cp's OEIS Frontend

Prime product form of primorial base expansion of n.

Sequence is a permutation of A048103. It maps the smallest prime not dividing n to the smallest prime dividing n, that is, A020639(a(n)) = A053669(n) holds for all n >= 1.

The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever A329041(x,y) = 1, that is, when adding x and y together will not generate any carries in the primorial base. Examples of such pairs of x and y are A328841(n) & A328842(n), and also A328770(n) (when added with itself). - Antti Karttunen, Oct 31 2019

From Antti Karttunen, Feb 18 2022: (Start)

The conjecture given in A327969 asks whether applying this function together with the arithmetic derivative (A003415) in some combination or another can eventually transform every positive integer into zero.

Another related open question asks whether there are any other numbers than n=6 such that when starting from that n and by iterating with A003415, one eventually reaches a(n). See comments in A351088.

This sequence is used in A351255 to list the terms of A099308 in a different order, by the increasing exponents of the successive primes in their prime factorization. (End)

From Bill McEachen, Oct 15 2022: (Start)

From inspection, the least significant decimal digits of a(n) terms form continuous chains of 30 as follows. For n == i (mod 30), i=0..5, there are 6 ordered elements of these 8 {1,2,3,6,9,8,7,4}. Then for n == i (mod 30), i=6..29, there are 12 repeated pairs = {5,0}.

Moreover, when the individual elements of any of the possible groups of 6 are transformed via (7*digit) (mod 10), the result matches one of the other 7 groupings (not all 7 may be seen). As example, {1,2,3,6,9,8} transforms to {7,4,1,2,3,6}. (End)

The least significant digit of a(n) in base 4 is given by A353486, and in base 6 by A358840. - Antti Karttunen, Oct 25 2022, Feb 17 2024

a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n).

Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a(A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022

The primes become the powers of 2 (2 -> 1, 3 -> 2, 5 -> 4, 7 -> 8); the composite numbers are formed by taking the values for the factors in the increasing order, multiplying them by the consecutive powers of 2, and summing. See the Example section.

From Antti Karttunen, Jun 27 2014: (Start)

The odd bisection (containing even terms) halved gives A244153.

The even bisection (containing odd terms), when one is subtracted from each and halved, gives this sequence back.

(End)

Question: Are there any other solutions that would satisfy the recurrence r(1) = 0; and for n > 1, r(n) = Sum_{d|n, d>1} 2^A033265(r(d)), apart from simple variants 2^k * A156552(n)? See also A297112, A297113. - Antti Karttunen, Dec 30 2017

A permutation of the squarefree numbers A005117. The missing positive numbers are in A013929. - Alois P. Heinz, Sep 06 2014

From Antti Karttunen, Apr 18 & 19 2017: (Start)

Because a(n) toggles the parity of n there are neither fixed points nor any cycles of odd length.

Conjecture: there are no finite cycles of any length. My grounds for this conjecture: any finite cycle in this sequence, if such cycles exist at all, must have at least one member that occurs somewhere in A285319, the terms that seem already to be quite rare. Moreover, any such a number n should satisfy in addition to A019565(n) < n also that A048675^{k}(n) is squarefree, not just for k=0, 1 but for all k >= 0. As there is on average a probability of only 6/(Pi^2) = 0.6079... that any further term encountered on the trajectory of A048675 is squarefree, the total chance that all of them would be squarefree (which is required from the elements of A019565-cycles) is soon minuscule, especially as A048675 is not very tightly bounded (many trajectories seem to skyrocket, at least initially). I am also assuming that usually there is no significant correlation between the binary expansions of n and A048675(n) (apart from their least significant bits), or, for that matter, between their prime factorizations.

See also the slightly stronger conjecture in A285320, which implies that there would neither be any two-way infinite cycles.

If either of the conjectures is false (there are cycles), then certainly neither sequence A285332 nor its inverse A285331 can be a permutation of natural numbers. (End)

The conjecture made in A087207 (see also A288569) implies the two conjectures mentioned above. A further constraint for cycles is that in any A019565-trajectory which starts from a squarefree number (A005117), every other term is of the form 4k+2, while every other term is of the form 6k+3. - Antti Karttunen, Jun 18 2017

The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever x and y do not have a 1-bit in the same position, i.e., when A004198(x,y) = 0. See also A283475. - Antti Karttunen, Oct 31 2019

The above identity becomes unconditional if binary exclusive OR, A003987(.,.), is substituted for addition, and A059897(.,.), a multiplicative equivalent of A003987, is substituted for multiplication. This gives us a(A003987(x,y)) = A059897(a(x), a(y)). - Peter Munn, Nov 18 2019

Also the Heinz number of the binary indices of n, where the Heinz number of a sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k), and a number's binary indices (A048793) are the positions of 1's in its reversed binary expansion. - Gus Wiseman, Dec 28 2022

Completely additive with a(p^e) = e * A002110(A000720(p)-1).

This is a left inverse of A276086 ("primorial base exp-function"), hence the name "primorial base log-function". When the domain is restricted to the terms of A048103, this works also as a right inverse, as A276086(a(A048103(n))) = A048103(n) for all n >= 1. - Antti Karttunen, Apr 24 2022

On average, every third term is a multiple of 4. See A369001. - Antti Karttunen, May 26 2024

Equivalently, sum of digits when n is written in factorial base (A007623).

Equivalently, a(0)...a(n!-1) give the total number of inversions of the permutations of n elements in lexicographic order (the factorial numbers in rising base are the inversion tables of the permutations and their sum of digits give the total number of inversions, see example and the Fxtbook link). - Joerg Arndt, Jun 17 2011

Also minimum number of adjacent transpositions needed to produce each permutation in the list A055089, or number of swappings needed to bubble sort each such permutation. (See A055091 for the minimum number of any transpositions.)

Numbers that are sums of distinct factorials (0! and 1! not treated as distinct).

Complement of A115945; A115944(a(n)) > 0; A115647 is a subsequence. - Reinhard Zumkeller, Feb 02 2006

A115944(a(n)) = 1. - Reinhard Zumkeller, Dec 04 2011

From Tilman Piesk, Jun 04 2012: (Start)

The inversion vector (compare A007623) of finite permutation a(n) (compare A055089, A195663) has only zeros and ones. Interpreted as a binary number it is 2*n (or n when the inversion vector is defined without the leading 0).

The inversion set of finite permutation a(n) interpreted as a binary number (compare A211362) is A211364(n).

(End)

Numbers not divisible by p^(1+A000720(p)) for any prime p, where A000720(p) gives the index of prime p: 1 for 2, 2 for 3, 3 for 5, and so on.

Also Heinz numbers of integer partitions where the multiplicity of i does not exceed i for any i (A052335). Differs from A048103 in lacking {625, 1250, 1875, 3750, 4375, 5625, 6875, 8125, 8750, ...}. - Gus Wiseman, Mar 09 2019

Asymptotic density is Product_{i>=1} 1-prime(i)^(-1-i) = 0.72102334... - Amiram Eldar, Oct 20 2020

The binary digits of a(n) encode the prime factorization of A007913(n), where the i-th digit from the right is 1 if and only if prime(i) divides A007913(n), otherwise 0. - Robert Israel, Jan 12 2015

Old name: a(1) = 0; a(A000040(n)) = 2^(n-1), and a(n*m) = a(n) XOR a(m).

XOR is the bitwise exclusive or operation (A003987).

a(k^2) = 0 for a natural number k.

Equivalently, the i-th binary digit from the right is 1 iff prime(i) divides n an odd number of times, otherwise zero. - Ethan Beihl, Oct 15 2016

When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443, with scheme explained in A206284), then A048675(n) gives the evaluation of that polynomial at x=2. This sequence is otherwise similar, except the polynomial is evaluated over the field GF(2), which implies also that all its coefficients are essentially reduced modulo 2. - Antti Karttunen, Dec 11 2015

Squarefree numbers (A005117) give the positions k where a(k) = A048675(k). - Antti Karttunen, Oct 29 2016

From Peter Munn, Jun 07 2021: (Start)

When we encode polynomials with nonnegative integer coefficients as described by Antti Karttunen above, polynomial addition is represented by integer multiplication, multiplication is represented by A297845(.,.), and this sequence represents a surjective semiring homomorphism to polynomials in GF(2)[x] (encoded as described in A048720). The mapping of addition operations by this homomorphism is part of the sequence definition: "a(n*m) = a(n) XOR a(m)". The mapping of multiplication is given by a(A297845(n, k)) = A048720(a(n), a(k)).

In a related way, A329329 defines a representation of a different set of polynomials as positive integers, namely polynomials in GF(2)[x,y].

Let P_n(x,y) denote the polynomial represented, as in A329329, by n >= 1. If 0 is substituted for y in P_n(x,y), we get a polynomial P'_n(x,y) (in which y does not appear, of course) that is equivalent to a polynomial P'_n(x) in GF(2)[x]. a(n) is the integer encoding of P'_n(x) (described in A048720).

Viewed as above, this sequence represents another surjective homomorphism, a homomorphism between polynomial rings, with A329329(.,.)/A059897(.,.) and A048720(.,.)/A003987(.,.) as the respective ring operations.

a(n) can be composed as a(n) = A048675(A007913(n)) and the effect of the A007913(.) component corresponds to different operations on the respective polynomial domains of the two homomorphisms described above. In the first homomorphism, coefficients are reduced modulo 2; in the second, 0 is substituted for y. This is illustrated in the examples.

(End)