A090748 -id:A090748 - cp's OEIS frontend

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

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 *)

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.

A319535 Primes of the form 2*6^k - 1.

Original entry on oeis.org

11, 71, 431, 2591, 15551, 4353564671, 5642219814911, 341163456359156416511, 2046980738154938499071, 20628849596981071092343898111, 26734989077687468135677691953151, 207891275068097752223029732627709951, 269427092488254686881046533485512097791

Offset: 1

Jianing Song, Sep 22 2018

nonn

Primes in A164559.

Companion sequence of A057472. There are 49 terms known in this sequence.

			2*6^1 - 1 = 11, 2*6^2 - 1 = 71, 2*6^3 - 1 = 431, 2*6^4 - 1 = 2591 and 2*6^5 - 1 = 15551 are primes, but 2*6^6 - 1 = 93311 = 23*4057 is not.

K. D. Bajpai, Table of n, a(n) for n = 1..26

Integers k such that 2*b^k - 1 is prime: A090748 (b=2), A003307 (b=3), A120375 (b=5), A057472 (b=6), A002959 (b=7), A002957 (b=10), A120378 (b=11).
Primes of the form 2*b^k - 1: A000668 (b=2), A079363 (b=3), A120376 (b=5), this sequence (b=6), A158795 (b=7), A055558 (b=10), A120377 (b=11).
Cf. also A000043, A002958, A068231, A164559.

[k: n in [1..100] | IsPrime(k) where k is 2*6^n-1];  // K. D. Bajpai, Nov 15 2019

A319535:= n-> (2*6^n-1): select(isprime, [seq((A319535(n), n=1..200))]);  # K. D. Bajpai, Nov 15 2019

Select[Table[2*6^k-1,{k,1600}], PrimeQ[#]&]  (* K. D. Bajpai, Nov 15 2019 *)

for(n=1, 99, my(t); if(ispseudoprime(t=2*6^n-1), print1(t", ")))

a(n) = 2*6^A057472(n) - 1.

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

A281622 Numbers k such that sigma(k-1) is a Mersenne prime (A000668).

Original entry on oeis.org

3, 5, 17, 26, 65, 4097, 65537, 262145, 1073741825

Offset: 1

Jaroslav Krizek, Jan 25 2017

Conjecture 1: the next terms are: 1152921504606846977, 309485009821345068724781057, 81129638414606681695789005144065, 85070591730234615865843651857942052865.
Conjecture 2: Union of 26 and A256438.
Conjecture 3: Mersenne prime 31 is the only prime p such that p = sigma(x-1) = sigma(y-1) for distinct numbers x and y; 31 = sigma(17-1) = sigma(26-1).

			65 is a term because sigma(64) = 127 (Mersenne prime).

Union of 26 and odd terms of A270413.
Prime terms are in A249759.
Subsequence of A270413.
Cf. A000203, A000668, A256438.

[n: n in[2..1000000], k in [1..20] | SumOfDivisors(n-1) eq 2^k-1 and IsPrime(2^k-1)];

isok(n) = my(s = sigma(n-1)); isprime(s) && ispower(s+1,,&p) && (p==2); \\ Michel Marcus, Jan 27 2017

Conjecture: a(n) = 2^A090748(n) + 1. - Daniel Suteu, Feb 08 2017

A358546 Least odd number m such that m mod 3 > 0 and m*3^n is an amicable number, and -1 if no such number exists.

Original entry on oeis.org

5480828320492525, 4865, 7735, 455, 131285, 849355, 11689795, 286385, 187047685, 104255, 32851039955, 2085985, 47942199242945, 189296520259, 349700961302721360788238344333849, 580068028631, 50392682631679406080371010751466781

Offset: 0

Jean-Marc Rebert, Nov 21 2022

If a(n) > -1 then a(n)*3^n is the least amicable number k such that A007949(k) = n.

			a(1) = 4865, because 4865 is an odd number and 4865 % 3 > 0 and 4865 * 3 = 14595 is an amicable number, and no lesser number has this property.

BOINC, Amicable Numbers

Cf. A347770, A000396, A001065, A002025, A259180, A262625.

Cf. A090748, A358415, A358320, A358022.

sigmap(k)=if(k,sigma(k)-k,0)
cycle(k, TT=2)=my(x=k, T=1); while(x>0&&T<=TT, x=sigmap(x); if(x==k, return(T)); T++)
a(n, TT=2)=my(p3n=3^n); forstep(m=1, +oo, 2, if(m%3&&cycle(p3n*m, TT)==2, return(m)))

cp's OEIS Frontend

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

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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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A319535 Primes of the form 2*6^k - 1.

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

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

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