cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A108408 Unique values found in A108407. Complement of A108409.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 17, 18, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 90
Offset: 1

Views

Author

Ralf Stephan, Jun 03 2005

Keywords

Crossrefs

Cf. A108407, A108409.

Programs

  • Mathematica
    terms = 71;
nmax = 3 terms;
A062854 = Table[u = If[n == 1, {}, Union[u, n Range[n]]]; Length[u], {n, 1, nmax + 1}] // Differences // Prepend[#, 1]&;
a108407[n_] := n + 1 - A062854[[n+1]];
Array[a108407, nmax] // Union // Rest // Take[#, terms]& (* Jean-François Alcover, Oct 02 2020 *)

A108409 Complement of A108408. Missing values in A108407.

Original entry on oeis.org

5, 7, 11, 13, 19, 20, 21, 26, 37, 48, 50, 51, 61, 65, 68, 75, 76, 80, 81, 93, 98, 101, 110, 113, 114, 121, 127, 128, 129, 135, 143, 146, 147, 149, 151, 163, 167, 171, 175, 179, 184, 185, 188, 194, 201, 205, 208, 209, 218, 219, 220, 221, 251, 254, 257, 263, 264
Offset: 1

Views

Author

Ralf Stephan, Jun 03 2005

Keywords

A027424 Number of distinct products ij with 1 <= i, j <= n (number of distinct terms in n X n multiplication table).

Original entry on oeis.org

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
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

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^2/(2 log n). - Richard C. 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

  • Hall, Richard Roxby, and Gérald Tenenbaum. Divisors. Cambridge University Press, 1988.
  • Y. V. Linnik and I. M. Vinogradov, An asymptotic inequality in the theory of numbers, Vestnik Leningrad. Univ. 15 (1960), 41-49 (in Russian).

Links

Crossrefs

Cf. A027384, A027417, A033677, A108407, A027426.
Equals A027384 - 1. First differences are in A062854.
Column 2 of A322967.
Cf. A074738 (constant in asymptotic).

Programs

  • Haskell
    import Data.List (nub)
a027424 n = length $ nub [i*j | i <- [1..n], j <- [1..n]]
-- Reinhard Zumkeller, Jan 01 2012
  • Maple
    A027424m := proc(d,n)
    local a,dvs ;
    a := 0 ;
    for dvs in numtheory[divisors](d) do
        if dvs <= n then
            a := max(a,dvs) ;
        end if;
    end do:
    a ;
end proc:
A027424 := proc(n)
    add(add(numtheory[mobius](L/d) *floor(A027424m(d,n) *n/L), d=numtheory[divisors](L)), L=1..n^2) ;
end proc:
seq(A027424(n),n=1..40) ; # R. J. Mathar, Oct 02 2020
  • Mathematica
    u = {}; Table[u = Union[u, n*Range[n]]; Length[u], {n, 100}] (* T. D. Noe, Jan 07 2012 *)
  • 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
  • Python
    def A027424(n): return len({i*j for i in range(1,n+1) for j in range(1,i+1)}) # Chai Wah Wu, Oct 13 2023

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
a(2^i-1) = A027417(i)-1. - N. J. A. Sloane, Sep 01 2018
From Mats Granvik, Nov 26 2019: (Start)
n^2 = Sum_{m=1..n^2} Sum_{k=1..n^2} [k|m]*[m <= n*k]*[k <= n]
a(n) = Sum_{m=1..n^2} sign( Sum_{k=1..n^2} [k|m]*[m <= n*k]*[k <= n] ), conjecture.
(End)

A333995 a(n) = number of distinct composite numbers in the n X n multiplication table that are not in the n-1 X n-1 multiplication table.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 6, 5, 6, 6, 10, 6, 12, 8, 9, 8, 16, 9, 18, 10, 12, 12, 22, 10, 16, 14, 15, 13, 28, 12, 30, 15, 18, 18, 20, 13, 36, 20, 21, 16, 40, 17, 42, 20, 21, 24, 46, 17, 31, 22, 27, 23, 52
Offset: 1

Views

Author

Charles Kusniec, Sep 05 2020

Keywords

Examples

			a(2) = 1 since the 1 X 1 and 2 X 2 multiplication tables are
---
  1
---
  1 2
  2 4
---
and the composite number 4 has appeared.
---
a(8)=5:
.1..2..3..4..5..6..7....8
....4..6..8.10.12.14...16
.......9.12.15.18.21...24
.........16.20.24.28...32 *
............25.30.35...40 *
...............36.42...48 *
..................49...56 *
.......................64 *

Crossrefs

Cf. A010051, A062854, A108407.

Programs

  • Python
    from itertools import takewhile
from sympy import divisors, isprime
def A333995(n): return sum(1 for i in range(1,n+1) if all(d<=i for d in takewhile(lambda d:d1 else 0 # Chai Wah Wu, Oct 14 2023

Formula

a(n) = n - A108407(n-1) - A010051(n), n > 1. - Corrected by R. J. Mathar, Oct 02 2020
a(n) = A062854(n) - A010051(n) for n > 1. - Chai Wah Wu, Oct 14 2023

A333996 Number of composite numbers in the triangular n X n multiplication table.

Original entry on oeis.org

0, 1, 3, 7, 11, 17, 23, 31, 40, 50, 60, 72, 84, 98, 113, 129, 145, 163, 181, 201, 222, 244, 266, 290, 315, 341, 368, 396, 424, 454, 484, 516, 549, 583, 618, 654, 690, 728, 767, 807, 847, 889, 931, 975, 1020, 1066, 1112, 1160, 1209, 1259, 1310, 1362, 1414
Offset: 1

Views

Author

Charles Kusniec, Sep 05 2020

Keywords

Comments

The number of pairs (i,j) with 1 <= i <= j <= n and i*j composite. - Peter Kagey, Sep 24 2020

Examples

			There are a(7) = 23 composite numbers in the 7x7 triangular multiplication table with the hypotenuse being the Square numbers:
1   2   3   4*  5   6*  7
    4*  6*  8* 10* 12* 14*
        9* 12* 15* 18* 21*
           16* 20* 24* 28*
               25* 30* 35*
                   36* 42*
                       49*

Crossrefs

Cf. A014684 (first differences), A333995, A108407, A000720, A000217, A256885, A334454.

Programs

  • Mathematica
    Array[Binomial[# + 1, 2] - PrimePi[#] - 1 &, 53] (* Michael De Vlieger, Nov 05 2020 *)
  • PARI
    a(n) = binomial(n+1, 2) - primepi(n)-1 \\ David A. Corneth, Sep 08 2020
  • Python
    from sympy import primepi
def A333996(n): return (n*(n+1)>>1)-primepi(n)-1 # Chai Wah Wu, Oct 14 2023

Formula

a(n) = A000217(n) - A000720(n) - 1. - David A. Corneth, Sep 08 2020
a(n) = A256885(n) - 1. - Michel Marcus, Sep 09 2020
a(n+1) - a(n) = A014684(n+1). - Bill McEachen, Oct 30 2020
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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