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A328400
Smallest number with the same set of distinct prime exponents as n.
9
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
1,2
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).
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); };
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
KEYWORD
nonn
AUTHOR
Antti Karttunen, Oct 15 2019
STATUS
approved