A001923 a(n) = Sum_{k=1..n} k^k. (Formerly M3968 N1639)

0, 1, 5, 32, 288, 3413, 50069, 873612, 17650828, 405071317, 10405071317, 295716741928, 9211817190184, 312086923782437, 11424093749340453, 449317984130199828, 18896062057839751444, 846136323944176515621

Offset

0,3

Comments

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

From Luan Alberto Ferreira, Aug 01 2017: (Start)

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

References

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).

Links

T. D. Noe, Table of n, a(n) for n=0..100

Mohammad K. Azarian, On the hyperfactorial function, hypertriangular function and the discriminants of certain polynomials, Int. J. Pure Appl. Math., Vol. 36, No. 2 (2007), pp. 251-257.

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.

G. W. Wishard (proposer) and F. Underwood (solution), Problem 4155: Bound for a Finite Sum, Amer. Math. Monthly, Vol. 53, No. 8 (1946), pp. 471-473.

Formula

a(n) = A062970(n) - 1.

a(n+1)/a(n) > e*n and a(n+1)/a(n) is asymptotic to e*n. - Benoit Cloitre, Sep 29 2002

For n > 0: a(n) = a(n-1) + A000312(n). - Reinhard Zumkeller, Jul 11 2014

Limit_{n->oo} (a(n+2)/a(n+1) - a(n+1)/a(n)) = e (Cusumano, 2007). - Amiram Eldar, Jan 03 2022

Mathematica

Accumulate[Join[{0}, Table[k^k, {k, 20}]]] (* Harvey P. Dale, Feb 11 2015 *)

Prog

(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
-- Reinhard Zumkeller, Jul 11 2014
(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)))
print(aupton(17)) # Michael S. Branicky, Feb 12 2022

Crossrefs

Cf. A073825, A062970 (another version).

Cf. A062815, A060946, A117887.

Cf. A000312, A002109.

Sequence in context: A320349 A354013 A331660 * A257710 A305305 A331339

Adjacent sequences: A001920 A001921 A001922 * A001924 A001925 A001926

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved