A083865 Sums of (one or more distinct) k-perfect numbers.

6, 28, 34, 120, 126, 148, 154, 496, 502, 524, 530, 616, 622, 644, 650, 672, 678, 700, 706, 792, 798, 820, 826, 1168, 1174, 1196, 1202, 1288, 1294, 1316, 1322, 8128, 8134, 8156, 8162, 8248, 8254, 8276, 8282, 8624, 8630, 8652, 8658, 8744, 8750, 8772, 8778

Offset: 1

Author

Torsten Klar (klar(AT)radbruch.jura.uni-mainz.de), Jun 18 2003

Keywords

• easy
• nonn

Comments

Each k-perfect number (A007691\{1}) appears once, and may also appear at most once in each sum of k-perfect numbers to create other terms in the sequence. [Harvey P. Dale, Feb 07 2012]

Examples

			a(3) = 34 because it is the sum of 6 + 28 both of which are perfect numbers.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A065997, A000695.

Programs

Maple

N:= 10000: # to get all terms <= N
Kperf:= select(t -> numtheory:-sigma(t) mod t = 0, [$2..N]):
S:= {0}:
for k in Kperf do S:= S union (k +~ S) od:
sort(convert(S minus {0}, list)); # Robert Israel, Nov 29 2016

Mathematica

With[{perf=Select[Range[10000],DivisorSigma[1,#]==2#&]},Rest[Union[Total/@ Subsets[perf]]]] (* Harvey P. Dale, Feb 07 2012 *)

PARI

a=[];n=1;until(50<#a=concat(a,vector(#a+1,i,n+if(i>1,a[i-1]))),while(sigma(n++)%n,));a  \\ M. F. Hasler, Feb 09 2012

Formula

Empirical observation: a(n) = 2*n + Sum_{k >= 1} 4^k*floor(2*n/2^k) for 1 <= n <= 15 and 32 <= n <= 47; a(n) = 2*n - 1344 + Sum_{k >= 1} 4^k*floor(2*n/2^k) for 16 <= n <= 31. Note 1344 = 4^3 + 4^4 + 4^5. Cf. A000695. - Peter Bala, Nov 29 2016

If b(n) = 2*n + Sum_{k >= 1} 4^k*floor(2*n/2^k) - a(n), we also have b(n) = 1344 for 48 <= n <= 63, then 2400 for 64 <= n <= 79, 3744 for 80 <= n <= 95, 8008 for 96 <= n <= 111, etc. The first case where b(n) is not constant on an interval 16*k <= n <= 16*k+15 is k=57214, where b(915431)=2747770287196 but b(915432)=2747770287312. - Robert Israel, Nov 29 2016

Extensions

Corrected by M. F. Hasler and others, Feb 07 2012