cp's OEIS Frontend

Primes and A030513(d(x)=4) are subsets; d(16k+4) and d(16k+12) have the form 3Q, so x=16k+4 or 16k-4 numbers are missing.

A number m is a term if and only if all its divisors are infinitary, or A000005(m) = A037445(m). - Vladimir Shevelev, Feb 23 2017

All exponents in the prime number factorization of a(n) have the form 2^k-1, k >= 1. So it is an S-exponential sequence (see Shevelev link) with S={2^k-1}. Using Theorem 1, we obtain that a(n) ~ C*n, where C = Product((1-1/p)*(1 + Sum_{i>=1} 1/p^(2^i-1))). - Vladimir Shevelev Feb 27 2017

This constant is C = 0.687827... . - Peter J. C. Moses, Feb 27 2017

From Peter Munn, Jun 18 2022: (Start)

1 and numbers j*m^2, j squarefree, m >= 1, such that all prime divisors of m divide j, and m is in the sequence.

Equivalently, the nonempty set of numbers whose squarefree part (A007913) and squarefree kernel (A007947) are equal, and whose square part's square root (A000188) is in the set.

(End)

Suppose that p(1), p(2), p(3), ... is an integer sequence satisfying 1 <= p(n) <= n for n >= 1. Define g(1)=(1) and for n > 1, form g(n) from g(n-1) by inserting n so that its position in the resulting n-tuple is p(n). The sequence f obtained by concatenating g(1), g(2), g(3), ... is clearly a fractal sequence, here introduced as the fractalization of p. The interspersion associated with f is here introduced as the interspersion fractally induced by p, denoted by I(p); thus, the k-th term in the n-th row of I(p) is the position of the k-th n in f. Regarded as a sequence, I(p) is a permutation of the positive integers; its inverse permutation is denoted by Q(p).

...

Example: Let p=(1,2,2,3,3,4,4,5,5,6,6,7,7,...)=A008619. Then g(1)=(1), g(2)=(1,2), g(3)=(1,3,2), so that

f=(1,1,2,1,3,2,1,3,4,2,1,3,5,4,2,1,3,5,6,4,2,1,...)=A194959; and I(p)=A057027, Q(p)=A064578.

The interspersion I(P) has the following northwest corner, easily read from f:

1 2 4 7 11 16 22

3 6 10 15 21 28 36

5 8 12 17 23 30 38

9 14 20 27 35 44 54

...

Following is a chart of selected p, f, I(p), and Q(p):

p f I(p) Q(p)

A000027 A002260 A000027 A000027

A008619 A194959 A057027 A064578

A194960 A194961 A194962 A194963

A053824 A194965 A194966 A194967

A053737 A194973 A194974 A194975

A019446 A194968 A194969 A194970

A049474 A194976 A194977 A194978

A194979 A194980 A194981 A194982

A194964 A194983 A194984 A194985

A194986 A194987 A194988 A194989

Count odd numbers up to n, then even numbers down from n. - Franklin T. Adams-Watters, Jan 21 2012

This sequence defines the square array A(n,k), n > 0 and k > 0, read by antidiagonals and the triangle T(n,k) = A(n+1-k,k) for 1 <= k <= n read by rows (see Formula and Example). - Werner Schulte, May 27 2018

A075680, which gives the values for odd n, isolates the essential behavior of this sequence. - T. D. Noe, Jun 01 2006

A033959 and A033958 give record values and where they occur. - Reinhard Zumkeller, Jan 08 2014

In the following, A stands for this sequence, A324400, and S -> T (where S and T are sequence A-numbers) indicates that for all i, j >= 1: S(i) = S(i) => T(i) = T(j).

For example, the following chains of implications hold:

A -> A286619 -> A005811,

and

A -> A003602 -> A286622 -> A000120,

-> A286378 -> A002487,

-> A323889,

-> A000593,

-> A001227,

among many others.

Let f(n) = A000265(n) be the odd part of n. Let p be the largest prime factor of k, and say k = p * m. Suppose that k is not a power of 2, i.e., p > 2, then f(k) = p * f(m). The iteration is k -> k + k/p = p*m + m = (p+1) * m. So, p * f(m) -> f(p+1) * f(m). Since for p > 2, f(p+1) < p, the odd part in each iteration decreases, until it becomes 1, i.e., until we reach a power of 2. - Amiram Eldar, Feb 19 2020

Any odd prime factor of k can be used at any step of the iteration, and the result will be same. Thus, like A329697, this is also fully additive sequence. - Antti Karttunen, Apr 29 2020

If and only if a(n) is equal to A005087(n), then sigma(2n) - sigma(n) is a power of 2. (See A336923, A046528). - Antti Karttunen, Mar 16 2021

a(n) is the length signature of a string plus its length.

The positive members of this sequence are exactly the numbers that can be expressed as the sum of two or more consecutive positive integers (cf. A138591). - David Wasserman, Jan 24 2002

Starting at 3, these are the positions of the data bits in the single-error-correcting Hamming code.

Except for the offset 0, sequence corresponds to numbers with at least an odd divisor > 1 (For largest odd divisor see A000265). - Lekraj Beedassy, Apr 12 2005

These are exactly the numbers n with the property that, given the n(n-1)/2 sums of pairs, the original numbers can be recovered uniquely. [Nick Reingold, see Winkler reference.]

Subsequence of A158581; A000120(a(n)) > 1. - Reinhard Zumkeller, Apr 16 2009

Range of A140977. - Reinhard Zumkeller, Aug 15 2010

A209229(a(n)) = 0. - Reinhard Zumkeller, Mar 07 2012

A001227(a(n)) > 1. - Reinhard Zumkeller, May 01 2012

Numbers that can be expressed as the sum of at least two consecutive integers; numbers that can be expressed as the difference of two nonconsecutive triangular numbers. - Charles R Greathouse IV, Jul 27 2012

Except for the 1st term 0, these are the integers k such that 2*(2*k-1) divides binomial(2*k-1,k). See Ihringer & Kupavskii. - Michel Marcus, Oct 02 2017

The terms 1 and 2 correspond to degenerate polygons.

These are also the numbers for which phi(n) is a power of 2: A209229(A000010(a(n))) = 1. - Olivier Gérard Feb 15 1999

From Stanislav Sykora, May 02 2016: (Start)

The sequence can be also defined as follows: (i) 1 is a member. (ii) Double of any member is also a member. (iii) If a member is not divisible by a Fermat prime F_k then its product with F_k is also a member. In particular, the powers of 2 (A000079) are a subset and so are the Fermat primes (A019434), which are the only odd prime members.

The definition is too restrictive (though correct): The Georg Mohr - Lorenzo Mascheroni theorem shows that constructibility using a straightedge and a compass is equivalent to using compass only. Moreover, Jean Victor Poncelet has shown that it is also equivalent to using straightedge and a fixed ('rusty') compass. With the work of Jakob Steiner, this became part of the Poncelet-Steiner theorem establishing the equivalence to using straightedge and a fixed circle (with a known center). A further extension by Francesco Severi replaced the availability of a circle with that of a fixed arc, no matter how small (but still with a known center).

Constructibility implies that when m is a member of this sequence, the edge length 2*sin(Pi/m) of an m-gon with circumradius 1 can be written as a finite expression involving only integer numbers, the four basic arithmetic operations, and the square root. (End)

If x,y are terms, and gcd(x,y) is a power of 2 then x*y is also a term. - David James Sycamore, Aug 24 2024

Number of ways to write n as a sum a_1 + ... + a_k where the a_i are positive integers and a_i = 2 * a_{i-1}, cf. A000929.

Number of divisors of n of the form 2^k - 1 (A000225) for k >= 1. - Jeffrey Shallit, Jan 23 2017

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Apart from missing 2, this sequence gives all numbers k such that the binary expansion of A156552(k) is a prefix of that of A156552(sigma(k)), that is, for k > 1, numbers k for which sigma(k) is a descendant of k in A005940-tree. This follows because of the two transitions x -> A005843(x) (doubling) and x -> A003961(x) (prime shift) used to generate descendants in A005940-tree, using A003961 at any step of the process will ruin the chances of encountering sigma(k) anywhere further down that subtree.

Proof: Any left child in A005940 (i.e., A003961(k) for k) is larger than sigma(k), for any k > 2 [see A286385 for a proof], and A003961(n) > n for all n > 1. Thus, apart from A003961(2) = 3 = sigma(2), A003961^t(k) > sigma(k), where A003961^t means t-fold application of prime shift, here with t >= 1. On the other hand, sigma(2n) > sigma(n) for all n, thus taking first some doubling steps before a run of one or more prime shift steps will not rescue us, as neither will taking further doubling steps after a bout of prime shifts.

The first terms of A325637 not included in this sequence are 154345556085770649600 and 9186050031556349952000, as they have abundancy index 6.

From Antti Karttunen, Nov 29 2021: (Start)

Odd terms of this sequence are given by the intersection of A349169 and A349174.

A064989 applied to the odd terms of this sequence gives the fixed points of A326042, i.e., the positions of zeros in A348736, and a subset of the positions of ones in A348941.

Odd terms of this sequence form a subsequence of A348943, but should occur neither in A348748 nor in A348749.

(End)