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.
Original entry on oeis.org
225, 3969, 1046529, 268402689, 4503599493152769, 295147905144993087489, 75557863725364567605249, 21267647932558653957237540927630737409, 28269553036454149273332760011886696242605918383730576346715242738439159809
Offset: 1
If p=2, then M_2=3, and a(1) = 3^2(3+2)^2 = 15^2 = 225.
-
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 *)
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
If n=1, a(1)=(3^2-1)^2=64.
-
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 *)
A330818
Numbers of the form 2^(2*p+1), where p is a Mersenne exponent, A000043.
Original entry on oeis.org
32, 128, 2048, 32768, 134217728, 34359738368, 549755813888, 9223372036854775808, 10633823966279326983230456482242756608, 766247770432944429179173513575154591809369561091801088
Offset: 1
a(1) = 2^(2*2+1) = 32. Since M_2=3, the number 2^5*3^2 has power-spectral basis {225,64}.
-
A330818:=[]:
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); A330818:=[op(A330818),x]; fi;
od;
od;
A330818;
-
2^(2 * MersennePrimeExponent[Range[10]] + 1) (* Amiram Eldar, Jan 03 2020 *)
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.
Original entry on oeis.org
3528, 1107072, 297289728, 5065312705708032, 332036326796518490112, 85002272432789680816128, 23926103901845565010319828907592777728, 31803247166010917904914435277786533840425989636087697369118739195223867392
Offset: 1
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.
Cf.
A000043,
A000668,
A133049,
A133051,
A152921,
A152922,
A330818,
A330819,
A330820, 9*
A133051, 9*
A330817,
A330837.
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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 *)
Showing 1-4 of 4 results.
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