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.

6, 18, 28, 30, 42, 54, 66, 78, 90, 102, 114, 126, 138, 150, 162, 174, 186, 196, 198, 210, 222, 234, 246, 258, 270, 282, 294, 306, 308, 318, 330, 342, 354, 364, 366, 378, 390, 402, 414, 426, 438, 450, 462, 474, 476, 486, 496, 498, 510, 522, 532, 534, 546, 558, 570, 582, 594

Offset: 1

Author

Hartmut F. W. Hoft, Mar 04 2022

Keywords

• nonn

Comments

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).

This sequence is a subsequence of A005835. It contains all even perfect numbers (A000396).

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.

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.

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.

Examples

			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).
a(51) = 532 = 2^2 * (2^3 - 1) * 19  since 7 is Mersenne prime A000668(2).
a(757) = 8128 = 2^6 * (2^7 - 1) = 2^6 * (2^A000043(4) - 1) = 2^6 * A000668(4) = A000396(4) is a perfect number.
		

Crossrefs

Cf. A000043, A000396, A000668, A005835, A237593, A352030.

Programs

Mathematica

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]
a352061[n_] := Select[Range[n], evenoddPartsQ]
a352061[600]