A152923
a(n) = 2^(2*p-1)/8, where p is A000043(n).
Original entry on oeis.org
1, 4, 64, 1024, 4194304, 1073741824, 17179869184, 288230376151711744, 332306998946228968225951765070086144, 23945242826029513411849172299223580994042798784118784, 1645504557321206042154969182557350504982735865633579863348609024
Offset: 1
A330836
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.
Original entry on oeis.org
4704, 1476096, 396386304, 6753750274277376, 442715102395357986816, 113336363243719574421504, 31901471869127420013759771876790370304, 42404329554681223873219247037048711787234652848116929825491652260298489856
Offset: 1
If p = 3, then a(1) = 2^(2*3-1)*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.
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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:
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f[p_] := 2^(2*p - 1)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 12 2020 *)
A330837
a(n) = M(n)^2*(M(n)+1)^2, where M(n) = A000668(n) is the n-th Mersenne prime.
Original entry on oeis.org
144, 3136, 984064, 264257536, 4502500182851584, 295143401596905324544, 75557575495813049614336, 21267647912751613342506514584526913536, 28269553036454149248812831358032474524823101898744619883661101506865659904
Offset: 1
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.
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a := proc(n::posint)
local p, m;
p:=NumberTheory[IthMersenne](n);
m:=2^p-1;
return m^2*(m+1)^2;
end:
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f[p_] := 2^(2p)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 12 2020 *)
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forprime(p=1,999,isprime(2^p-1)&&print1((2^p-1)^2<<(2*p)",")) \\ M. F. Hasler, Feb 07 2020
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