A090748 -id:A090748 - cp's OEIS frontend

A119591 Least k such that 2*n^k - 1 is prime.

1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 2, 1, 10, 1, 1, 6, 1, 2, 6, 1, 2, 136, 1, 1, 6, 6, 1, 6, 1, 1, 2, 2, 1, 2, 1, 2, 4, 1, 2, 4, 4, 1, 2, 1, 1, 44, 1, 1, 2, 1, 3, 2, 5, 3, 2, 2, 1, 4, 1, 768, 4, 1, 1, 52, 34, 2, 132, 1, 1, 14, 7, 1, 2, 2, 1, 8, 1, 2, 10, 1, 24, 60, 1, 1, 2, 3, 5, 2, 1, 1, 2, 1, 1

Offset: 2

Pierre CAMI, Jun 01 2006

From Eric Chen, Jun 01 2015: (Start)
Conjecture: a(n) is defined for all n.
a(303) > 10000, a(304)..a(360) = {1, 2, 11, 1, 990, 1, 1, 2, 2, 4, 74, 5, 1, 10, 6, 6, 4, 1, 1, 2, 1, 9, 12, 1, 80, 2, 1, 1, 2, 14, 3, 2, 3, 1, 12, 1, 60, 36, 1, 8, 4, 34, 1, 522, 3, 15, 14, 1, 6, 2, 3, 1, 4, 5, 4, 10, 1}.
a(n) = 1 if and only if n is in A006254. (End)
From Eric Chen, Sep 16 2021: (Start)
Now a(303) is known to be 40174, also other terms > 10000: a(383) = 20956, a(515) = 58466, a(522) = 62288, a(578) = 129468, a(581) > 400000, a(590) = 15526, a(647) = 21576, a(662) = 16590, a(698) = 127558, a(704) = 62034, see the a-file and the references.
a(n) = 2 if and only if n is in A066049 but not in A006254.
a(n) = 3 if and only if n is in A214289 but not in A006254 or A066049. (End)

Eric Chen, Table of n, a(n) for n = 2..580
Gary Barnes, Riesel conjectures and proofs
Eric Chen, Table of n, a(n) for n = 2..2050 status
Prime Wiki, Riesel prime small bases least n

Cf. A119624, A253178.

Numbers r such that 2*k^r-1 is prime: A090748 (k=2), A003307 (k=3), A146768 (k=4), A120375 (k=5), A057472 (k=6), A002959 (k=7), ... (k=8), ... (k=9), A002957 (k=10), A120378 (k=11), ... (k=12), A174153 (k=13), A273517 (k=14), ... (k=15), ... (k=16), A193177 (k=17), A002958 (k=25).

f[n_] := Block[{k = 0}, While[ ! PrimeQ[2*n^k - 1], k++ ]; k ]; Table[f[n], {n, 2, 106}] (* Ray Chandler, Jun 08 2006 *)

a(n) = for(k=1, 2^24, if(ispseudoprime(2*n^k-1), return(k))) \\ Eric Chen, Jun 01 2015

From Eric Chen, Sep 16 2021: (Start)
a(6*n) = A098873(n).
a(2^n) = A279095(n).
a(A006254(n)) = 1.
a(A066049(n)) <= 2.
a(A214289(n)) <= 3. (End)

Corrected and extended by Ray Chandler, Jun 08 2006

A120376 Primes of the form 2*5^k - 1.

Original entry on oeis.org

1249, 31249, 305175781249, 119209289550781249, 1862645149230957031249, 111022302462515654042363166809082031249, 25243548967072377773175314089049159349542605923488736152648925781249

Offset: 1

Walter Kehowski, Jun 28 2006

nonn

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 1249 is 881 in base 12.

The next term has 125 digits. - Harvey P. Dale, Jan 26 2019

			a(1) = 4 since 2*5^4 - 1 = 1249 is the first prime.

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), this sequence (b=5), A158795 (b=7), A055558 (b=10), A120377 (b=11).
Cf. also A000043, A002958.

for w to 1 do for k from 1 to 2000 do n:=2*5^k-1; if isprime(n) then printf("%d, %d",k,n) fi od od;

Select[2*5^Range[100]-1,PrimeQ] (* Harvey P. Dale, Jan 26 2019 *)

for(k=1, 1e3, if(ispseudoprime(p=2*5^k-1), print1(p, ", "))); \\ Altug Alkan, Sep 22 2018

a(n) = 2*5^A120375(n) - 1 = 2*5^(2*A002958(n)) - 1. - Jianing Song, Sep 22 2018

A138831 n-th perfect number minus 1, written in base 2.

Original entry on oeis.org

101, 11011, 111101111, 1111110111111, 1111111111110111111111111, 111111111111111101111111111111111, 1111111111111111110111111111111111111

Offset: 1

Omar E. Pol, Apr 08 2008, Apr 09 2008, Apr 14 2008

Subset of A138148, cyclops numbers with binary digits, only.
Subset of A002113, palindromes in base 10.
a(n) has 2*A090748(n) digits 1.
The number of digits of a(n) is 2*A000043(n)-1, equal to A133033(n), the number of proper divisors of n-th perfect number.
a(n) = (A135627(n) written in base 2).

			n ... A000396(n) - 1 = A135627(n) ............. a(n)
1 ............ 6 - 1 = ...... 5 ............... 101
2 ........... 28 - 1 = ..... 27 .............. 11011
3 .......... 496 - 1 = .... 495 ............ 111101111
4 ......... 8128 - 1 = ... 8127 .......... 1111110111111
5 ..... 33550336 - 1 = 33550335 .... 1111111111110111111111111

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

Cf. A000043, A000396, A090748, A133033, A134808, A135627, A138131, A138148.

a(n) = A138148(A090748(n)).

A291901 Numbers n such that the sum of sums of elements of subsets of divisors of n is a perfect number (A000396).

Original entry on oeis.org

2, 4, 13, 16, 64, 4096, 65536, 262144, 3145341, 932181397, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864, 75603657215035519123837860069507929970384679

Offset: 1

Jaroslav Krizek, Nov 02 2017

nonn

Numbers n such that A229335(n) is in the sequence of perfect numbers, A000396.
Corresponding values of perfect numbers: 6, 28, 28, 496, 8128, 33550336, 8589869056, 137438691328, 33550336, ...
All even superperfect numbers A061652(n) are terms in this sequence.
Primes q of the form 2^(p-2) * (2^p - 1) - 1 where p is a Mersenne exponent (A000043) are terms: 2, 13, ...

			Divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; sum of sums of elements of subsets = 1 + 2 + 4 + 3 + 5 + 6 + 7 = 28 (perfect number).
sigma(16) * 2^(tau(16) - 1) = 31 * 16 = 496 (perfect number).

Max Alekseyev, Table of n, a(n) for n = 1..71

Cf. A000043, A000396, A061652, A090748, A229335.

[n: n in [1..10^6]  | SumOfDivisors(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1))) eq 2*(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1)))];

isA000396 := proc(n)
    numtheory[sigma](n)=2*n ;
    simplify(%) ;
end proc:
for n from 1 do
    if isA000396(A229335(n)) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2017

Select[Range[2^20], DivisorSigma[1, DivisorSigma[1, #] 2^(DivisorSigma[0, #] - 1)] == 2 (DivisorSigma[1, #] 2^(DivisorSigma[0, #] - 1)) &] (* Michael De Vlieger, Nov 02 2017 *)

Terms a(10) onward added by Max Alekseyev, Sep 18 2024

A309530 Number of power-of-two-divisors of sum of divisors of sum of divisors of powers of two: a(n) = A001511(A051027(A000079(n))).

Original entry on oeis.org

1, 3, 4, 4, 6, 4, 8, 5, 5, 10, 5, 6, 14, 12, 12, 6, 18, 10, 20, 11, 9, 9, 6, 8, 8, 18, 6, 15, 7, 16, 32, 7, 11, 22, 17, 14, 7, 24, 22, 13, 5, 13, 11, 12, 20, 10, 7, 11, 9, 16, 33, 22, 6, 10, 15, 17, 28, 12, 6, 20, 62, 36, 12, 9, 24, 16, 5, 26, 12, 26, 10, 18, 6, 12, 16, 28, 19, 26

Offset: 0

Juri-Stepan Gerasimov, Aug 06 2019

nonn

			a(0) = A001511(A051027(A000079(0))) = A001511(A051027(A000079(2^0))) = A001511(A051027(1)) = A001511(1) = 1.

Juri-Stepan Gerasimov, x = A001511(A051027(A000079(x))), SeqFan list, Aug 19 2019.

Cf. A000043 (numbers m such that m - 1 divides a(m - 1) - 2), A000079, A001511, A051027, A090748.

[Valuation(2*SumOfDivisors(SumOfDivisors(2^n)),2): n in [0..89]];

a(n) = valuation(2*sigma(sigma(2^n)), 2); \\ Michel Marcus, Aug 06 2019

from sympy import divisor_sigma
def A309530(n): return ((m:=int(divisor_sigma((1<Chai Wah Wu, Jul 13 2022

a(n) = A001511(A051027(A000079(n))).

A358320 Least odd number m such that m*2^n is a perfect, amicable or sociable number, and -1 if no such number exists.

Original entry on oeis.org

12285, 3, 7, 779, 31, 37, 127, 651, 2927269, 93, 25329329, 7230607, 8191, 66445153, 7613527, 18431675687, 131071, 264003743, 524287, 59592560831, 949755039781

Offset: 0

Jean-Marc Rebert, Nov 09 2022

For n in {1,2,4,6,12,16,18}, a(n)*2^n is a perfect number. See A090748.
For n in {0,3,5,8,10,11,13,14,15,17,19}, a(n)*2^n is an amicable number.
For n in {7,9} a(n)*2^n is a sociable number of order 28.
That is, h_k(m*2^n) = m*2^n for some k > 0, where h_{k+1}(n) = h_k(h(n)) and h(n) = A001065(n), the sum of aliquot parts of n. - Charles R Greathouse IV, Nov 09 2022
Least m such that m*2^n is in A347770. - Charles R Greathouse IV, Nov 09 2022

			a(1) = 3, because 3 is an odd number and 3 * 2^1 = 6 is a perfect number, and no lesser number has this property.

BOINC, Amicable Numbers
HandWiki, List of perfect numbers
David Moews, A list of currently known aliquot cycles of length greater than 2

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

Cf. A090748, A358415.

sigmap(n)=if(n<=1, return(0)); sigma(n)-n
cycle(n,TT=28)=my(x=n,T=1); while(x>0&&T<=TT, x=sigmap(x); if(x==n, return(T)); T++)
a(n,TT=28)=my(p2n=2^n); forstep(m=1, +oo, 2, if(cycle(p2n*m,TT), return(m)))

a(0), a(15)-a(20) from Jean-Marc Rebert, Nov 17 2022

A134712 Base-2 logarithm of (n-th even superperfect number divided by 2^n).

Original entry on oeis.org

0, 0, 1, 2, 7, 10, 11, 22, 51, 78, 95, 114, 507, 592, 1263, 2186, 2263, 3198, 4233, 4402, 9667, 9918, 11189, 19912, 21675, 23182, 44469, 86214, 110473, 132018, 216059, 756806, 859399, 1257752, 1398233, 2976184, 3021339, 6972554, 13466877

Offset: 1

Omar E. Pol, Nov 07 2007

nonn

			a(5) = 7 because the 5th even superperfect number is 4096, 2^5 = 32, 4096/32 = 128 and log_2(128) = 7 (because 2^7 = 128).

Amiram Eldar, Table of n, a(n) for n = 1..48
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A000668, A019279, A061652, A090748, A133028.

With[{max = 48}, MersennePrimeExponent[Range[max]] - Range[max] - 1] (* Amiram Eldar, Oct 21 2024 *)

a(n) = log_2(A061652(n)/(2^n)) = A000043(n) - n - 1 = A090748(n) - n.

A134713 Base-2 logarithm of (n-th even superperfect number divided by 2^n), plus 1.

Original entry on oeis.org

1, 1, 2, 3, 8, 11, 12, 23, 52, 79, 96, 115, 508, 593, 1264, 2187, 2264, 3199, 4234, 4403, 9668, 9919, 11190, 19913, 21676, 23183, 44470, 86215, 110474, 132019, 216060, 756807, 859400, 1257753, 1398234, 2976185, 3021340, 6972555, 13466878

Offset: 1

Omar E. Pol, Nov 07 2007

nonn

			a(5) = 8 because the 5th even superperfect number is 4096, 2^5 = 32, 4096/32 = 128, log_2(128) = 7 (because 2^7 = 128) and 7+1 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..48
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A000668, A019279, A061652, A090748, A133028.

With[{max = 48}, MersennePrimeExponent[Range[max]] - Range[max]] (* Amiram Eldar, Oct 21 2024 *)

a(n) = 1 + log_2(A061652(n)/(2^n)) = A000043(n) - n = A090748(n) - n + 1.

A135651 Even superperfect numbers written in base 2.

Original entry on oeis.org

10, 100, 10000, 1000000, 1000000000000, 10000000000000000, 1000000000000000000, 1000000000000000000000000000000

Offset: 1

Omar E. Pol, Feb 23 2008

Also, superperfect numbers (A019279) written in base 2 (If there are no odd perfect numbers).
Also, concatenation of "1" and A090748(n) digits "0".
The number of digits of a(n) is equal to A000043(n) and also equal to the number of digits of n-th Mersenne prime written in base 2 (see A117293, A135650).

			a(3)=10000 because the 3rd even superperfect number A061652(3)=16 and 16 written in base 2 is equal to 10000.

Cf. A000043, A000668, A019279, A061652, A090748, A117293, A135650.

A135656 Perfect numbers divided by 2, written in base 2.

Original entry on oeis.org

11, 1110, 11111000, 111111100000, 111111111111100000000000, 11111111111111111000000000000000, 111111111111111111100000000000000000, 111111111111111111111111111111100000000000000000000000000000

Offset: 1

Omar E. Pol, Feb 28 2008

The number of divisors of a(n) is equal to the number of its digits. This number is equal to 2*A000043(n)-2. The number of divisors of a(n) that are powers of 2 is equal to the number of divisors that are multiples of n-th Mersenne prime A000668(n) and this number of divisors is equal to A090748(n). The first digits of a(n) are "1". For n>1 the last digits are "0". The number of digits "1" is equal to A000043(n). The number of digits "0" is equal to A000043(n)-2. The concatenation of digits "1" gives the n-th Mersenne prime written in binary (see A117293(n)). The structure of divisors of a(n) represent a triangle (see example).

			a(4)=111111100000 because the 4th. perfect number is 8128 and 8128/2=4064 and 4064 written in base 2 is 111111100000. Note that 1111111 is the 4th. Mersenne prime A000668(4)=127, written in base 2.
The structure of divisors of a(4)=111111100000

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

Perfect numbers divided by 2: A133028. Cf. A000396, A000668, A019279, A090748, A117293, A135650.

a(n)=A133028(n) written in base 2.

cp's OEIS Frontend

A119591 Least k such that 2*n^k - 1 is prime.

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A120376 Primes of the form 2*5^k - 1.

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A138831 n-th perfect number minus 1, written in base 2.

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A291901 Numbers n such that the sum of sums of elements of subsets of divisors of n is a perfect number (A000396).

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A309530 Number of power-of-two-divisors of sum of divisors of sum of divisors of powers of two: a(n) = A001511(A051027(A000079(n))).

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A358320 Least odd number m such that m*2^n is a perfect, amicable or sociable number, and -1 if no such number exists.

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PARI

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A134712 Base-2 logarithm of (n-th even superperfect number divided by 2^n).

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