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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.
1
3528, 1107072, 297289728, 5065312705708032, 332036326796518490112, 85002272432789680816128, 23926103901845565010319828907592777728, 31803247166010917904914435277786533840425989636087697369118739195223867392
OFFSET
1,1
COMMENTS
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.
LINKS
G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.
FORMULA
a(n) = A152922(n+1) * 9 * A133049(n+1).
EXAMPLE
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.
MAPLE
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;
MATHEMATICA
f[p_] := 9*2^(2*p - 3)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Feb 07 2020 *)
KEYWORD
nonn
AUTHOR
Walter Kehowski, Jan 25 2020
STATUS
approved