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A356016
Consider the exponents in the prime factorization of n, and replace each run of k consecutive e's by a unique k; the resulting list corresponds to the exponents in the prime factorization of a(n).
1
1, 2, 6, 2, 12, 4, 24, 2, 6, 30, 48, 6, 96, 90, 18, 2, 192, 6, 384, 30, 210, 270, 768, 6, 12, 810, 6, 90, 1536, 8, 3072, 2, 1050, 2430, 36, 4, 6144, 7290, 5250, 30, 12288, 60, 24576, 270, 30, 21870, 49152, 6, 24, 30, 26250, 810, 98304, 6, 420, 90, 131250
OFFSET
1,2
COMMENTS
We ignore the exponents (all 0's) for the prime numbers beyond the greatest prime factor of n.
There are only two fixed points: a(1) = 1 and a(2) = 2.
Iterating the sequence starting from any n > 1 will always eventually reach the fixed point 2.
FORMULA
a(n) = A319522(A356008(n)).
a(n^k) = a(n) for any k > 0.
a(n) = 2 iff n is a power of 2 > 1.
a(n) = 4 iff n is a power of 6 > 1.
a(n) = 2^k iff n is a power of A002110(k) > 1 (with k > 0).
a(prime(n)) = 3*2^(n-1) for any n > 1.
EXAMPLE
For n = 99:
- 99 = 11^1 * 7^0 * 5^0 * 3^2 * 2^0,
- the list of exponents is: 1 0 0 2 0,
- the run lengths are: 1 2 1 1,
- so a(99) = 7^1 * 5^2 * 3^1 * 2^1 = 1050.
PROG
(PARI) a(n) = { my (v=1, e=-1, k=0, r=0); forprime (p=2, oo, if (n==1, return (v*if (r, prime(k++)^r, 1)), if (e!=e=valuation(n, p), if (r, v*=prime(k++)^r; r=0)); r++; n/=p^e)) }
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Jul 23 2022
STATUS
approved