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A001923
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a(n) = Sum_{k=1..n} k^k.
(Formerly M3968 N1639)
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34
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0, 1, 5, 32, 288, 3413, 50069, 873612, 17650828, 405071317, 10405071317, 295716741928, 9211817190184, 312086923782437, 11424093749340453, 449317984130199828, 18896062057839751444, 846136323944176515621
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OFFSET
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0,3
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COMMENTS
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Starting from the second term, 1, the terms could be described as the special case (n=1; j=1) of the following general formula: a(n) = <from k=j to k=i> Sum [(n + k - 1)]^(k) n=1; j=1; i=1,2,3,...,... For (n=0; j=1) the formula yields A062815 n=0; j=1; i=2,3,4,... For (n=2; j=0) we get A060946 and for (n=3; j=0) A117887. - Alexander R. Povolotsky, Sep 01 2007
If n == 0 or 3 (mod 4), then a(n) == 0 (mod 4).
If n == 0, 4, 7, 14, 15 or 17 (mod 18), then a(n) == 0 (mod 3). (End)
Called the hypertriangular function by M. K. Azarian. - Light Ediand, Nov 19 2021
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REFERENCES
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József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, p. 308.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Andrew Cusumano, Problem H-656, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 45, No. 2 (2007), p. 187; A Sequence Tending To e, Solution to Problem H-656, ibid., Vol. 46-47, No. 3 (2008/2009), pp. 285-287.
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FORMULA
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a(n+1)/a(n) > e*n and a(n+1)/a(n) is asymptotic to e*n. - Benoit Cloitre, Sep 29 2002
Limit_{n->oo} (a(n+2)/a(n+1) - a(n+1)/a(n)) = e (Cusumano, 2007). - Amiram Eldar, Jan 03 2022
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MATHEMATICA
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Accumulate[Join[{0}, Table[k^k, {k, 20}]]] (* Harvey P. Dale, Feb 11 2015 *)
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PROG
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(PARI) for(n=1, 20, print1(sum(x=1, n, x^x), ", ")) \\ Jorge Coveiro, Dec 24 2004
(Haskell)
a001923 n = a001923_list !! n
a001923_list = scanl (+) 0 $ tail a000312_list
(Python) # generates initial segment of sequence
from itertools import accumulate
def f(k): return 0 if k == 0 else k**k
def aupton(nn): return list(accumulate(f(k) for k in range(nn+1)))
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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