A031435 -id:A031435

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A003320

a(n) = max_{k=0..n} k^(n-k).
(Formerly M1198)

+10
11

1, 1, 1, 2, 4, 9, 27, 81, 256, 1024, 4096, 16384, 78125, 390625, 1953125, 10077696, 60466176, 362797056, 2176782336, 13841287201, 96889010407, 678223072849, 4747561509943, 35184372088832, 281474976710656, 2251799813685248

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

For n > 0, a(n+1) = largest term of row n in triangles A051129 and A247358. - Reinhard Zumkeller, Sep 14 2014

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

I. Tomescu, Introducere in Combinatorica. Editura Tehnica, Bucharest, 1972, p. 231.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..599 (terms 0..100 from T. D. Noe).

D. Easdown, Minimal faithful permutation and transformation representations of groups and semigroups, Contemporary Math. (1992), Vol. 131 (Part 3), 75-84.

R. Gray and J. D. Mitchell, Largest subsemigroups of the full transformation monoid, Discrete Math., 308 (2008), 4801-4810.

W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.

R. K. Guy, Letter to N. J. A. Sloane, Mar 1974

I. Tomescu, Excerpts from "Introducese in Combinatorica" (1972), pp. 230-1, 44-5, 128-9. (Annotated scanned copy)

FORMULA

a(n) = A056155(n-1)^(n - A056155(n-1)), for n >= 2. - Ridouane Oudra, Dec 09 2020

EXAMPLE

a(5) = max(5^0, 4^1, 3^2, 2^3, 1^4, 0^5) = max(1,4,9,8,1,0) = 9.

MATHEMATICA

Join[{1}, Max[#]&/@Table[k^(n-k), {n, 25}, {k, n}]] (* Harvey P. Dale, Jun 20 2011 *)

PROG

(Haskell)

a003320 n = maximum $ zipWith (^) [0 .. n] [n, n-1 ..]

-- Reinhard Zumkeller, Jun 24 2013

(PARI) a(n) = vecmax(vector(n+1, k, (k-1)^(n-k+1))); \\ Michel Marcus, Jun 13 2017

CROSSREFS

Cf. A003992, A031435, A056155.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, R. K. Guy

EXTENSIONS

Easdown reference from Michail Kats (KatsMM(AT)info.sgu.ru)

More terms from James A. Sellers, Aug 21 2000

STATUS

approved

A265899

After a(1) = 1, positions of descents in A265894.

+10
5

1, 3, 6, 8, 11, 14, 17, 20, 24, 27, 31, 34, 38, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 94, 98, 102, 106, 111, 115, 120, 124, 128, 133, 137, 142, 146, 151, 156, 160, 165, 169, 174, 179, 184, 188, 193, 198, 202, 207, 212, 217, 222, 227, 231, 236, 241, 246, 251, 256, 261, 266, 271, 276, 281, 286, 291, 296, 301

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Numbers n for which A099563(A001813(n)) <= A099563(A001813(n-1)), where A001813(n) = (2n)! / n!, and A099563 gives the most significant digit in the factorial base representation (A007623) of n.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..1503

PROG

(PARI)

A099563(n) = { my(i=2, dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); };

A265894 = n -> A099563((2*n)! / n!);

i=0; p=1; n=0; while(i < 5000, n++; k = A265894(n); if(k <= p, i++; write("b265899.txt", i, " ", n)); p = k; );

(Scheme, with Antti Karttunen's IntSeq-library)

(define A265899 (MATCHING-POS 1 1 (lambda (n) (<= (A265894 n) (A265894 (- n 1))))))

CROSSREFS

Cf. A001813, A007623, A099563, A265894.

Cf. A265898 (a subsequence), A266119 (first differences), A266120 (terms immediately before descents).

Cf. also A031435.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 24 2015

STATUS

approved

A307175

Smallest power to which 1+1/n must be raised in order for an interval [k,k+1], with k an integer, to be skipped.

+10
0

4, 7, 9, 13, 15, 18, 22, 26, 27, 32, 33, 40, 42, 48, 51, 55, 58, 62, 66, 71, 75, 80, 85, 85, 91, 97, 103, 105, 111, 112, 120, 121, 129, 131, 139, 142, 143, 153, 156, 158, 168, 172, 175, 178, 181, 193, 197, 201, 206, 210, 215, 220, 225, 230, 235, 241, 246, 252

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

Here the skipping of an interval means that the interval falls strictly between (1+1/n)^(a(n)-1) and (1+1/n)^a(n).

The sequence is not monotonically increasing; a(24) = a(25) and a(62) > a(63) are the first counterexamples.

Asymptotic to n * log(n), and as such also to the prime numbers (A000040).

LINKS

Table of n, a(n) for n=2..59.

EXAMPLE

1.1^26 = 11.918... and 1.1^27 = 13.109...; [12,13] is skipped, and this is the first time this happens, thus a(10)=27.

MATHEMATICA

a[n_, m_] := Reduce[(1+1/n)^(m-1) < k < k+1 < (1+1/n)^m, k, Integers];

a[n_] := For[m = 1, True, m++, If[a[n, m] =!= False, Return[m]]];

Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jul 07 2019 *)

PROG

(PARI) a(n) = my(k=2, last=1+1/n); while(floor(new = (1+1/n)^k) - ceil(last) != 1, k++; last = new); k; \\ Michel Marcus, Mar 30 2019

(Python 3)

from math import floor, log

def get_a_of_n(i):

x=1+1/i

j=i

while floor(log(j, x))!=floor(log(j+1, x)):

j+=1

return floor(log(j, x))+1

def main():

step=1

i=2

while True:

y=get_a_of_n(i)

print(y, end=", ")

i+=step

CROSSREFS

Cf. A031435.

KEYWORD

nonn

AUTHOR

Alex Costea, Mar 27 2019

STATUS

approved

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