A328400 - OEIS

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

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OFFSET

1,2

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).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from exponents in factorization of n

FORMULA

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

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

EXAMPLE

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.

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 *)

PROG

(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)));

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).

Sequence in context: A360460 A057000 A348044 * A239676 A295639 A374587

Adjacent sequences: A328397 A328398 A328399 * A328401 A328402 A328403

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 15 2019

STATUS

approved