A356014 -id:A356014 - cp's OEIS frontend

A356008 A variant of Look and Say sequence (A005150) based on exponents in prime factorization of n (see Comments section for precise definition).

1, 6, 105, 12, 315, 18, 945, 24, 525, 6006, 2835, 420, 8505, 42042, 735, 48, 25515, 1050, 76545, 12012, 440895, 294294, 229635, 840, 1575, 2060058, 2625, 84084, 688905, 54, 2066715, 96, 5731635, 14420406, 2205, 36, 6200145, 100942842, 74511255, 24024, 18600435

Offset: 1

Rémy Sigrist, Jul 23 2022

nonn

To compute a(n):
- a(1) = 1,
- for n > 1:
    - consider the prime factorization of n:
          n = Product_{i = 1..k} prime(i)^e_i
          (where e_k > 0 and prime(i) denotes the i-th prime number),
    - apply the Look and Say procedure to the list (e_k, ..., e_1),
    - the result, say (f_m, ..., f_1), gives the prime exponents for a(n):
         a(n) = Product_{i = 1..m} prime(i)^f_i.
There are only two fixed points: a(1) = 1 and a(36) = 36.
All terms are distinct and belong to A244990 (but some terms of A244990, like 210 = 7*5*3*2, do not appear here).

			For n = 99:
- 99 = 11^1 * 7^0 * 5^0 * 3^2 * 2^0,
- the list of exponents is: 1 0 0 2 0,
- applying the Look and Say procedure, we obtain: 1 1 2 0 1 2 1 0,
- so a(99) = 19^1 * 17^1 * 13^2 * 11^0 * 7^1 * 5^2 * 3^1 * 2^0 = 28658175.

Rémy Sigrist, PARI program

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

PARI
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See Links section.
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a(n) = n mod 2.
A007814(a(n)) = A007814(n).
a(prime(n)) = 7*5*3^(n-1) for any n > 1.
a(A002110(n)) = 2*3^n = A008776(n) for any n > 0.

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).

Original entry on oeis.org

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

Rémy Sigrist, Jul 23 2022

nonn

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.

			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.

Cf. A002110, A067255, A318928, A319522, A356008, A356014.

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)) }

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.

A356021 Positive numbers k such that, for any consecutive prime numbers p, q <= A006530(n), the p-adic and q-adic valuations of n are different.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 10, 12, 16, 18, 20, 21, 24, 27, 32, 40, 45, 48, 50, 54, 63, 64, 72, 75, 80, 81, 84, 90, 96, 100, 108, 110, 126, 128, 135, 144, 147, 160, 162, 168, 189, 192, 200, 220, 243, 250, 256, 270, 273, 288, 300, 320, 324, 336, 350, 360, 375, 378, 384

Offset: 1

Rémy Sigrist, Jul 23 2022

nonn

Equivalently, these are fixed points of A356014.
This sequence is infinite as it contains A066205 and A066206.
If m is a term, then m^k is a term (for any k >= 0).

Cf. A006530, A066205, A066206, A356014.

is(n) = { my (v=-1); forprime (p=2, oo, if (n==1, return (1), v==v=valuation(n,p), return (0), n\=p^v)) }

cp's OEIS Frontend

A356008 A variant of Look and Say sequence (A005150) based on exponents in prime factorization of n (see Comments section for precise definition).

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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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A356021 Positive numbers k such that, for any consecutive prime numbers p, q <= A006530(n), the p-adic and q-adic valuations of n are different.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI