A330824 -id:A330824 - cp's OEIS frontend

A330837 a(n) = M(n)^2*(M(n)+1)^2, where M(n) = A000668(n) is the n-th Mersenne prime.

144, 3136, 984064, 264257536, 4502500182851584, 295143401596905324544, 75557575495813049614336, 21267647912751613342506514584526913536, 28269553036454149248812831358032474524823101898744619883661101506865659904

Offset: 1

Walter Kehowski, Jan 12 2020

nonn

a(n+1) is the second element of the power-spectral basis of both A330836(n) and A330838(n). Also, a(n) = A139256(n)^2, where A139256(n) is the sum of the divisors of the n-th perfect number, A000396(n).

Also: squares of twice the perfect numbers. - M. F. Hasler, Feb 07 2020

			If p=3, then a(2) = (7*2^3)^2 = 56^2, and the spectral basis of A330836(1) = 4704 and A330838(1) = 9408 is {63^2, 56^2, 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, A000396, A000668, A133049, A139306, A139256, A330819, A330820, A330836.

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

f[p_] := 2^(2p)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 12 2020 *)

forprime(p=1,999,isprime(2^p-1)&&print1((2^p-1)^2<<(2*p)",")) \\ M. F. Hasler, Feb 07 2020

a(n) = A330824(n) * A133049(n).

a(n) = (2*A000396(n))^2 = (2^p-1)^2*4^p with p = A000043(n). - M. F. Hasler, Feb 07 2020

A330838 Numbers of the form 2^(2p)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

9408, 2952192, 792772608, 13507500548554752, 885430204790715973632, 226672726487439148843008, 63802943738254840027519543753580740608, 84808659109362447746438494074097423574469305696233859650983304520596979712

Offset: 1

Walter Kehowski, Jan 17 2020

nonn

a(n) has the same spectral basis as A330836(n), namely {M_p^2*(M_p+2)^2, M_p^2*(M_p+1)^2, (M_p^2-1)^2}, so the two numbers are isospectral as well as power-spectral, that is, they have the same spectral basis and that basis consists of powers. The spectral sum of a(n), that is, the sum of the elements of its spectral basis, is 1*a(n)+1, while the spectral sum of A330836(n) is 2*A330836(n)+1. We say that a(n) and A330836(n) form an isospectral pair, with a(n) of index 1 and A330836(n) of index 2.

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

			If p = 3, then M_3 = 7 and a(1) = 2^(2*3)*3*7^2 = 9408, with spectral basis {63^2, 56^2, 48^2}, and spectral sum equal to 1*9408 + 1 = 9409. However, {63^2, 56^2, 48^2} is also the spectral basis of A330836(1) = 4704, with spectral sum equal to 2*4704+1.

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, A330819, A330820, A330824, A330825, A330826, A330836, A330837.

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

f[p_] := 2^(2p)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9]  (* Amiram Eldar, Jan 17 2020 *)

a(n) = A330824(n+1) * 3 * A133049(n+1).

A330840 a(n) = 4M(n)^2(M(n)+1)^2, where M(n) is the n-th Mersenne prime, A000668.

Original entry on oeis.org

576, 12544, 3936256, 1057030144, 18010000731406336, 1180573606387621298176, 302230301983252198457344, 85070591651006453370026058338107654144, 113078212145816596995251325432129898099292407594978479534644406027462639616

Offset: 1

Walter Kehowski, Jan 23 2020

nonn

Also a(n+1) is the second element of the power-spectral basis of A330839(n), where by power-spectral we mean that the spectral basis consists of primes and powers.

			a(2) = 4*7^2*2^(2*3) = 2^8*7^2 = 112^2, and the spectral basis of A330839(1) = 18816 is {63^2, 112^2, 48^2}, consisting only of powers.

Garret Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A000043, A000668, A133049, A330818, A330819, A330820, A330839.

A330840 := proc(n::posint)
  local p, m;
  p:=NumberTheory[IthMersenne](n);
  m:=2^p-1;
  return 4*m^2*(m+1)^2;
end:

f[p_] := 2^(2*p + 2)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[9] (* Amiram Eldar, Jan 24 2020 *)

a(n) = 4 * A133049(n) * A330824(n).

cp's OEIS Frontend

A330837 a(n) = M(n)^2*(M(n)+1)^2, where M(n) = A000668(n) is the n-th Mersenne prime.

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A330838 Numbers of the form 2^(2p)3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

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Programs

Maple

Mathematica

Formula

A330840 a(n) = 4M(n)^2(M(n)+1)^2, where M(n) is the n-th Mersenne prime, A000668.

Views

Author

Keywords

Comments

Examples

Links

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Programs

Maple

Mathematica

Formula

cp's OEIS Frontend

A330837 a(n) = M(n)^2*(M(n)+1)^2, where M(n) = A000668(n) is the n-th Mersenne prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A330838 Numbers of the form 2^(2*p)*3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A330840 a(n) = 4*M(n)^2*(M(n)+1)^2, where M(n) is the n-th Mersenne prime, A000668.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A330838 Numbers of the form 2^(2p)3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

A330840 a(n) = 4M(n)^2(M(n)+1)^2, where M(n) is the n-th Mersenne prime, A000668.