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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.
1
18816, 5904384, 1585545216, 27015001097109504, 1770860409581431947264, 453345452974878297686016, 127605887476509680055039087507161481216, 169617318218724895492876988148194847148938611392467719301966609041193959424
OFFSET
1,1
COMMENTS
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
LINKS
G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.
FORMULA
a(n) = A330818(n+1) * 3 * A133049(n+1).
EXAMPLE
a(1) = 2^(2*3+1) * 3 * 7^2 = 18816, and 18816 has spectral basis {63^2, 112^2, 48^2}, consisting of powers.
MAPLE
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:
MATHEMATICA
f[p_] := 2^(2p + 1)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 22 2020 *)
KEYWORD
nonn
AUTHOR
Walter Kehowski, Jan 21 2020
STATUS
approved