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.

A340058 Composite numbers c such that phi(c)/phi(mind(c)) mod phi(c)/phi(maxd(c)) = 0, where phi is the Euler function, mind(c) is the smallest nontrivial divisor of c, maxd(c) is the largest nontrivial divisor of c.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96, 98, 99
Offset: 1

Views

Author

Maxim Karimov, Dec 27 2020

Keywords

Comments

This equivalence criterion splits a set of composite numbers into two classes and can be used to count certain combinatorial objects.

Crossrefs

Cf. A000010, A002808, A335902.

Programs

  • MATLAB
    n=100; % gives all terms of the sequence not exceeding n
A=[];
for i=1:n
   dn=divisors(i);
   if size(dn,2)>2 && mod(totient(i)/totient(dn(2)),totient(i)/totient(dn(end-1)))==0
      A=[A i];
   end
end
function [res] = totient(n)
res=0;
    for i=1:n
        if gcd(i,n)==1
            res=res+1;
        end
    end
end
  • PARI
    isok(c) = if ((c>1) && !isprime(c), my(t=eulerphi(c), d=divisors(c)); ((t/eulerphi(d[2])) % (t/eulerphi(d[#d-1]))) == 0); \\ Michel Marcus, Dec 28 2020

A340269 Numbers k > 1 such that lpf(k)-1 does not divide d-1 for at least one divisor d of k, where lpf(k) is the least prime factor of k (A020639).

Original entry on oeis.org

35, 55, 77, 95, 115, 119, 143, 155, 161, 175, 187, 203, 209, 215, 221, 235, 245, 247, 253, 275, 287, 295, 299, 319, 323, 329, 335, 355, 371, 377, 385, 391, 395, 403, 407, 413, 415, 437, 455, 473, 475, 493, 497, 515, 517, 527, 533, 535, 539, 551, 559, 575, 581
Offset: 1

Views

Author

Maxim Karimov, Jan 02 2021

Keywords

Comments

No terms are divisible by 2 or 3; no terms are in A000961. - Robert Israel, Oct 10 2023

Links

Crossrefs

Cf. A000010, A000961, A020639, A335902, A340058, A340268.

Programs

  • MATLAB
    n=300; % gives all terms of the sequence not exceeding n
A=[];
for i=2:n
    lpf=2;
    while mod(i,lpf)~=0
        lpf=lpf+1;
    end
    for d=1:i
        if mod(i,d)==0 && mod(d-1,lpf-1)~=0
            A=[A i];
            break
        end
    end
end
  • Maple
    with(numtheory):
q:= n-> (f-> ormap(d-> irem(d-1, f)>0, divisors(n)))(min(factorset(n))-1):
select(q, [$2..600])[];  # Alois P. Heinz, Feb 12 2021
  • Mathematica
    Select[Range[2, 600], Function[{d, k}, AnyTrue[d, Mod[#, k] != 0 &]] @@ {Divisors[#] - 1, FactorInteger[#][[1, 1]] - 1} &] (* Michael De Vlieger, Feb 12 2021 *)

A340268 Composite numbers k>1 such that (s-1) | (d-1) for each d | k, where s = lpf(k) = A020639(k).

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96
Offset: 1

Views

Author

Maxim Karimov, Jan 02 2021

Keywords

Comments

Not a duplicate of A340058 because the complements A335902 and A340269 differ. - R. J. Mathar, Feb 16 2021

Crossrefs

Cf. A000010, A000961, A020639, A340058, A335902, A340269 (complement).
Contains all composite terms of at least A003586, A003591, A003592, A003593, A003596.

Programs

  • MATLAB
    n=300; % gives all terms of the sequence not exceeding n
A=[];
for i=2:n
    lpf=2;
    while mod(i,lpf)~=0
        lpf=lpf+1;
    end
    for d=1:floor(i/2)
        if mod(i,d)==0 && mod(d-1,lpf-1)~=0
            break
        elseif d==floor(i/2)
            A=[A i];
        end
    end
end
  • Maple
    with(numtheory):
q:= n-> (f-> andmap(d-> irem(d-1, f)=0, divisors(n)))(min(factorset(n))-1):
select(not isprime and q, [$2..96])[];  # Alois P. Heinz, Feb 12 2021
  • Mathematica
    Select[Range[2, 96], Function[{n, s}, And[! PrimeQ@ n, AllTrue[Divisors[n] - 1, Mod[#, s] == 0 &]]] @@ {#, FactorInteger[#][[1, 1]] - 1} &] (* Michael De Vlieger, Feb 12 2021 *)
  • PARI
    isok(c) = if ((c>1) && !isprime(c), my(f=factor(c)[,1]); for (k=1, #f~, if ((f[k]-1) % (f[1]-1), return(0))); return(1)); \\ Michel Marcus, Jan 03 2021
Showing 1-3 of 3 results.

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