A000043 -id:A000043 - cp's OEIS frontend

A330841 Numbers of the form 2^(2p-3)9*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

3528, 1107072, 297289728, 5065312705708032, 332036326796518490112, 85002272432789680816128, 23926103901845565010319828907592777728, 31803247166010917904914435277786533840425989636087697369118739195223867392

Offset: 1

Walter Kehowski, Jan 25 2020

nonn

a(1) = 3528 has power-spectral basis {21^2, 28^2, 48^2}, of index 1. If n > 1, then a(n) has power-spectral basis {M^2*(M+2)^2, (1/4)*M^2*(M+1)^2, (M^2-1)^2}, with index 2, where M=A000668(n+1) is the (n+1)-st Mersenne prime. The first element of the spectral basis of a(n), n > 1, is A330819(n+1), the second element is A133051(n+1), and the third element is A330820(n+1). Generally, a power-spectral basis is a spectral basis that consists of primes and powers.
The spectral sum of a(n), that is, the sum of the elements of its spectral basis, is a(1) + 1 whenever n = 1, and 2*a(n)+1 whenever n > 1. In this case, we say that a(n) has index 1 and index 2, respectively.
a(n), n > 1, is also isospectral with 9*A133051(n), that is, a(n) and 9*A133051(n) have the same spectral basis, but 9*A133051(n) has index 1. Thus 9*A133051(n) and a(n) form an isospectral pair.

			a(2) = 2^(2*5-3)*9*31^2 = 2^7*9*31^2 = 1107072 has spectral basis {1023^2, 496^2, 960^2}, consisting of powers. The spectral sum of a(2), that is, the sum of the elements of its spectral basis, is 2*a(2)+1 = 2214145. In this case we say that a(2) has index 2. The number 9 * A330817(2) = 2^(2*5-2)*9*31^2 = 2^8*9*31^2 = 2214144 has the same spectral basis as a(2), but with index 1. We say that 9 * A330817(2) and a(2) are isospectral and form an isospectral pair.

G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A000043, A000668, A133049, A133051, A152921, A152922, A330818, A330819, A330820, 9*A133051, 9*A330817, A330837.

a := proc(n::posint)
local p, m;
p:=NumberTheory[IthMersenne](n+1);
m:=2^p-1;
return 2^(2*p-3)*9*m^2;
end;

f[p_] := 9*2^(2*p - 3)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Feb 07 2020 *)

a(n) = A152922(n+1) * 9 * A133049(n+1).

A366818 Let p = A000043(n) be the n-th Mersenne exponent, then a(n) = ((2^p-1)^2-1)/p.

Original entry on oeis.org

4, 16, 192, 2304, 5160960, 1010565120, 14467203072, 148764064972013568, 87162491526879729295140036437606400, 4304762755241260838085244444377946703587691074682880, 246056756234946697892331840382404519263272106760845744463151104, 227937183538024006739312962615527377661237903932985846822055286571232395264

Offset: 1

Jianing Song, Oct 24 2023

nonn

a(n) is the largest k such that 2 is a k-th power in the finite field F_{2^p-1}(i), where i^2 = -1.

			In F_9 = F_3(i), we have 2 = (1+i)^2.
Jn F_49 = F_7(i), we have 2 = (3+i)^16.
In F_961 = F_31(i), we have 2 = (5+4*i)^192.

Amiram Eldar, Table of n, a(n) for n = 1..15

Cf. A000043, A000668.

A366818(lim) = my(q); forprime(p=2, lim, if(isprime(q=2^p-1), print1((q^2-1)/p, ", ")))

A065406 Mersenne prime exponents (A000043) which are also Sophie Germain primes (A005384).

Original entry on oeis.org

2, 3, 5, 89, 9689, 21701, 859433, 43112609

Offset: 1

Labos Elemer, Nov 06 2001

From Gord Palameta, Jul 19 2018: (Start)
All terms after the first two are congruent to 1 modulo 4, because if p is a Sophie Germain prime that is congruent to 3 modulo 4 then 2p + 1 divides 2^p - 1.
Boklan and Conway conjecture that this sequence is finite.
(End)

			31 = 2^5 - 1 and 11 = 2 * 5 + 1 are primes.

David W. Ash, Ian F. Blake, and Scott A. Vanstone, Low Complexity Normal Bases, Discrete Applied Mathematics, 25(1989), 206.
Kent D. Boklan, John H. Conway, Expect at most one billionth of a new Fermat Prime!, arXiv:1605.01371 [math.NT], 2016, p. 6.

Cf. A000043, A005384.

Select[Prime[Range[1000]], PrimeQ[2# + 1] && PrimeQ[2^# - 1] &] (* Alonso del Arte, Jul 20 2018 *)
Select[Prime@ Range[10^6], And[PrimeQ[2 # + 1], MersennePrimeExponentQ@ #] &] (* Michael De Vlieger, Jul 20 2018 *)

a(8) = 43112609, since the ordinal position of this term in A000043 is now confirmed. - Gord Palameta, Jul 19 2018

A126045 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 5.

Original entry on oeis.org

2, 3, 0, 2, 3, 2, 4, 1, 1, 4, 2, 2, 1, 2, 4, 3, 1, 2, 3, 3, 4, 1, 3, 2, 1, 4, 2, 3, 3, 4, 1, 4, 3, 2, 4, 1, 2, 3, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A010874, A126043-A126059.

Array[Mod[MersennePrimeExponent@ #, 5] &, 45] (* Michael De Vlieger, Apr 07 2018 *)

a(n) = A010874(A000043(n)). - Michel Marcus, Apr 07 2018

a(45)-a(47) from Ivan Panchenko, Apr 08 2018

a(48) from Amiram Eldar, Oct 14 2024

A126046 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 6.

Original entry on oeis.org

2, 3, 5, 1, 1, 5, 1, 1, 1, 5, 5, 1, 5, 1, 1, 1, 1, 1, 5, 1, 5, 5, 5, 5, 5, 1, 1, 5, 1, 1, 1, 5, 5, 1, 5, 5, 5, 5, 1, 1, 1, 5, 1, 5, 5, 1, 5, 5

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A010875, A126043-A126059.

Array[Mod[MersennePrimeExponent@ #, 6] &, 45] (* Michael De Vlieger, Apr 07 2018 *)

a(n) = A010875(A000043(n)). - Michel Marcus, Apr 07 2018

a(45)-a(46) from Ivan Panchenko, Apr 07 2018
a(47) from Ivan Panchenko, Apr 09 2018
a(48) from Max Alekseyev, Sep 19 2023

A126047 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 7.

Original entry on oeis.org

2, 3, 5, 0, 6, 3, 5, 3, 5, 5, 2, 1, 3, 5, 5, 5, 6, 4, 4, 6, 1, 1, 6, 1, 1, 4, 5, 3, 1, 1, 1, 6, 1, 6, 5, 3, 2, 5, 2, 1, 4, 5, 1, 2, 2, 4, 1, 5

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A010876, A126043-A126059.

Mod[MersennePrimeExponent[Range[48]], 7] (* Amiram Eldar, Oct 14 2024 *)

a(n) = A010876(A000043(n)). - Ivan Panchenko, Apr 07 2018

a(45)-a(46) from Ivan Panchenko, Apr 07 2018
a(47) from Ivan Panchenko, Apr 09 2018
a(48) from Amiram Eldar, Oct 14 2024

A126048 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 8.

Original entry on oeis.org

2, 3, 5, 7, 5, 1, 3, 7, 5, 1, 3, 7, 1, 7, 7, 3, 1, 1, 5, 7, 1, 5, 5, 1, 5, 1, 1, 3, 7, 1, 3, 7, 1, 3, 5, 5, 1, 1, 5, 3, 7, 7, 1, 1, 3, 1, 1, 1

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A010877, A126043-A126059.

Mod[MersennePrimeExponent[Range[47]],8] (* Harvey P. Dale, Apr 18 2019 *)

a(n) = A010877(A000043(n)). - Ivan Panchenko, Apr 07 2018

a(45)-a(46) from Ivan Panchenko, Apr 07 2018
a(47) from Ivan Panchenko, Apr 09 2018
a(48) from Amiram Eldar, Oct 14 2024

A126049 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 9.

Original entry on oeis.org

2, 3, 5, 7, 4, 8, 1, 4, 7, 8, 8, 1, 8, 4, 1, 7, 4, 4, 5, 4, 5, 5, 8, 2, 2, 7, 1, 5, 1, 1, 1, 2, 5, 1, 2, 2, 5, 5, 1, 1, 4, 5, 7, 2, 5, 1, 8, 5

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A010878, A126043-A126059.

Array[Mod[MersennePrimeExponent@ #, 9] &, 45] (* Michael De Vlieger, Apr 10 2018 *)

a(n) = A010878(A000043(n)). - Ivan Panchenko, Apr 07 2018

a(45)-a(46) from Ivan Panchenko, Apr 07 2018
a(47) from Ivan Panchenko, Apr 09 2018
a(48) from Amiram Eldar, Oct 14 2024

A126050 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 10.

Original entry on oeis.org

2, 3, 5, 7, 3, 7, 9, 1, 1, 9, 7, 7, 1, 7, 9, 3, 1, 7, 3, 3, 9, 1, 3, 7, 1, 9, 7, 3, 3, 9, 1, 9, 3, 7, 9, 1, 7, 3, 7, 1, 3, 1, 7, 7, 7, 1, 9, 1

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A010879, A126043-A126059.

Array[Mod[MersennePrimeExponent@ #, 10] &, 45] (* Michael De Vlieger, Apr 10 2018 *)

a(n) = A010879(A000043(n)). - Ivan Panchenko, Apr 07 2018

a(45)-a(46) from Ivan Panchenko, Apr 07 2018
a(47) from Ivan Panchenko, Apr 09 2018
a(48) from Amiram Eldar, Oct 14 2024

A126051 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 11.

Original entry on oeis.org

2, 3, 5, 7, 2, 6, 8, 9, 6, 1, 8, 6, 4, 2, 3, 3, 4, 5, 7, 1, 9, 8, 4, 5, 9, 10, 2, 3, 8, 5, 7, 6, 3, 3, 4, 6, 7, 1, 2, 3, 10, 1, 8, 8, 9, 2, 1, 4

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A010880, A126043-A126059.

Array[Mod[MersennePrimeExponent@ #, 11] &, 45] (* Michael De Vlieger, Apr 10 2018 *)

a(n) = A010880(A000043(n)). - Ivan Panchenko, Apr 07 2018

a(45)-a(46) from Ivan Panchenko, Apr 07 2018
a(47) from Ivan Panchenko, Apr 09 2018
a(48) from Amiram Eldar, Oct 14 2024

cp's OEIS Frontend

A330841 Numbers of the form 2^(2*p-3)*9*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

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A366818 Let p = A000043(n) be the n-th Mersenne exponent, then a(n) = ((2^p-1)^2-1)/p.

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A065406 Mersenne prime exponents (A000043) which are also Sophie Germain primes (A005384).

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A126045 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 5.

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A126046 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 6.

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A126047 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 7.

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Mathematica

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Extensions

A126048 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 8.

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Mathematica

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Extensions

A126049 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 9.

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Author

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Crossrefs

Programs

Mathematica

Formula

Extensions

A126050 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 10.

A330841 Numbers of the form 2^(2p-3)9*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.