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A083865
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Sums of (one or more distinct) k-perfect numbers.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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]
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LINKS
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FORMULA
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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
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EXAMPLE
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a(3) = 34 because it is the sum of 6 + 28 both of which are perfect numbers.
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MAPLE
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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:
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MATHEMATICA
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With[{perf=Select[Range[10000], DivisorSigma[1, #]==2#&]}, Rest[Union[Total/@ Subsets[perf]]]] (* Harvey P. Dale, Feb 07 2012 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Torsten Klar (klar(AT)radbruch.jura.uni-mainz.de), Jun 18 2003
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EXTENSIONS
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STATUS
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approved
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