

A330836


Numbers of the form 2^(2*p1)*3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.


3



4704, 1476096, 396386304, 6753750274277376, 442715102395357986816, 113336363243719574421504, 31901471869127420013759771876790370304, 42404329554681223873219247037048711787234652848116929825491652260298489856
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OFFSET

1,1


COMMENTS

Also numbers with powerspectral basis {M_p^2*(M_p+2)^2, M_p^2*(M_p+1)^2, (M_p^21)^2}. The first element of the spectral basis of a(n) is A330819(n+1), the second element is A330837(n+1), and the third element is A330820(n+1). Generally, a powerspectral 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 2*a(n)+1. In this case, we say that a(n) has index 2.
a(n) is also isospectral with A330838(n), that is, a(n) and A330838(n) have the same spectral basis, but A330838(n) has index 1. Thus, A330838(n) and a(n) form an isospectral pair.
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


LINKS

Table of n, a(n) for n=1..8.
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


FORMULA

a(n) = A139306(n+1) * 3 * A133049(n+1).


EXAMPLE

If p = 3, then a(1) = 2^(2*31)*3*7^2 = 4704, and the spectral basis of 4704 is {63^2, 56^2, 48^2}, consisting of powers. The spectral sum of a(1), that is, the sum of the elements of its spectral basis, is 2*4704+1 = 9409. In this case, we say that a(1) has index 2. The number A330838(1) = 9704 has the same spectral basis as a(1), but with index 1. We say that A330838(1) and a(1) are isospectral and form an isospectral pair.


MAPLE

a := proc(n::posint)
local p, m;
p:=NumberTheory[IthMersenne](n+1);
m:=2^p1;
return 2^(2*p1)*3*m^2;
end:


MATHEMATICA

f[p_] := 2^(2*p  1)*3*(2^p  1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 12 2020 *)


CROSSREFS

Cf. A000043, A000668, A133049, A139306, A330819, A330820, A330837, A330838.
Sequence in context: A237699 A235022 A223246 * A107544 A092375 A252239
Adjacent sequences: A330833 A330834 A330835 * A330837 A330838 A330839


KEYWORD

nonn


AUTHOR

Walter Kehowski, Jan 12 2020


STATUS

approved



