A328400 Smallest number with the same set of distinct prime exponents as n.

1, 2, 2, 4, 2, 2, 2, 8, 4, 2, 2, 12, 2, 2, 2, 16, 2, 12, 2, 12, 2, 2, 2, 24, 4, 2, 8, 12, 2, 2, 2, 32, 2, 2, 2, 4, 2, 2, 2, 24, 2, 2, 2, 12, 12, 2, 2, 48, 4, 12, 2, 12, 2, 24, 2, 24, 2, 2, 2, 12, 2, 2, 12, 64, 2, 2, 2, 12, 2, 2, 2, 72, 2, 2, 12, 12, 2, 2, 2, 48, 16, 2, 2, 12, 2, 2, 2, 24, 2, 12, 2, 12, 2, 2, 2, 96, 2, 12, 12, 4, 2, 2, 2, 24, 2

Offset: 1

Author

Antti Karttunen, Oct 15 2019

Keywords

• nonn

Comments

A variant of A046523 which gives the smallest number with the same prime signature as n. However, in this sequence, if any prime exponent occurs multiple times in n, the extra occurrences are removed and the signature is that of one of the numbers where only distinct values of prime exponents occur (A130091).

Examples

			90 = 2^1 * 3^2 * 5^1 has prime signature (1,1,2). The smallest number with prime signature (1,2) is 12 = 2^2 * 3, thus a(90) = 12.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A007947, A046523, A181819, A181821, A328401 (rgs-transform).

Cf. A005117 (gives indices of terms <= 2), A062503 (after its initial 1, gives indices of 4's in this sequence).

Programs

Mathematica

Array[Times @@ MapIndexed[Prime[#2[[1]]]^#1 &, Reverse[Flatten[Cases[FactorInteger[#], {p_, k_} :> Table[PrimePi[p], {k}]]]]] &[Times @@ FactorInteger[#][[All, 1]]] &@ If[# == 1, 1, Times @@ Prime@ FactorInteger[#][[All, -1]]] &, 105] (* Michael De Vlieger, Oct 17 2019, after Gus Wiseman at A181821 *)

PARI

A007947(n) = factorback(factorint(n)[, 1]);
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); };
A328400(n) = A181821(A007947(A181819(n)));

Formula

a(n) = A181821(A007947(A181819(n))).

For all n, a(n) = a(A046523(n)).