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A139306 Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).

%I A139306 #22 Oct 17 2024 08:24:49
%S A139306 8,32,512,8192,33554432,8589934592,137438953472,2305843009213693952,
%T A139306 2658455991569831745807614120560689152,
%U A139306 191561942608236107294793378393788647952342390272950272
%N A139306 Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).
%C A139306 Sum of n-th even perfect number and n-th even superperfect number.
%C A139306 Also, sum of n-th perfect number and n-th superperfect number, if there are no odd perfect and odd superperfect numbers, then the n-th perfect number is the difference between a(n) and the n-th superperfect number (see A135652, A135653, A135654 and A135655).
%H A139306 Amiram Eldar, <a href="/A139306/b139306.txt">Table of n, a(n) for n = 1..15</a>
%H A139306 Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%F A139306 a(n) = 2^(2*A000043(n) - 1). Also, a(n) = 2^A133033(n), if there are no odd perfect numbers. Also, a(n) = A000396(n) + A019279(n), if there are no odd perfect and odd superperfect numbers. Also, a(n) = A000396(n) + A061652(n), if there are no odd perfect numbers, then we can write: perfect number A000396(n) = a(n) - A061652(n).
%F A139306 a(n) = A061652(n)*(A000668(n)+1) = A061652(n)*A072868(n). - _Omar E. Pol_, Apr 13 2008
%e A139306 a(5) = 33554432 because A000043(5) = 13 and 2^(2*13 - 1) = 2^25 = 33554432.
%e A139306 Also, if there are no odd perfect and odd superperfect numbers then we can write a(5) = A000396(5) + A019279(5) = A000396(5) + A061652(5) = 33554432.
%t A139306 2^(2 * MersennePrimeExponent[Range[10]] - 1) (* _Amiram Eldar_, Oct 17 2024 *)
%Y A139306 Cf. A000079, A000396, A019279, A061645, A061652, A133033, A135652, A135653, A135654, A135655, A139286, A139294, A139307.
%Y A139306 Cf. A000043, A000668, A072868.
%K A139306 nonn
%O A139306 1,1
%A A139306 _Omar E. Pol_, Apr 13 2008