (see A139231).

Also, first differences of superperfect numbers A019279, divided by 2, if there are no odd superperfect numbers.

Paolo Xausa, Table of n, a(n) for n = 1..17

Cf. A000668, A019279, A061652, A139228, A139229, A139230, A139231, A139232, A139233, A139234, A139236, A139237.

Differences[2^MersennePrimeExponent[Range[14]]-1]/4 (* Paolo Xausa, Oct 20 2023 *)

a(n) = A139234(n)/2.

a(8)-a(11) from Jinyuan Wang, Mar 04 2020

A139236 Second differences of even superperfect numbers A061652.

Original entry on oeis.org

10, 36, 3984, 57408, 135168, 1073283072, 1152921502459625472, 309485007515502060584828928, 81129019444588191927156162428928, 85070429471267271662301605348656545792

Offset: 1

Omar E. Pol, Apr 18 2008

nonn

Second differences of Mersenne primes A000668, divided by 2 (see A139232).

Also, second differences of superperfect numbers A019279, if there are no odd superperfect numbers.

Jinyuan Wang, Table of n, a(n) for n = 1..16

Cf. A000668, A019279, A061652, A139228, A139229, A139230, A139231, A139232, A139233, A139234, A139235, A139237.

Differences[2^(Select[Range[512],PrimeQ[2^#-1]&]-1),2] (* Harvey P. Dale, Oct 15 2017 *)

a(n) = A139234(n+1) - A139234(n).

a(6) corrected and more terms from Joerg Arndt, Jul 09 2017

A173898 Decimal expansion of sum of the reciprocals of the Mersenne primes.

Original entry on oeis.org

5, 1, 6, 4, 5, 4, 1, 7, 8, 9, 4, 0, 7, 8, 8, 5, 6, 5, 3, 3, 0, 4, 8, 7, 3, 4, 2, 9, 7, 1, 5, 2, 2, 8, 5, 8, 8, 1, 5, 9, 6, 8, 5, 5, 3, 4, 1, 5, 4, 1, 9, 7, 0, 1, 4, 4, 1, 9, 3, 1, 0, 6, 5, 2, 7, 3, 5, 6, 8, 7, 0, 1, 4, 4, 0, 2, 1, 2, 7, 2, 3, 4, 9, 9, 1, 5, 4, 8, 8, 3, 2, 9, 3, 6, 6, 6, 2, 1, 5, 3, 7, 4, 0, 3, 2, 4

Offset: 0

Jonathan Vos Post, Mar 01 2010

We know this a priori to be strictly less than the Erdős-Borwein constant (A065442), which Erdős (1948) showed to be irrational. This new constant would also seem to be irrational.

			Decimal expansion of (1/3) + (1/7) + (1/31) + (1/127) + (1/8191) + (1/131071) + (1/524287) + ... = .5164541789407885653304873429715228588159685534154197.
This has continued fraction expansion 0 + 1/(1 + 1/(1 + 1/(14 + 1/(1 + ...)))) (see A209601).

Peter B. Borwein, On the Irrationality of Certain Series, Math. Proc. Cambridge Philos. Soc. 112, 141-146, 1992.
Paul Erdős, On Arithmetical Properties of Lambert Series, J. Indian Math. Soc. 12, 63-66, 1948.
Yoshihiro Tanaka, On the Sum of Reciprocals of Mersenne Primes, American Journal of Computational Mathematics, Vol. 7, No. 2 (2017), pp. 145-148.
Eric Weisstein's World of Mathematics, Erdos-Borwein Constant.
Marek Wolf, Computer experiments with Mersenne primes, arXiv preprint arXiv:1112.2412 [math.NT], 2011.

Cf. A209601, A000668, A065442 (decimal expansion of Erdos-Borwein constant), A000043, A001348, A046051, A057951-A057958, A034876, A124477, A135659, A019279, A061652, A000225.

Digits := 120 ; L := [ 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917 ] ;
x := 0 ; for i from 1 to 30 do x := x+1.0/(2^op(i,L)-1 ); end do ;

RealDigits[Sum[1/(2^p - 1), {p, MersennePrimeExponent[Range[14]]}], 10, 100][[1]] (* Amiram Eldar, May 24 2020 *)

isM(p)=my(m=Mod(4,2^p-1));for(i=1,p-2,m=m^2-2);!m
s=1/3;forprime(p=3,default(realprecision)*log(10)\log(2), if(isM(p), s+=1./(2^p-1)));s \\ Charles R Greathouse IV, Mar 22 2012

Sum_{i>=1} 1/A000668(i).

Entry revised by N. J. A. Sloane, Mar 10 2012

cp's OEIS Frontend

A135655 Divisors of 33550336 (the 5th perfect number), written in base 2.

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A139237 Second differences of even superperfect numbers A061652, divided by 2.

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A019281 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,3)-perfect numbers.

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A019282 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,4)-perfect numbers.

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A134708 Even superperfect numbers divided by 2.

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A135652 Divisors of 28 (the 2nd perfect number), written in base 2.

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A139232 Second differences of Mersenne primes A000668.

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A139235 First differences of even superperfect numbers A061652, divided by 2.

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