A000043 -id:A000043 - cp's OEIS frontend

A126044 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 4.

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

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A010873, A126043-A126059.

Array[Mod[MersennePrimeExponent@ #, 4] &, 45] (* Michael De Vlieger, Apr 07 2018 *)

a(n) = A010873(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 Amiram Eldar, Oct 14 2024

A126058 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 18.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 1, 13, 7, 17, 17, 1, 17, 13, 1, 7, 13, 13, 5, 13, 5, 5, 17, 11, 11, 7, 1, 5, 1, 1, 1, 11, 5, 1, 11, 11, 5, 5, 1, 1, 13, 5, 7, 11, 5, 1, 17, 5

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A126043-A126059.

Array[Mod[MersennePrimeExponent@ #, 18] &, 45] (* Michael De Vlieger, Apr 10 2018 *)

a(45)-a(47) from Ivan Panchenko, Apr 09 2018

a(48) from Amiram Eldar, Oct 15 2024

A139117 Triangular numbers (A000217) with indices A000043.

Original entry on oeis.org

3, 6, 15, 28, 91, 153, 190, 496, 1891, 4005, 5778, 8128, 135981, 184528, 818560, 2427706, 2602621, 5176153, 9046131, 9783676, 46943205, 49416711, 62871291, 198751953, 235477551, 269340445, 990013753, 3718970646, 6105511756, 8718535225, 23347768186, 286403014380

Offset: 1

Omar E. Pol, May 08 2008

nonn

			a(4) = 28 because A000043(4) = 7 and the 7th triangular number A000217(7) is 28.

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

Cf. A000217, A000396, A034953.

Map[#*(#+1)/2 &, MersennePrimeExponent[Range[48]]] (* Amiram Eldar, Oct 21 2024 *)

a(n) = A000217(A000043(n)).

A159585 Nearest k to j such that k*(2^j-1)+1 is prime where j=A000043(n) and 2^j-1 = Mersenne-prime(n) = A000668(n). If there are two k values equidistant from j, each of which produces a prime, the larger of the two gets added to the sequence.

Original entry on oeis.org

2, 4, 10, 4, 46, 22, 16, 46, 66, 136, 166, 124, 636, 550, 1474, 3066, 1656, 1816, 3708, 9436, 1746, 3696, 11262, 40138, 25900, 20808, 60340, 58818

Offset: 1

Ray Chandler, Apr 16 2009

			n=6, j=A000043(6)=17, A000668(6)=131071, then k=12 or k=22 are the nearest values to j which produce primes so we take the larger of the two k values for a(6)=22.

Cf. A000043, A000668, A098556, A101416.

A177876 a(n) is the number of distinct prime factors in Lucas-Lehmer numbers A003010(k-2)/(2^k-1), where k = A000043(n+1).

Original entry on oeis.org

2, 3, 3, 4

Offset: 1

G. L. Honaker, Jr., Dec 13 2010

a(5) of this sequence will require the factorization of A003010(15), an 18742-digit integer divisible by 2^17-1.

			a(1)=2 because A003010(1)=14, contains exactly 2 distinct prime factors;
a(2)=3 because A003010(3)=37634, contains exactly 3 distinct prime factors;
a(3)=3 because A003010(5)=2005956546822746114, contains exactly 3 distinct prime factors;
a(4)=4 because A003010(11)=the associated 1172-digit Lucas-Lehmer number, contains exactly 4 distinct prime factors.

Cf. A000040, A000668, A003010, A177874, A177875, A177879, A177892

Definition corrected by Max Alekseyev, Jul 29 2024

A233008 p mod 24, where p is such that 2^p - 1 is prime (see Mersenne primes, A000043).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 7, 13, 17, 11, 7, 17, 7, 7, 19, 1, 1, 5, 7, 17, 5, 5, 17, 5, 1, 1, 11, 7, 1, 19, 23, 17, 19, 5, 5, 17, 17, 13, 19, 7, 23, 1, 17, 11, 1, 17, 17

Offset: 1

Freimut Marschner, Dec 03 2013

Chris K. Caldwell, Mersenne Primes.
Chris K. Caldwell, Recent Mersenne primes.
Walter Wegscheider, Liste der Mersenne-Primzahlen, Pädagogische Hochschule für Niederösterreich (in German).

Cf. A000043, A000668, A124477, A126043-A126059, A233009.

Mod[#, 24] &@ MersennePrimeExponent@ Range@ 45 (* Michael De Vlieger, Jul 22 2018 *)

a(n) = A000043(n) mod 24.

a(46)-a(47) corrected and a(48) removed by Gord Palameta, Jul 21 2018

a(48) from Amiram Eldar, Oct 15 2024

A233009 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 23.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 8, 15, 20, 15, 12, 15, 9, 14, 18, 4, 20, 21, 7, 6, 5, 12, 19, 12, 2, 15, 16, 11, 6, 6, 1, 15, 9, 7, 21, 5, 5, 3, 1, 19, 21, 22, 6, 6, 7, 6, 3

Offset: 1

Freimut Marschner, Dec 03 2013

			For n = 9, the 9th Mersenne prime index is A000043(9) = 61 and a(9) = 61 mod 23 = 15.

Chris K. Caldwell, Recent Mersenne primes.
Walter Wegscheider, Liste der Mersenne-Primzahlen, Pädagogische Hochschule für Niederösterreich (in German).

Cf. A000043, A126043-A126059, A233008.

Array[ Mod[ MersennePrimeExponent@#, 23] &, 44] (* Robert G. Wilson v, Aug 06 2018 *)

a(n) = A000043(n) mod 23.

a(46)-a(47) corrected and a(48) removed by Gord Palameta, Aug 06 2018

a(48) from Amiram Eldar, Oct 15 2024

A330163 Even perfect numbers m from A000396 such that w = (m + 2^(k(m) - 1) - 1) * 2^(2*(k(m) - 1)) is also an even perfect number, where k(m) is the Mersenne exponent A000043(m).

Original entry on oeis.org

6, 28, 8128, 2305843008139952128

Offset: 1

Jaroslav Krizek, Dec 04 2019

Corresponding values of even perfect numbers w: 28, 496, 33550336, 2658455991569831744654692615953842176, ... (A330164).

Corresponding values of Mersenne exponents k(m) and k(w): (2, 3, 7, 31, ...), (3, 5, 13, 61, ...), where k(w) = 2*k(m) - 1.

Cf. A000043, A000396, A330164.

[(2^k - 1) * (2^(k - 1)): k in [1..100] | SumOfDivisors((2^k - 1) * (2^(k - 1))) / ( (2^k - 1) * (2^(k - 1))) eq 2 and SumOfDivisors(((2^k - 1) * (2^(k - 1)) + (2^(k - 1) - 1)) * (2^(2*(k - 1)))) / (((2^k - 1) * (2^(k - 1)) + (2^(k - 1) - 1)) * (2^(2*(k - 1)))) eq 2]

f[n_] := 2^(n - 1)*(2^n - 1); g[n_] := 2^n - 2^((n - 1)/2); mers = MersennePrimeExponent[Range[10]]; g /@ Select[mers, MemberQ[f /@ mers, g[#]] &] (* Amiram Eldar, Dec 06 2019 *)

A330164 Even perfect numbers w from A000396 such that number m = w / 2^(k(w) - 1) - 2^((k(w) - 1)/2) + 1 = 2^k(w) - 2^((k(w) - 1)/2) is also an even perfect number, where k(w) is the Mersenne exponent (A000043) for number w.

Original entry on oeis.org

28, 496, 33550336, 2658455991569831744654692615953842176

Offset: 1

Jaroslav Krizek, Dec 04 2019

nonn

Corresponding values of even perfect numbers m: 6, 28, 8128, 2305843008139952128, ... (A330163).

Corresponding values of Mersenne exponents k(w) and k(m): (3, 5, 13, 61, ...), (2, 3, 7, 31, ...), where k(m) = (k(w) + 1)/2.

Cf. A000043, A000396, A330163.

[(2^k - 1) * 2^(k - 1): k in [1..100] | SumOfDivisors((2^k - 1) * 2^(k - 1)) / ((2^k - 1) * 2^(k - 1)) eq 2 and SumOfDivisors(2^k - 2^((k-1) div 2)) / (2^k - 2^((k-1) div 2) ) eq 2]

f[n_] := 2^(n - 1)*(2^n - 1); g[n_] := 2^n - 2^((n - 1)/2); mers = MersennePrimeExponent[Range[10]]; f /@ Select[mers, MemberQ[f /@ mers, g[#]] &] (* Amiram Eldar, Dec 06 2019 *)

A330839 Numbers of the form 2^(2p+1)3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

Original entry on oeis.org

18816, 5904384, 1585545216, 27015001097109504, 1770860409581431947264, 453345452974878297686016, 127605887476509680055039087507161481216, 169617318218724895492876988148194847148938611392467719301966609041193959424

Offset: 1

Walter Kehowski, Jan 21 2020

nonn

Also numbers with power-spectral basis {M_p^2*(M_p+2)^2, 4*M_p^2*(M_p+1)^2, (M_p^2-1)^2}, where by power-spectral basis we mean a spectral basis that consists of primes and powers. The first element of the power-spectral basis is A330819(n+1), the second element is A330840(n+1), and the third element is A330820(n+1).

Subsequence of Zumkeller numbers (A083207), since a(n) = 2^r * 3 * s, where s is relatively prime to 6. - Ivan N. Ianakiev, Feb 03 2020

			a(1) = 2^(2*3+1) * 3 * 7^2 = 18816, and 18816 has spectral basis {63^2, 112^2, 48^2}, consisting of powers.

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, A330818, A330819, A330820, A330840.

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

f[p_] := 2^(2p + 1)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 22 2020 *)

a(n) = A330818(n+1) * 3 * A133049(n+1).

cp's OEIS Frontend

A126044 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 4.

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A126058 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 18.

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A139117 Triangular numbers (A000217) with indices A000043.

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A159585 Nearest k to j such that k*(2^j-1)+1 is prime where j=A000043(n) and 2^j-1 = Mersenne-prime(n) = A000668(n). If there are two k values equidistant from j, each of which produces a prime, the larger of the two gets added to the sequence.

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A177876 a(n) is the number of distinct prime factors in Lucas-Lehmer numbers A003010(k-2)/(2^k-1), where k = A000043(n+1).

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A233008 p mod 24, where p is such that 2^p - 1 is prime (see Mersenne primes, A000043).

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A233009 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 23.

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A330163 Even perfect numbers m from A000396 such that w = (m + 2^(k(m) - 1) - 1) * 2^(2*(k(m) - 1)) is also an even perfect number, where k(m) is the Mersenne exponent A000043(m).

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A330164 Even perfect numbers w from A000396 such that number m = w / 2^(k(w) - 1) - 2^((k(w) - 1)/2) + 1 = 2^k(w) - 2^((k(w) - 1)/2) is also an even perfect number, where k(w) is the Mersenne exponent (A000043) for number w.

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Magma

A330839 Numbers of the form 2^(2p+1)3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.