A191743 Smallest numbers with a given factorization pattern in their sequence of divisors.

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 60, 64, 72, 80, 84, 88, 90, 96, 100, 108, 120, 126, 128, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 210, 216, 220, 240, 252, 256, 264, 270, 272, 280, 288, 294, 300, 312, 315, 320, 324, 330

Offset

1,2

Comments

To get the factorization pattern of the divisors of n, take the list of divisors of n, and factor each one, using p,q,r,... to represent the prime divisors of n in order. E.g., when factoring 14 as a divisor of 84, the prime divisors of 84 are p=2, q=3, r=7, so 14 => p*r.

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Example

The factors of any prime p are 1,p, so this is the factorization pattern for all primes. The first prime, 2, is thus in the sequence, and no other primes are. Semiprimes have either the pattern 1,p,p^2 or 1,p,q,p*q, so the semiprimes in this sequence are the first instances of each of these, respectively 4 and 6.
For numbers which are the product of the square of a prime and a different prime (A054753), there are three possible patterns: 1,p,q,p^2,p*q,p^2*q, 1,p,q,p*q,q^2,p*q^2, and 1,p,p^2,q,p*q,p*q^2; the exemplars in the sequence are 12, 18, and 20 respectively.

Mathematica

f[n_] := If[n==1, 1, Block[{p = First /@ FactorInteger@n, z}, z = Table[p[[i]] -> x[i], {i, Length@p}]; Times @@ (((#[[1]] /. z)^#[[2]]) & /@ FactorInteger[#]) & /@ Divisors[n]]]; A = <||>; L={}; Do[k = f[n]; If[! KeyExistsQ[A, k], AppendTo[L, n]; A[k] = 1], {n, 330}]; L (* Giovanni Resta, Jul 20 2017 *)

Prog

(PARI)
vecfnd(v, x)={ for(k=1, #v, if(v[k]==x, return(k))); return(0); }
vecfndn(v, x, n)={ for(k=1, n, if(v[k]==x, return(k))); return(0); }
factfmt(k, ps)=
{
    local(r, fm); r=""; fm=factor(k);
    for(i=1, matsize(fm)[1],
        if(i>1, r=Str(r"*"));
        r=Str(r, vecfnd(ps, fm[i, 1]));
        if(fm[i, 2]>1, r=Str(r"^"fm[i, 2]))
   );
   return(r);
} /* end factfmt() */
factpatt(n)=
{
    local(ps, ds, r); r=""; ps=factor(n)[, 1]~; ds=divisors(n);
    for(k=1, #ds, if(k>1, r=Str(r", ")); r=concat(r, factfmt(ds[k], ps)));
    return(r);
} /* end factpatt() */
al(n)=
{
    local(k, r, st, m, pt); k=1; r=vector(n); st=vector(n);
    while(m<n, pt=factpatt(k); if(!vecfndn(st, pt, m), m++; r[m]=k; st[m]=pt); k++);
    return(r);
} /* end al() */
al(66) /* show first 66 terms */

Crossrefs

A025487 and A055932 are subsequences.

Sequence in context: A359420 A325781 A231565 * A308115 A331828 A258118

Adjacent sequences: A191740 A191741 A191742 * A191744 A191745 A191746

Keyword

nonn

Author

Franklin T. Adams-Watters, Jun 13 2011

Status

approved