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 1-3 of 3 results.

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

A338713 Number of numbers of the form i*n with 1 <= i <= n and tau(i*n) = 4.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 1, 1, 1, 4, 0, 5, 1, 1, 0, 6, 0, 7, 0, 1, 1, 8, 0, 1, 1, 1, 0, 9, 0, 10, 0, 1, 1, 1, 0, 11, 1, 1, 0, 12, 0, 13, 0, 0, 1, 14, 0, 1, 0, 1, 0, 15, 0, 1, 0, 1, 1, 16, 0, 17, 1, 0, 0, 1, 0, 18, 0, 1, 0, 19, 0, 20, 1, 0, 0, 1, 0, 21, 0, 0, 1, 22, 0, 1, 1, 1, 0, 23
Offset: 1

Views

Author

N. J. A. Sloane, Nov 11 2020

Keywords

Links

Crossrefs

tau is A000005.
For tau(i*n) = 2, 3, 6, see A010051, A296084, A338714.
Inspired by A333995.

Programs

  • Mathematica
    Table[Count[Table[n i,{i,n}],?(DivisorSigma[0,#]==4&)],{n,90}] (* _Harvey P. Dale, Jun 14 2022 *)
  • PARI
    a(n) = sum(i=1, n, numdiv(i*n)==4); \\ Michel Marcus, Nov 11 2020
  • Python
    from collections import Counter
from sympy import factorint
def A338713(n):
    f = Counter(factorint(n))
    return sum(1 for i in range(1,n+1) if (l:=tuple((f+Counter(factorint(i))).values()))==(1,1) or l==(3,)) # Chai Wah Wu, Oct 14 2023

A338714 Number of numbers of the form i*n with 1 <= i <= n and tau(i*n) = 6.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 1, 1, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 0, 8, 2, 1, 1, 3, 0, 3, 1, 2, 2, 2, 0, 3, 2, 2, 0, 3, 0, 3, 1, 1, 2, 3, 0, 14, 1, 2, 1, 4, 0, 2, 0, 2, 2, 4, 0, 4, 2, 1, 0, 2, 0, 4, 1, 2, 0, 4, 0, 4, 2, 1, 1, 2, 0, 4, 0, 1, 2, 4, 0, 2, 2, 2, 0, 4, 0, 2, 1, 2, 2, 2
Offset: 1

Views

Author

N. J. A. Sloane, Nov 11 2020

Keywords

Links

Crossrefs

tau is A000005.
For tau(i*n) = 2, 3, 4, see A010051, A296084, A338713.
Inspired by A333995.

Programs

  • PARI
    a(n) = sum(i=1, n, numdiv(i*n)==6); \\ Michel Marcus, Nov 11 2020
  • Python
    from collections import Counter
from sympy import factorint
def A338714(n):
    f = Counter(factorint(n))
    return sum(1 for i in range(1,n+1) if (l:=tuple(sorted((f+Counter(factorint(i))).values())))==(1,2) or l==(5,)) # Chai Wah Wu, Oct 14 2023
Showing 1-3 of 3 results.

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

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