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A352061 Numbers n = 2^m * q, m > 0 and q > 1 odd, where the smallest odd divisor p > 1 is the m-th Mersenne prime 2^(m+1) - 1.

%I A352061 #25 Mar 05 2022 00:35:35
%S A352061 6,18,28,30,42,54,66,78,90,102,114,126,138,150,162,174,186,196,198,
%T A352061 210,222,234,246,258,270,282,294,306,308,318,330,342,354,364,366,378,
%U A352061 390,402,414,426,438,450,462,474,476,486,496,498,510,522,532,534,546,558,570,582,594
%N A352061 Numbers n = 2^m * q, m > 0 and q > 1 odd, where the smallest odd divisor p > 1 is the m-th Mersenne prime 2^(m+1) - 1.
%C A352061 All numbers in the sequence are pseudoperfect numbers since n = Sum_{i=0..m-1} (2^i * q) + Sum_{i=0..m} (2^i * q/p).
%C A352061 This sequence is a subsequence of A005835. It contains all even perfect numbers (A000396).
%C A352061 The first pseudoperfect number not in this sequence is A005835(2) = 12 = 2^2 * 3  since 3 is the first, not the second Mersenne prime.
%C A352061 The first pseudoperfect number in this sequence that is not in A352030 is 90 = 2*3*3*5 since its symmetric representation of sigma consists of one part with maximum width 3.
%C A352061 Since p = 2^(m+1) - 1 < 2^(m+1) the maximum width of the symmetric representation of sigma(a(n)) is at least 2, for all n.
%e A352061 a(2) = 18 = 2 * 9 =  2^1 * (2^2 - 1) * 3  and a(9) = 90 = 2^1 * (2^2 - 1) * 15 since 3 is Mersenne prime A000668(1).
%e A352061 a(51) = 532 = 2^2 * (2^3 - 1) * 19  since 7 is Mersenne prime A000668(2).
%e A352061 a(757) = 8128 = 2^6 * (2^7 - 1) = 2^6 * (2^A000043(4) - 1) = 2^6 * A000668(4) = A000396(4) is a perfect number.
%t A352061 evenoddPartsQ[n_] := Module[{dL=Select[Divisors[n], OddQ], fL=First[FactorInteger[n]], evenE}, evenE=If[First[fL]==2, Last[fL], 0]; n/2^evenE>1&&dL[[2]]==2^(evenE+1)-1]
%t A352061 a352061[n_] := Select[Range[n], evenoddPartsQ]
%t A352061 a352061[600]
%Y A352061 Cf. A000043, A000396, A000668, A005835, A237593, A352030.
%K A352061 nonn
%O A352061 1,1
%A A352061 _Hartmut F. W. Hoft_, Mar 04 2022