A073825 -id:A073825 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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.

Cf. A073825, A062970 (another version).
Cf. A062815, A060946, A117887.
Cf. A000312, A002109.

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

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

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

# 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

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

A073826 Primes of the form Sum_{k=1..n} k^k, i.e., primes in A001923.

Original entry on oeis.org

5, 3413, 50069, 10405071317, 208492413443704093346554910065262730566475781

Offset: 1

Rick L. Shepherd, Aug 13 2002

nonn

a(3) = A001923(10) = 10405071317 and the 45-digit a(4) = A001923(30) have been certified prime with Primo. Any additional terms are too big to include here.

The next term would have over 20000 digits; see A073825 for more information and updates.

			a(1) = 5 = 1^1 + 2^2 is the smallest prime of the form A001923(n) = sum_{k=1..n} k^k, namely for n = 2 = A073825(1).
a(2) = sum_{k=1..A073825(2)} k^k = 1^1 + 2^2 + 3^3 + 4^4 + 5^5 = 3413, a prime, so 3413 is in this sequence (A073825(2) = 5).

Cf. A073825 (corresponding n), A001923 (sum_{k=1..n} k^k).

Cf. A122166 (indices of primes in A062970 (sum_{k=0..n} k^k)).

Select[s=0;Table[s+=n^n,{n,5!}],PrimeQ[ # ]&] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

s=0; for(k=1,1320, s=s+k^k; if(isprime(s), print1(s,",")))

a(j) = A001923(A073825(j)) = sum_{k=1..A073825(j)} k^k.

Intersection of A001923 with A000040.

Edited by M. F. Hasler, Mar 22 2008

Typo in comment corrected by Jonathan Vos Post, Mar 23 2008

A122166 Numbers n such that 1 + Sum k^k (k=1..n) is prime.

Original entry on oeis.org

1, 52, 124, 431

Offset: 1

Alexander Adamchuk, Aug 23 2006

nonn

Primes of the form A001923[n] = Sum k^k (k=1..n) are given in A073826[n] and their indices are given in A073825[n] = {2,5,6,10,30,...}.

Cf. A073825, A001923.

s=1; Do[s=s+k^k;If[PrimeQ[s],Print[{k,s}]],{k,1,500}]

A097349 Numbers n such that (Sum (2k)^k, k=1..n) + 1 is prime.

Original entry on oeis.org

1, 2, 13, 72, 318, 13583

Offset: 1

Rick L. Shepherd, Aug 07 2004

Some of the larger entries may only correspond to probable primes.

The numbers produced by 72 and 318 have now been certified prime by Primo. 13583, found by PrimeForm using recurrence mode, corresponds to a 60228-digit probable prime. - Rick L. Shepherd, Apr 29 2006

			13 is a term as 2^1 + 4^2 + 6^3 + 8^4 + 10^5 + 12^6 + 14^7 + 16^8 + 18^9 + 20^10 + 22^11 + 24^12 + 26^13 + 1 = 2518267981703965963, which is prime (certified with Primo).

Cf. A073825 (Sum k^k, k=1..n, is prime), A097350 ((Sum (2k)^k, k=1..n) - 1 is prime).

s=1; for(k=1,700, s=s+(2*k)^k; if(isprime(s), print1(k,",")))

One more term from Rick L. Shepherd, Apr 29 2006

A097350 Numbers n such that (Sum (2k)^k, k=1..n) - 1 is prime.

Original entry on oeis.org

2, 3, 6, 49

Offset: 1

Rick L. Shepherd, Aug 07 2004

nonn

Some of the larger entries may only correspond to probable primes.

Term 49 corresponds to a certified prime (Primo 2.2.0 beta). Next term is greater than 5400. - Ryan Propper, Aug 23 2005

			6 is a term as 2^1 + 4^2 + 6^3 + 8^4 + 10^5 + 12^6 - 1 = 3090313, which is prime.

Cf. A073825 (Sum k^k, k=1..n, is prime), A097349 ((Sum (2k)^k, k=1..n) + 1 is prime).

s=-1; for(k=1,830, s=s+(2*k)^k; if(isprime(s), print1(k,",")))

cp's OEIS Frontend

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

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A073826 Primes of the form Sum_{k=1..n} k^k, i.e., primes in A001923.

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A122166 Numbers n such that 1 + Sum k^k (k=1..n) is prime.

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A097349 Numbers n such that (Sum (2k)^k, k=1..n) + 1 is prime.

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A097350 Numbers n such that (Sum (2k)^k, k=1..n) - 1 is prime.

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