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 1-4 of 4 results.

A069900 Numbers k such that the integer quotient of largest and smallest prime factors of k is greater than one.

Original entry on oeis.org

10, 14, 20, 21, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120
Offset: 1

Views

Author

Labos Elemer, Apr 10 2002

Keywords

Comments

Numbers k such that A069897(k) = floor(P(k)/p(k)) > 1, where P(k) and p(k) are largest and least prime factor of k, respectively.
Also numbers having at least one prime factor greater than twice the smallest prime factor: complement of A081306. - Reinhard Zumkeller, Mar 17 2003

Examples

			Composites with at least two and sufficiently deviating prime factors are here, like 2q, where q = prime >= 5: {10, ..., 254}.
Numbers with such divisors like 30 are also included.

Links

Crossrefs

Cf. A006530, A020639, A069897, A069899, A081303, A081306.

Programs

  • Mathematica
    Select[Range@ 120, #[[-1]] > 2 #[[1]] &@ FactorInteger[#][[All, 1]] &] (* Michael De Vlieger, Dec 08 2018 *)
  • PARI
    is(k) = if(k == 1, 0, my(p = factor(k)[,1]); p[#p] > 2*p[1]); \\ Amiram Eldar, Feb 10 2025

Formula

A081303(a(n)) > 0. - Reinhard Zumkeller, Mar 17 2003

Extensions

More terms from Reinhard Zumkeller, Mar 17 2003

A069859 (Largest prime factor of n) modulo (smallest prime factor of n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 1, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 3, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 4, 1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 1, 0, 1, 6, 1, 1, 1, 4, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 23 2002

Keywords

Comments

a(n) = A006530(n) - A069897(n)*A020639(n).

Links

Crossrefs

Cf. A006530, A020639, A046665, A069897.

Programs

  • Mathematica
    lpfmspf[n_]:=Module[{fs=Transpose[FactorInteger[n]][[1]]},Mod[Last[fs], First[fs]]]; Array[lpfmspf,100] (* Harvey P. Dale, Jul 10 2015 *)
  • PARI
    A069859(n) = if(1==n,0, my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); (gpf%lpf)); \\ Antti Karttunen, Sep 25 2018

Formula

a(n) = A006530(n) mod A020639(n).

Extensions

More terms from Antti Karttunen, Sep 25 2018

A069899 Numbers k such that the integer quotient of largest and smallest prime factors of k is 1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 59, 61, 64, 67, 71, 72, 73, 75, 77, 79, 81, 83, 89, 91, 96, 97, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 135, 137, 139, 143, 144
Offset: 1

Views

Author

Labos Elemer, Apr 10 2002

Keywords

Comments

Numbers k such that A069897(k) = floor(P(k)/p(k)) = 1, where P(k) and p(k) are largest and least prime factor of k, respectively.
Numbers k > 1 such that A006530(k) < 2*A020639(k). - Amiram Eldar, Feb 10 2025

Examples

			Beside primes and prime powers, composite numbers like 96 are terms because floor(3/2) = 1.

Links

Crossrefs

Cf. A006530, A020639, A069897, A069900, A081306.

Programs

  • Mathematica
    q[k_] := Module[{p = FactorInteger[k][[;;, 1]]}, p[[-1]] < 2*p[[1]]]; Select[Range[2, 150], q] (* Amiram Eldar, Feb 10 2025 *)
  • PARI
    isok(k) = if(k == 1, 0, my(p = factor(k)[,1]); p[#p] < 2*p[1]); \\ Amiram Eldar, Feb 10 2025

Formula

a(n) = A081306(n+1). - Amiram Eldar, Feb 10 2025

A120454 a(n) = ceiling(GPF(n)/LPF(n)) where GPF is greatest prime factor, LPF is least prime factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 2, 1, 1, 2, 1, 3, 3, 6, 1, 2, 1, 7, 1, 4, 1, 3, 1, 1, 4, 9, 2, 2, 1, 10, 5, 3, 1, 4, 1, 6, 2, 12, 1, 2, 1, 3, 6, 7, 1, 2, 3, 4, 7, 15, 1, 3, 1, 16, 3, 1, 3, 6, 1, 9, 8, 4, 1, 2, 1, 19, 2, 10, 2, 7, 1, 3, 1, 21, 1, 4, 4, 22, 10, 6, 1, 3, 2, 12, 11, 24, 4, 2, 1, 4, 4
Offset: 1

Views

Author

Jonathan Vos Post, Aug 16 2006

Keywords

Comments

Given GPF(n) and LPF(n), the sum is A074320, the difference is A046665 and the product is A066048. a(n) = 1 iff n is p^k iff n is in A000961.

Examples

			a(26) = ceiling(GPF(26)/LPF(26)) = ceiling(13/2) = 7.

Links

Crossrefs

Cf. A000961, A006530, A020639, A024619, A046665, A066048, A069897, A074320.

Programs

  • Maple
    A120454 := proc(n) local ifs ; if n = 1 then RETURN(1) ; else ifs := ifactors(n)[2] ; RETURN( ceil(op(1,op(-1,ifs))/op(1,op(1,ifs))) ) ; fi ; end ; for n from 1 to 100 do printf("%d, ",A120454(n)) ; od ; # R. J. Mathar, Dec 16 2006
  • Mathematica
    a[n_] := Module[{p = FactorInteger[n][[;;, 1]]}, Ceiling[p[[-1]] / p[[1]]]]; Array[a, 100] (* Amiram Eldar, Oct 24 2024 *)
  • PARI
    A120454(n) = if(1==n,1, my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); ceil(gpf/lpf)); \\ Antti Karttunen, Sep 06 2018

Formula

a(n) = ceiling(A006530(n)/A020639(n)).
a(n) = A069897(n) + 1 if n is not a power of a prime (A024619), and 1 otherwise. - Amiram Eldar, Oct 24 2024

Extensions

Corrected and extended by R. J. Mathar, Dec 16 2006
Showing 1-4 of 4 results.

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