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.

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A295075 Numbers k such that d + k/d is never prime for any divisor d of k.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 25, 26, 27, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99
Offset: 1

Views

Author

Michel Marcus, Nov 13 2017

Keywords

Comments

Numbers k such that A282849(k) = 0.

Links

Crossrefs

Cf. A282849 (number of divisors of n such that d + n/d is prime), A080715 (d + n/d is prime for every divisor d of n).
Includes all elements > 2 of A047255.

Programs

  • Maple
    remove(n -> ormap(t -> isprime(t+n/t), numtheory:-divisors(n)), [$1..100]); # Robert Israel, Nov 14 2017
  • Mathematica
    Select[Range@ 100, Function[k, NoneTrue[Divisors@ k, PrimeQ[# + k/#] &]]] (* Michael De Vlieger, Nov 13 2017 *)
  • PARI
    isok(n) = sumdiv(n, d, isprime(d+n/d)) == 0;

A350230 Numbers m such that for all factorizations m=a*b with positive integers (a, b), a*b+a+b is a prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 11, 14, 15, 21, 23, 26, 29, 30, 33, 35, 41, 51, 53, 65, 74, 83, 86, 89, 111, 113, 131, 141, 155, 158, 173, 179, 186, 191, 194, 209, 215, 221, 230, 231, 233, 239, 251, 254, 278, 281, 293, 321, 323, 326, 329, 341, 345, 359, 371, 398, 413, 419, 426
Offset: 1

Views

Author

Michel Lagneau, Dec 21 2021

Keywords

Comments

From Marius A. Burtea, Dec 26 2021: (Start)
If p is a prime number in A005384 (Sophie Germain prime), then it is a term. Indeed, p = p*1 and p + p + 1 = 2*p + 1 is prime.
All terms (m >= 2) are squarefree numbers (A005117). Indeed, if m = p^2*k, p >= 2, k >= 1, then m = p*(p*k) and p^2*k + p + p*k = p*(p*k + 1 + k ) is not prime. (End).

Examples

			1 is in the sequence because 1 = 1*1 and 1*1 + 1 + 1 = 3 is prime;
30 is in the sequence because A038548(30) = 4 has 4 factorizations:
30 = 1*30 = 2*15 = 3*10 = 5*6 and
30 + 1 + 30 = 61 is prime;
30 + 2 + 15 = 47 is prime;
30 + 3 + 10 = 43 is prime;
30 + 5 + 6 = 41 is prime.

Crossrefs

Cf. A005117 (squarefree numbers), A005384 (Sophie Germain primes), A038548, A080715.

Programs

  • Magma
    [n:n in [1..450]|forall{d: d in Divisors(n)| IsPrime(n+d+n div d)}]; // Marius A. Burtea, Dec 26 2021
  • Maple
    A350230 := proc(n)
    local a,b ;
    for a in numtheory[divisors](n) do
        b := n/a ;
        if not isprime(a*b+b+a) then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 1 to 500 do
    if isA350230(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 24 2022
  • Mathematica
    t={};Do[ds=Divisors[n];If[EvenQ[Length[ds]],ok=True;k=1;While[k<=Length[ds]/2&&(ok=PrimeQ[ds[[k]]*ds[[-k]]+ds[[k]]+ds[[-k]]]),k++];If[ok,AppendTo[t,n]]],{n,2,4000}];t
  • PARI
    isok(m) = sumdiv(m, d, isprime(m+d+m/d)) == numdiv(m); \\ Michel Marcus, Dec 25 2021
  • Python
    from sympy import divisors, isprime
def ok(n):
    divs = divisors(n)
    if n == 0: return False
    return all(isprime(a*b+a+b) for a, b in ((d, n//d) for d in divs))
print([k for k in range(427) if ok(k)]) # Michael S. Branicky, Dec 21 2021

A358816 Numbers k such that d + k/d is prime for any unitary divisor d of k.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 52, 58, 60, 70, 72, 78, 82, 88, 100, 102, 108, 112, 130, 148, 162, 172, 190, 192, 196, 198, 210, 228, 232, 240, 250, 256, 268, 270, 280, 310, 312, 316, 330, 352, 358, 372, 378, 382, 388, 396, 400, 408, 432, 442, 448
Offset: 1

Views

Author

Amiram Eldar, Dec 02 2022

Keywords

Comments

A unitary divisor d of k is a number d such that d|k and gcd(d, k/d) = 1.
A080715 is the subsequence of the squarefree terms of this sequence.

Links

Crossrefs

Cf. A005117, A077610, A080715.
Subsequence of A006093.

Programs

  • Mathematica
    q[n_] := AllTrue[Select[Divisors[n], #^2 < n && CoprimeQ[#, n/#] &], PrimeQ[# + n/#] &]; Select[Prime[Range[90]] - 1, q]
  • PARI
    is(n) = fordiv(n, d, if(d < n^2 && gcd(d, n/d) == 1 && !isprime(d+n/d), return(0))); return(1);
Previous Showing 11-13 of 13 results.

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