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A348717 a(n) is the least k such that A003961^i(k) = n for some i >= 0 (where A003961^i denotes the i-th iterate of A003961).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 10, 2, 12, 2, 14, 6, 16, 2, 18, 2, 20, 10, 22, 2, 24, 4, 26, 8, 28, 2, 30, 2, 32, 14, 34, 6, 36, 2, 38, 22, 40, 2, 42, 2, 44, 12, 46, 2, 48, 4, 50, 26, 52, 2, 54, 10, 56, 34, 58, 2, 60, 2, 62, 20, 64, 14, 66, 2, 68, 38, 70, 2, 72, 2
Offset: 1

Views

Author

Rémy Sigrist, Oct 31 2021

Keywords

Comments

All terms except a(1) = 1 are even.
To compute a(n) for n > 1:
- if n = Product_{j = 1..o} prime(p_j)^e_j (where prime(i) denotes the i-th prime number, p_1 < ... < p_o and e_1 > 0)
- then a(n) = Product_{j = 1..o} prime(p_j + 1 - p_1)^e_j.
This sequence has similarities with A304776: here we shift down prime indexes, there prime exponents.
Smallest number generated by uniformly decrementing the indices of the prime factors of n. Thus, for n > 1, the smallest m > 1 such that the first differences of the indices of the ordered prime factors (including repetitions) are the same for m and n. As a function, a(.) preserves properties such as prime signature. - Peter Munn, May 12 2022

Links

Crossrefs

Positions of particular values (see formula section): A000040, A001248, A006094, A030078, A030514, A046301, A050997, A090076, A090090, A166329, A251720.
Also see formula section for the relationship to: A000265, A003961, A004277, A005940, A020639, A046523, A055396, A071364, A122111, A156552, A243055, A243074, A297845, A322993.
Sequences with comparable definitions: A304776, A316437.
Cf. A246277 (terms halved), A305897 (restricted growth sequence transform), A354185 (Möbius transform), A354186 (Dirichlet inverse), A354187 (sum with it).

Programs

  • Mathematica
    a[1] = 1; a[n_] := Module[{f = FactorInteger[n], d}, d = PrimePi[f[[1, 1]]] - 1; Times @@ ((Prime[PrimePi[#[[1]]] - d]^#[[2]]) & /@ f)]; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)
  • PARI
    a(n) = { my (f=factor(n)); if (#f~>0, my (pi1=primepi(f[1,1])); for (k=1, #f~, f[k,1] = prime(primepi(f[k,1])-pi1+1))); factorback(f) }

Formula

a(n) = n iff n belongs to A004277.
A003961^(A055396(n)-1)(a(n)) = n for any n > 1.
a(n) = 2 iff n belongs to A000040 (prime numbers).
a(n) = 4 iff n belongs to A001248 (squares of prime numbers).
a(n) = 6 iff n belongs to A006094 (products of two successive prime numbers).
a(n) = 8 iff n belongs to A030078 (cubes of prime numbers).
a(n) = 10 iff n belongs to A090076.
a(n) = 12 iff n belongs to A251720.
a(n) = 14 iff n belongs to A090090.
a(n) = 16 iff n belongs to A030514.
a(n) = 30 iff n belongs to A046301.
a(n) = 32 iff n belongs to A050997.
a(n) = 36 iff n belongs to A166329.
a(1) = 1, for n > 1, a(n) = 2*A246277(n). - Antti Karttunen, Feb 23 2022
a(n) = A122111(A243074(A122111(n))). - Peter Munn, Feb 23 2022
From Peter Munn and Antti Karttunen, May 12 2022: (Start)
a(1) = 1; a(2n) = 2n; a(A003961(n)) = a(n). [complete definition]
a(n) = A005940(1+A322993(n)) = A005940(1+A000265(A156552(n))).
Equivalently, A156552(a(n)) = A000265(A156552(n)).
A297845(a(n), A020639(n)) = n.
A046523(a(n)) = A046523(n).
A071364(a(n)) = A071364(n).
a(n) >= A071364(n).
A243055(a(n)) = A243055(n).
(End)

A243057 If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 1 and for k>=1, p_{k} = A000040(k).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 15, 11, 12, 13, 35, 10, 2, 17, 6, 19, 45, 21, 77, 23, 24, 5, 143, 3, 175, 29, 30, 31, 2, 55, 221, 14, 12, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 7, 15, 187, 1573, 53, 6, 33, 875, 247, 667, 59, 90, 61, 899, 63, 2, 65, 385
Offset: 1

Views

Author

Antti Karttunen, May 31 2014

Keywords

Comments

A243058 gives all n such that a(n) = n (the fixed points of this sequence, which include primes).
A102750 gives such n that a(a(n)) = n. A243286 is the self-inverse permutation induced when the domain is restricted to A102750. Cf. also A242420.
A070003 gives all n such that a(a(n)) <> n. Another variant, A243059, is defined to be zero when n is one of the terms of A070003.

Examples

			For n = 9 = 3*3 = p_2 * p_2, we have a(n) = p_{3-3} * p_3 = 1*3 = 3. [Like all terms in A070003 this is an example of "degenerate case", where some p's in the product get index 0, and thus are set to 1 by convention used here.]
For n = 10 = 2*5 = p_1 * p_3, we have a(n) = p_{3-1} * p_3 = 3*5 = 15.
For n = 12 = 2*2*3 = p_1 * p_1 * p_2, we have a(n) = p_{2-1} * p{2-1} * p_2 = p_1^2 * p_2 = 12.
For n = 15 = 3*5 = p_2 * p_3, we have a(n) = p_{3-2} * p_3 = 2*5 = 10.
For n = 2200 = 2*2*2*5*5*11 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5, we have a(n) = p_{5-3} * p_{5-3} * p_{5-1} * p_{5-1} * p_{5-1} * p_5 = 3*3*7*7*7*11 = 33957.
For n = 33957 = 3*3*7*7*7*11 = p_2 * p_2 * p_4 * p_4 * p_4 * p_5, we have a(n) = p_{5-4} * p_{5-4} * p_{5-4} * p_{5-2} * p_{5-2} * p_5 = 2*2*2*5*5*11 = 2200.

Links

Crossrefs

Fixed points: A243058 (includes primes).
Cf. A243059, A000040, A032742, A243056, A242419, A242420, A243286, A070003, A102750, A243074.

Formula

If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, where p_a <= p_b <= ... <= p_k are (not necessarily distinct) primes (sorted into nondescending order) in the prime factorization of n, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 1 and for k>=1, p_{k} = A000040(k).
a(1)=1, and for n>1, a(n) = p_{A243056(n)} * a(A032742(n)). Here p_{k} stands for 1 when k=0, and otherwise for the k-th prime, A000040(k).
For all n, a(n) = a(A243074(n)).
For all k in A102750, a(k) = A242420(k).
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