A330818 -id:A330818 - cp's OEIS frontend

A330819 Numbers of the form M_p^2(M^p+2)^2, where M_p is a Mersenne prime (A000668) and p is a Mersenne exponent (A000043). Also the first element of the spectral basis of A330817.

225, 3969, 1046529, 268402689, 4503599493152769, 295147905144993087489, 75557863725364567605249, 21267647932558653957237540927630737409, 28269553036454149273332760011886696242605918383730576346715242738439159809

Offset: 1

Walter Kehowski, Jan 01 2020

nonn

The second element of the spectral basis of A330817 is A330820.

			If p=2, then M_2=3, and a(1) = 3^2(3+2)^2 = 15^2 = 225.

Walter Kehowski, Table of n, a(n) for n = 1..12

Cf. A000043, A000668, A132794, A133049, A330817, A330818, A330820.

A330819:=[]:
for www to 1 do
for i from 1 to 31 do
  #ithprime(31)=127
  p:=ithprime(i);
  q:=2^p-1;
  if isprime(q) then x:=2^(2*p+1)*q^2; A330819:=[op(A330819),x]; fi;
od;
od;
A330819;

(m = 2^MersennePrimeExponent[Range[9]] - 1)^2 * (m + 2)^2 (* Amiram Eldar, Jan 03 2020 *)

a(n) = A000668(n)^2*(A000668(n)+2)^2.

A330820 Numbers of the form (M_p^2-1)^2, where M_p is a Mersenne prime, A000668. Also the second element of the power-spectral basis of A330817.

Original entry on oeis.org

64, 2304, 921600, 260112384, 4501400872550400, 295138898048817561600, 75557287266261531623424

Offset: 1

Walter Kehowski, Jan 06 2020

nonn

The first element of the power-spectral basis of A330817 is A330819.

			If n=1, a(1)=(3^2-1)^2=64.

Cf. A000043, A000668, A330817, A330818, A330819.

A330820:=[]:
for www to 1 do
for i from 1 to 31 do
#ithprime(31)=127
  p:=ithprime(i);
  q:=2^p-1;
if isprime(q) then x:=(q^2-1)^2; A330820:=[op(A330820),x] fi;
od;
od;
A330820;

Array[((2^MersennePrimeExponent[#] - 1)^2 - 1)^2 &, 10] (* Amiram Eldar, Jan 07 2020 *)

a(n) = (A000668(n)^2-1)^2.

A330817 Numbers of the form 2^(2p+1)M_p^2, where M_p is a Mersenne prime, A000668, with Mersenne exponent p, A000043.

Original entry on oeis.org

288, 6272, 1968128, 528515072, 9005000365703168, 590286803193810649088, 151115150991626099228672, 42535295825503226685013029169053827072, 56539106072908298497625662716064949049646203797489239767322203013731319808

Offset: 1

Walter Kehowski, Jan 01 2020

nonn

Also numbers with power-spectral basis {M_p^2*(M_p+2)^2,(M_p^2-1)^2}.

The first factor of a(n) is A330818. The first element of the spectral basis of a(n) is A330819, and the second element is A330820.

			Since p=2 and M_2=3, then a(1)=2^(2*2+1)*3^3=288, and 288 has spectral basis {15^2, 2^6}, consisting of powers.

Walter Kehowski, Table of n, a(n) for n = 1..12

Cf. A000043, A000668, A132794, A133049, A330818, A330819, A330820.

A330817:=[]:
for www to 1 do
for i from 1 to 31 do
  #ithprime(31)=127
  p:=ithprime(i);
  q:=2^p-1;
  if isprime(q) then x:=2^(2*p+1)*q^2; A330817:=[op(A330817),x]; fi;
od;
od;
A330817;

2^(2 * (p = MersennePrimeExponent[Range[9]]) + 1) * (2^p - 1)^2 (* Amiram Eldar, Jan 03 2020 *)

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).

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).

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.

Original entry on oeis.org

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).

cp's OEIS Frontend

A330819 Numbers of the form M_p^2(M^p+2)^2, where M_p is a Mersenne prime (A000668) and p is a Mersenne exponent (A000043). Also the first element of the spectral basis of A330817.

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A330820 Numbers of the form (M_p^2-1)^2, where M_p is a Mersenne prime, A000668. Also the second element of the power-spectral basis of A330817.

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A330817 Numbers of the form 2^(2*p+1)*M_p^2, where M_p is a Mersenne prime, A000668, with Mersenne exponent p, A000043.

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

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A330839 Numbers of the form 2^(2*p+1)*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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Maple

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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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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A330817 Numbers of the form 2^(2p+1)M_p^2, where M_p is a Mersenne prime, A000668, with Mersenne exponent p, A000043.

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

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.

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.