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A083865 Sums of (one or more distinct) k-perfect numbers. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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]

LINKS

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

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Cf. A065997, A000695.

Sequence in context: A105402 A145551 A259917 * A185351 A272971 A117948

Adjacent sequences:  A083862 A083863 A083864 * A083866 A083867 A083868

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified June 22 07:47 EDT 2017. Contains 288605 sequences.