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

%I A191743 #11 Jul 20 2017 02:24:04
%S A191743 1,2,4,6,8,12,16,18,20,24,30,32,36,40,42,48,54,60,64,72,80,84,88,90,
%T A191743 96,100,108,120,126,128,140,144,150,156,160,162,168,176,180,192,198,
%U A191743 200,210,216,220,240,252,256,264,270,272,280,288,294,300,312,315,320,324,330
%N A191743 Smallest numbers with a given factorization pattern in their sequence of divisors.
%C A191743 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.
%H A191743 Giovanni Resta, <a href="/A191743/b191743.txt">Table of n, a(n) for n = 1..10000</a>
%e A191743 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.
%e A191743 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.
%t A191743 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 *)
%o A191743 (PARI)
%o A191743 vecfnd(v, x)={ for(k=1, #v, if(v[k]==x, return(k))); return(0); }
%o A191743 vecfndn(v, x, n)={ for(k=1, n, if(v[k]==x, return(k))); return(0); }
%o A191743 factfmt(k, ps)=
%o A191743 {
%o A191743     local(r, fm); r=""; fm=factor(k);
%o A191743     for(i=1, matsize(fm)[1],
%o A191743         if(i>1, r=Str(r"*"));
%o A191743         r=Str(r, vecfnd(ps, fm[i, 1]));
%o A191743         if(fm[i, 2]>1, r=Str(r"^"fm[i, 2]))
%o A191743    );
%o A191743    return(r);
%o A191743 } /* end factfmt() */
%o A191743 factpatt(n)=
%o A191743 {
%o A191743     local(ps, ds, r); r=""; ps=factor(n)[, 1]~; ds=divisors(n);
%o A191743     for(k=1, #ds, if(k>1, r=Str(r", ")); r=concat(r, factfmt(ds[k], ps)));
%o A191743     return(r);
%o A191743 } /* end factpatt() */
%o A191743 al(n)=
%o A191743 {
%o A191743     local(k, r, st, m, pt); k=1; r=vector(n); st=vector(n);
%o A191743     while(m<n, pt=factpatt(k); if(!vecfndn(st, pt, m), m++; r[m]=k; st[m]=pt); k++);
%o A191743     return(r);
%o A191743 } /* end al() */
%o A191743 al(66) /* show first 66 terms */
%Y A191743 A025487 and A055932 are subsequences.
%K A191743 nonn
%O A191743 1,2
%A A191743 _Franklin T. Adams-Watters_, Jun 13 2011