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A097621
In canonical prime factorization of n replace p^e with its index in A000961.
7
1, 2, 3, 4, 5, 6, 6, 7, 8, 10, 9, 12, 10, 12, 15, 11, 12, 16, 13, 20, 18, 18, 14, 21, 15, 20, 16, 24, 17, 30, 18, 19, 27, 24, 30, 32, 20, 26, 30, 35, 21, 36, 22, 36, 40, 28, 23, 33, 24, 30, 36, 40, 25, 32, 45, 42, 39, 34, 26, 60, 27, 36, 48, 28, 50, 54, 29, 48, 42, 60, 30, 56, 31
OFFSET
1,2
COMMENTS
The definition of the sequence has been corrected, given that it uses A095874, the indices in the list A000961 of "powers of primes" starting with A000961(1) = 1, rather than A322981, index of p^e in the list of prime powers A246655, as written in the original definition. See A333235 for the variant of this sequence which uses A322981 and A246655 instead, maybe the more natural choice given that the factorization of integers consists of prime powers > 1. - M. F. Hasler, Jun 15 2021
LINKS
FORMULA
Multiplicative with: p^e -> A095874(p^e), p prime.
a(A000961(n)) = n; a(a(n)) = A097622(n); a(a(a(n))) = A097623(n);
a(n) <= n; a(n) = n iff 60 mod n = 0: a(A018266(n)) = A018266(n);
a(A097624(n)) = n and a(m) <> n for n < A097624(n).
EXAMPLE
n=600 = 2^3 * 3 * 5^2 -> A095874(8)*A095874(3)*A095874(25) = 7 * 3 * 15 = 315.
MAPLE
N:= 1000: # to get a(1) to a(N)
Primes:= select(isprime, [2, seq(2*i+1, i=1..(N-1)/2)]):
PP:= sort([1, seq(seq(p^k, k=1..floor(log[p](N))), p=Primes)]):
for n from 1 to nops(PP) do B[PP[n]]:= n od:
seq(mul(B[f[1]^f[2]], f=ifactors(n)[2]), n=1..N); # Robert Israel, Sep 02 2015
MATHEMATICA
pp = Select[Range@100, Length[FactorInteger[#]] == 1 &]; a = Table[Times @@ (Position[pp, #][[1, 1]] & /@ (#[[1]]^#[[2]] & /@ FactorInteger[n])), {n, 73}] (* Ivan Neretin, Sep 02 2015 *)
PROG
(PARI) f(n) = if(isprimepower(n), sum(i=1, logint(n, 2), primepi(sqrtnint(n, i)))+1, n==1); \\ A095874
a(n) = my(fr=factor(n)); for (k=1, #fr~, fr[k, 1] = f(fr[k, 1]^fr[k, 2]); fr[k, 2] = 1); factorback(fr); \\ Michel Marcus, May 29 2021
A097621(n)=vecprod([A095874(f[1]^f[2])|f<-factor(n)~]) \\ M. F. Hasler, Jun 15 2021
CROSSREFS
Cf. A000961 (powers of primes), A246655 (prime powers), A003963, A018266, A095874 (index of n = p^e in A000961).
Cf. A322981 (index of n = p^e in A246655), A333235 (variant of this sequence).
Sequence in context: A302780 A064524 A153249 * A017851 A345424 A359513
KEYWORD
nonn,mult
AUTHOR
Reinhard Zumkeller, Aug 17 2004
EXTENSIONS
Definition corrected by M. F. Hasler, Jun 16 2021
Example corrected by Ray Chandler, Jun 30 2021
STATUS
approved