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A variant of Look and Say sequence (A005150) based on exponents in prime factorization of n (see Comments section for precise definition).
3

%I #7 Jul 25 2022 10:43:34

%S 1,6,105,12,315,18,945,24,525,6006,2835,420,8505,42042,735,48,25515,

%T 1050,76545,12012,440895,294294,229635,840,1575,2060058,2625,84084,

%U 688905,54,2066715,96,5731635,14420406,2205,36,6200145,100942842,74511255,24024,18600435

%N A variant of Look and Say sequence (A005150) based on exponents in prime factorization of n (see Comments section for precise definition).

%C To compute a(n):

%C - a(1) = 1,

%C - for n > 1:

%C - consider the prime factorization of n:

%C n = Product_{i = 1..k} prime(i)^e_i

%C (where e_k > 0 and prime(i) denotes the i-th prime number),

%C - apply the Look and Say procedure to the list (e_k, ..., e_1),

%C - the result, say (f_m, ..., f_1), gives the prime exponents for a(n):

%C a(n) = Product_{i = 1..m} prime(i)^f_i.

%C There are only two fixed points: a(1) = 1 and a(36) = 36.

%C All terms are distinct and belong to A244990 (but some terms of A244990, like 210 = 7*5*3*2, do not appear here).

%H Rémy Sigrist, <a href="/A356008/a356008.gp.txt">PARI program</a>

%F a(n) = n mod 2.

%F A007814(a(n)) = A007814(n).

%F a(prime(n)) = 7*5*3^(n-1) for any n > 1.

%F a(A002110(n)) = 2*3^n = A008776(n) for any n > 0.

%e For n = 99:

%e - 99 = 11^1 * 7^0 * 5^0 * 3^2 * 2^0,

%e - the list of exponents is: 1 0 0 2 0,

%e - applying the Look and Say procedure, we obtain: 1 1 2 0 1 2 1 0,

%e - so a(99) = 19^1 * 17^1 * 13^2 * 11^0 * 7^1 * 5^2 * 3^1 * 2^0 = 28658175.

%o (PARI) See Links section.

%Y Cf. A002110, A005150, A007814, A008776, A067255, A244990, A356014, A356016.

%K nonn

%O 1,2

%A _Rémy Sigrist_, Jul 23 2022