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A027424 Number of distinct products ij with 1 <= i, j <= n (number of distinct terms in n X n multiplication table). 24
1, 3, 6, 9, 14, 18, 25, 30, 36, 42, 53, 59, 72, 80, 89, 97, 114, 123, 142, 152, 164, 176, 199, 209, 225, 239, 254, 267, 296, 308, 339, 354, 372, 390, 410, 423, 460, 480, 501, 517, 558, 575, 618, 638, 659, 683, 730, 747, 778, 800, 827, 850, 903 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

As n->infinity what is an asymptotic expression for a(n)? Reply from Carl Pomerance: Erdős showed that a(n) is o(n^2). Linnik and Vinogradov (1960) showed it is O(n^2/(log n)^c) for some c > 0. Finer estimations were achieved in the book Divisors by Hall and Tenenbaum (Cambridge, 1988), see Theorem 23 on p. 33.

An easy lower bound is to consider primes p > n/2, times anything < n. This gives n * (n/2 logn) - ((n/2 log n)^2)/2, after subtracting double counting of p*p'; or roughly n^2/2 log n. - Rich Schroeppel, Jul 05 2003

A033677(n) is the smallest k such that n appears in the k X k multiplication table and a(k) is the number of n with A033677(n) <= k.

Erdős showed in 1955 that a(n)=O(n^2/(log n)^c) for some c>0. In 1960, Erdős proved a(n) = n^2/(log n)^(b+o(1)), where b = 1-(1+loglog 2)/log 2 = 0.08607... In 2004, Ford proved a(n) is bounded between two positive constant multiples of n^2/((log n)^b (log log n)^(3/2)). - Kevin Ford (ford(AT)math.uiuc.edu), Apr 20 2006

REFERENCES

P. Erdős, An asymptotic inequality in the theory of numbers, Vestnik Leningrad. Univ. 15 (1960), 41-49 (Russian).

Y. V. Linnik and I. M. Vinogradov, Vestnik Leningrad Univ. 13 (1960), 41-49.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

R. P. Brent and C. Pomerance, The multiplication table, and random factored integers, 2012.

P. Erdős, Some remarks on number theory, Riveon Lematematika 9 (1955), 45-48 (Hebrew. English summary).

K. Ford, The distribution of integers with a divisor in a given interval. Annals of Math. 168(2), 367-433. arXiv:math/0401223, (2008).

D. Koukoulopoulos, On the number of integers in a generalized multiplication table. Journal für die reine und angewandte Mathematik, 2012.

C. Pomerance (1998) Paul Erdős, Number Theorist Extraordinaire, Notices Amer. Math. Soc. 45(1), 19-23.

M. Hassani, Approximation of the Multiplication Table Function. Preprint arXiv:math/0603644, 2006.

FORMULA

a(n) = Sum_{l=1..n^2} Sum_{d|l} moebius(l/d) * floor( m(d,n) * n / l ), where m(d,n) is the maximum divisor of d not exceeding n. - Max Alekseyev, Jul 14 2011

MATHEMATICA

u = {}; Table[u = Union[u, n*Range[n]]; Length[u], {n, 100}] (* T. D. Noe, Jan 07 2012 *)

PROG

(PARI) multab(N)=local(v, cv, s, t); v=vector(N); cv=vector(N*N); v[1]=cv[1]=1; for(k=2, N, s=0:for(l=1, k, t=k*l: if(cv[t]==0, s++); cv[t]++); v[k]=v[k-1]+s); v \\ Ralf Stephan

(PARI) A027424(n)={my(u=0); sum(j=1, n, sum(i=1, j, !bittest(u, i*j) & u+=1<<(i*j)))} \\ M. F. Hasler, Oct 08 2012

(PARI) a(n)=#Set(concat(Vec(matrix(n, n, i, j, i*j)))) \\ Charles R Greathouse IV, Mar 27 2014

(PARI) a(n) = #setbinop((x, y)->x*y, vector(n, i, i); ); \\ Michel Marcus, Jun 19 2015

(Haskell)

import Data.List (nub)

a027424 n = length $ nub [i*j | i <- [1..n], j <- [1..n]]

-- Reinhard Zumkeller, Jan 01 2012

CROSSREFS

Cf. A027384, A027417, A033677, A108407, A027426.

Equals A027384 - 1. First differences are in A062854.

Sequence in context: A187263 A230876 A000791 * A258087 A191130 A134031

Adjacent sequences:  A027421 A027422 A027423 * A027425 A027426 A027427

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 10 11:58 EST 2016. Contains 279001 sequences.