A097272 Least integer with same "mod 2 prime signature" as n.

1, 2, 3, 4, 3, 6, 3, 8, 9, 6, 3, 12, 3, 6, 15, 16, 3, 18, 3, 12, 15, 6, 3, 24, 9, 6, 27, 12, 3, 30, 3, 32, 15, 6, 15, 36, 3, 6, 15, 24, 3, 30, 3, 12, 45, 6, 3, 48, 9, 18, 15, 12, 3, 54, 15, 24, 15, 6, 3, 60, 3, 6, 45, 64, 15, 30, 3, 12, 15, 30, 3, 72, 3, 6, 45, 12, 15, 30, 3, 48, 81, 6, 3, 60

Offset: 1

Author

Ray Chandler, Aug 22 2004

Keywords

• nonn

Comments

For n = 2^e_0 * p_1^e_1 * ... * p_n^e_n where p_i is odd prime and e_1 >= e_2 >= ... >= e_n, define "mod 2 prime signature" to be ordered prime exponents (e_0,e_1,...,e_n).

Least integer with a given "mod 2 prime signature" is obtained by replacing p_1 with 3, p_2 with 5,..., p_n with n-th odd prime.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A046523, A097273, A097274, A097275.

Programs

Mathematica

a[n_] := Module[{m = IntegerExponent[n, 2], e}, 2^m * If[n == 2^m, 1, e = FactorInteger[n/2^m][[;; , 2]]; Times @@ (Prime[Range[2, Length[e] + 1]]^ReverseSort[e])]]; Array[a, 100] (* Amiram Eldar, Jul 23 2024 *)

PARI

A000265(n) = (n/2^valuation(n, 2));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A006519(n) = (1<A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A097272(n) = A006519(n)*A003961(A046523(A000265(n))); \\ Antti Karttunen, Sep 27 2018

Formula

a(n) = A006519(n)*A003961(A046523(A000265(n))). - Antti Karttunen, Sep 27 2018

Extensions

Offset corrected by Antti Karttunen, Sep 27 2018