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).

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

1,2

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

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from indices in prime factorization

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)

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 *)

Prog

(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) }

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).

Sequence in context: A320389 A378447 A046801 * A316437 A137502 A318885

Adjacent sequences: A348714 A348715 A348716 * A348718 A348719 A348720

Keyword

nonn

Author

Rémy Sigrist, Oct 31 2021

Status

approved