

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
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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

Ivan Neretin, Table of n, a(n) for n = 1..10000


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..(N1)/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. A097622, A097623, A097624.
Cf. A322981 (index of n = p^e in A246655), A333235 (variant of this sequence).
Sequence in context: A302780 A064524 A153249 * A017851 A345424 A358506
Adjacent sequences: A097618 A097619 A097620 * A097622 A097623 A097624


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



