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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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) = 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.

Crossrefs

Cf. A073825, A062970 (another version).

Cf. A062815, A060946, A117887.

Cf. A000312, A002109.

Programs

Haskell

a001923 n = a001923_list !! n
a001923_list = scanl (+) 0 $ tail a000312_list
-- Reinhard Zumkeller, Jul 11 2014

Mathematica

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

PARI

for(n=1,20,print1(sum(x=1,n,x^x), ", ")) \\ Jorge Coveiro, Dec 24 2004

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

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