cp's OEIS Frontend

Cubefull numbers (A036966) all of whose nonunitary divisors are not cubefull (A362147). - Amiram Eldar, May 13 2023

a(n) = r(n,m) with m such that r(n,m)=r(n,m+1), where r(n,k) = A097246(r(n,k-1)), r(n,0)=n. (The original definition.)

A097248(n) = r(n,a(n)).

From Antti Karttunen, Nov 15 2016: (Start)

The above remark could be interpreted to mean that A097249(n) <= a(n).

All terms are squarefree, and the squarefree numbers are the fixed points.

These are also fixed points eventually reached when iterating A277886.

(End)

This sequence is a permutation of A050376, so every positive integer is the product of a unique subset, S_factors, of its terms. If we restrict S_factors to be chosen from a subset, S_0, consisting of numbers from specified rows and/or columns of this array, there are notable sequences among those that may be generated. See the examples. Other notable sequences can be generated if we restrict the intersection of S_factors with specific rows/columns to have even cardinality. In any of the foregoing cases, the numbers in the resulting sequence form a group under the binary operation A059897(.,.).

Shares with array A246278 the property that columns grow downward by iterating A003961, and indeed, this array can be obtained from A246278 by selecting its columns 1, 2, 8, 128, ..., 2^((2^k)-1), for k >= 0.

A(n,k) is the image of the lattice point with coordinates X=n and Y=k under the inverse of the bijection f defined in the first comment of A306697. This geometric relationship can be used to construct an isomorphism from the polynomial ring GF(2)[x,y] to a ring over the positive integers, using methods similar to those for constructing A297845 and A306697. See A329329, the ring's multiplicative operator, for details.

Square array A(n,k), n >= 1, k >= 1, read by descending antidiagonals.

The group defined by the binary operation A059897(.,.) over the positive integers is commutative with all elements self-inverse, and isomorphic to the additive group of GF(2) polynomial rings such as GF(2)[x,y]. There is a unique isomorphism extending each bijective mapping between respective minimal generating sets. The lexicographically earliest minimal generating set for the A059897 group is A050376, often called the Fermi-Dirac primes. This set has a natural arrangement in a square array, given by A329050(i,j) = prime(i+1)^(2^j), i >= 0, j >= 0. The most meaningful generating set for the additive group of GF(2)[x,y] is {x^i * y^j: i >= 0, j >= 0}, which similarly forms a square array. All this makes A329050(i,j) especially appropriate to be the image (under an isomorphism) of the GF(2) polynomial x^i * y^j.

Using g to denote the intended isomorphism, we specify g(x^i * y^j) = A329050(i,j). This maps minimal generating sets of the additive groups, so the definition of g is completed by specifying g(a+b) = A059897(g(a), g(b)). We then calculate the image under g of polynomial multiplication in GF(2)[x,y], giving us this sequence as the matching multiplicative operator for an isomorphic ring over the positive integers. Using f to denote the inverse of g, A[n,k] = g(f(n) * f(k)).

See the formula section for an alternative definition based on the A329050 array, independent of GF(2)[x,y].

Closely related to A306697 and A297845. If A059897 is replaced in the alternative definition by A059896 (and the definition is supplemented by the derived identity for the absorbing element, shown in the formula section), we get A306697; if A059897 is similarly replaced by A003991 (integer multiplication), we get A297845. This sequence and A306697, considered as multiplicative operators, are carryless arithmetic equivalents of A297845. A306697 uses a method analogous to binary-OR when there would be a multiplicative carry, while this sequence uses a method analogous to binary exclusive-OR. In consequence A(n,k) <> A297845(n,k) exactly when A306697(n,k) <> A297845(n,k). This relationship is not symmetric between the 3 sequences: there are n and k such that A(n,k) = A306697(n,k) <> A297845(n,k). For example A(54,72) = A306697(54,72) = 273375000 <> A297845(54,72) = 22143375000.

a(n) is the number of terms in the unique factorization of n into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between this factorization, canonical (prime power) factorization and A225546.

The result depends only on the prime signature of n.

a(n) is the number of distinct bit-positions where there is a 1-bit in the binary representation of an exponent in the prime factorization of n. - Antti Karttunen, Feb 05 2020

The first 3 is a(128) = a(2^1 * 2^2 * 2^4) = 3 and in general each m occurs first at position 2^(2^m-1) = A058891(m+1). - Peter Munn, Mar 07 2022

We alternatively refer to this sequence as a triangle T(.,.), with T(n,k) = A(k,n-k+1) = prime(k)^(n-k+1).

The monotonic ordering of this sequence, prefixed by 1, is A000961.

The joint-rank array of this sequence is A182869.

Main diagonal gives A062457. - Omar E. Pol, Sep 11 2018

x^(2^n) - a(n) is the minimal polynomial over Q for the algebraic number sqrt(p(1)*sqrt(p(2)*...*sqrt(p(n-1)*sqrt(p(n)))...)), where p(k) is the k-th prime. Each such monic polynomial is irreducible by Eisenstein's Criterion (using p = p(n)).

A prime version of Somos's quadratic recurrence sequence A052129(n) = A052129(n-1)^2 * n = Product_{k=1..n} k^(2^(n-k)). - Jonathan Sondow, Mar 29 2014

All positive integers have unique factorizations into powers of distinct primes, and into powers of squarefree numbers with distinct exponents that are powers of 2. (See A329332 for a description of the relationship between the two.) a(n) is the least number such that both factorizations have n factors. - Peter Munn, Dec 15 2019

From Peter Munn, Jan 24 2020 to Feb 06 2020: (Start)

For n >= 0, a(n+1) is the n-th power of 12 in the monoid defined by A306697.

a(n) is the least positive integer that cannot be expressed as the product of fewer than n terms of A072774 (powers of squarefree numbers).

All terms that are less than the order of the Monster simple group (A003131) are divisors of the group's order, with a(6) exceeding its square root.

(End)

It is remarkable that 4 of the first 5 terms are factorials. - Hal M. Switkay, Jan 21 2025

See A329332 for a description of the relationship between the two factorizations. From this relationship we get the formula a(n) = min(A001221(n), A001221(A225546(n))).

The result depends only on the prime signature of n.

k first appears at A191555(k).