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A374958 Median, as a function of n, of the probability mass function s(k-1,n-1)/k!, where k>=n and s(a,b) denotes the unsigned Stirling number of the first kind.

2, 7, 21, 57, 157, 430, 1173, 3199, 8717

Offset: 2

Matthew McMullen, Aug 05 2024

This probability mass function occurs in the statistics of records, specifically, it is the probability that the n-th record is set on the k-th race.

The ratio of successive terms seems to be converging to e.

M. Ahsanullah and V. B. Nevzorov, Records via Probability Theory, Atlantis Press, 2015.

Cf. A094638.

Solve[Sum[Abs[StirlingS1[k-1,n-1]]/k!,{k,n,x}]>=.5 && x>=n, x, Integers]

a(n)=my(s=0); for(x=n, oo, s+=abs(stirling(x-1,n-1,1))/x!; if(2*s >= 1, return(x))) \\ Andrew Howroyd, Aug 09 2024

a(9)-a(10) from Sean A. Irvine, Aug 26 2024

A268631 Number of ordered pairs (a,b) of positive integers less than n with the property that n divides ab.

Original entry on oeis.org

0, 0, 0, 1, 0, 4, 0, 5, 4, 8, 0, 17, 0, 12, 16, 17, 0, 28, 0, 33, 24, 20, 0, 53, 16, 24, 28, 49, 0, 76, 0, 49, 40, 32, 48, 97, 0, 36, 48, 101, 0, 112, 0, 81, 100, 44, 0, 145, 36, 96, 64, 97, 0, 136, 80, 149, 72, 56, 0, 241, 0, 60, 148, 129, 96, 184, 0, 129, 88, 212

Offset: 1

Matthew McMullen, Feb 09 2016

nonn

a(n)=0 iff n is prime or 1. a(n) is odd iff n is a multiple of 4.

			For n=10 the a(10)=8 ordered pairs are (2,5), (5,2), (4,5), (5,4), (5,6), (6,5), (5,8), and (8,5).

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006579.

a[n_] := Sum[Sum[1, {i, Divisors[n*k]}] - 2*Sum[1, {i, TakeWhile[Divisors[n*k], # <= k &]}], {k, 1, n - 1}]

a(n) = sum(k=1, n-1, sumdiv(n*k, d, (d > k) && (d < n))); \\ Michel Marcus, Feb 09 2016

from math import prod
from sympy import factorint
def A268631(n): return 1 - 2*n + prod(p**(e-1)*((p-1)*e+p) for p, e in factorint(n).items()) # Chai Wah Wu, May 15 2022

a(n) = Sum_{k=1..n-1} (number of divisors of nk that are between k and n, exclusive).
a(n) = Sum_{k=1..n-1} (number of divisors of nk - 2*(number of divisors of nk that are <= k)).
a(n) = A006579(n) - (n-1). - Michel Marcus, Feb 09 2016
a(p^k) = (p(k-1)-k)*p^(k-1)+1 for prime p. - Chai Wah Wu, May 15 2022

A154429 a(n) is the least k such that the greedy algorithm (for Egyptian fractions) on 4k/(24n+1) terminates in at most three steps.

Original entry on oeis.org

2, 2, 2, 5, 3, 4, 13, 2, 2, 7, 5, 51, 4, 4, 5, 2, 3, 5, 5, 7, 5, 6, 2, 5, 11, 4, 3, 5, 5, 2, 2, 7, 4, 5, 29, 2, 2, 2, 5, 8, 4, 11, 2, 2, 6, 4, 11, 5, 3, 11, 2, 5, 5, 5, 7, 4, 37, 2, 3, 3, 4, 7, 5, 5, 2, 2, 17, 5, 5, 54, 2, 2, 2, 5, 7, 4, 11, 2, 2, 6, 5, 3, 4, 5, 10, 2, 7, 5, 5, 7, 5, 12, 2, 3, 10, 4, 7, 5, 5, 2

Offset: 1

Matthew McMullen (mmcmullen(AT)otterbein.edu), Jan 09 2009

nonn

			For n=3, the Greedy Algorithm gives 8/73=1/10+1/105+1/15330

J. Steuding, Diophantine Analysis, Chapman & Hall/CRC, 2005, pp. 39-40, 50.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
D. Eppstein, Egyptian fractions

GreedyPart[q_Integer] := 0;
GreedyPart[Rational[1, y_]] := 0;
GreedyPart[q_Rational] := q - If[q < 0 || q > 1, Floor[q], Rational[1, 1 + Quotient[1, q]]];
SubtractShifted[l_] := Drop[l, -2] - Take[l, {2, -2}];
EgyptGreedy[q_] := SubtractShifted[FixedPointList[GreedyPart, q]];
terms := 200;
For[i = 25, i <= 24*terms + 1, i = i + 24,k = 2;While[Length[EgyptGreedy[4k/i]]> 3, k++ ];Print[k]]

More terms from Seiichi Manyama, Sep 21 2022

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User: Matthew McMullen

A374958 Median, as a function of n, of the probability mass function s(k-1,n-1)/k!, where k>=n and s(a,b) denotes the unsigned Stirling number of the first kind.

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A268631 Number of ordered pairs (a,b) of positive integers less than n with the property that n divides ab.

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A154429 a(n) is the least k such that the greedy algorithm (for Egyptian fractions) on 4k/(24n+1) terminates in at most three steps.

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