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A097272
Least integer with same "mod 2 prime signature" as n.
5
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
1,2
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
FORMULA
a(n) = A006519(n)*A003961(A046523(A000265(n))). - Antti Karttunen, Sep 27 2018
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 *)
PROG
(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<<valuation(n, 2));
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Ray Chandler, Aug 22 2004
EXTENSIONS
Offset corrected by Antti Karttunen, Sep 27 2018
STATUS
approved