cp's OEIS Frontend

Number of edges of the complete bipartite graph of order 3n, K_{n,2n}. - Roberto E. Martinez II, Jan 07 2002

"If each period in the periodic system ends in a rare gas ..., the number of elements in a period can be found from the ordinal number n of the period by the formula: L = ((2n+3+(-1)^n)^2)/8..." - Nature, Jun 09 1951; Nature 411 (Jun 07 2001), p. 648. This produces the present sequence doubled up.

Let z(1) = i = sqrt(-1), z(k+1) = 1/(z(k)+2i); then a(n) = (-1)*Imag(z(n+1))/Real(z(n+1)). - Benoit Cloitre, Aug 06 2002

Maximum number of electrons in an atomic shell with total quantum number n. Partial sums of A016825. - Jeremy Gardiner, Dec 19 2004

Arithmetic mean of triangular numbers in pairs: (1+3)/2, (6+10)/2, (15+21)/2, ... . - Amarnath Murthy, Aug 05 2005

These numbers form a pattern on the Ulam spiral similar to that of the triangular numbers. - G. Roda, Oct 20 2010

Integral areas of isosceles right triangles with rational legs (legs are 2n and triangles are nondegenerate for n > 0). - Rick L. Shepherd, Sep 29 2009

Even squares divided by 2. - Omar E. Pol, Aug 18 2011

Number of stars when distributed as in the U.S.A. flag: n rows with n+1 stars and, between each pair of these, one row with n stars (i.e., n-1 of these), i.e., n*(n+1)+(n-1)*n = 2*n^2 = A001105(n). - César Eliud Lozada, Sep 17 2012

Apparently the number of Dyck paths with semilength n+3 and an odd number of peaks and the central peak having height n-3. - David Scambler, Apr 29 2013

Sum of the partition parts of 2n into exactly two parts. - Wesley Ivan Hurt, Jun 01 2013

Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c (A020882) and respective odd leg a (A180620); sequence gives values c-a, sorted with duplicates removed. - K. G. Stier, Nov 04 2013

Number of roots in the root systems of type B_n and C_n (for n > 1). - Tom Edgar, Nov 05 2013

Area of a square with diagonal 2n. - Wesley Ivan Hurt, Jun 18 2014

This sequence appears also as the first and second member of the quartet [a(n), a(n), p(n), p(n)] of the square of [n, n, n+1, n+1] in the Clifford algebra Cl_2 for n >= 0. p(n) = A046092(n). See an Oct 15 2014 comment on A147973 where also a reference is given. - Wolfdieter Lang, Oct 16 2014

a(n) are the only integers m where (A000005(m) + A000203(m)) = (number of divisors of m + sum of divisors of m) is an odd number. - Richard R. Forberg, Jan 09 2015

a(n) represents the first term in a sum of consecutive integers running to a(n+1)-1 that equals (2n+1)^3. - Patrick J. McNab, Dec 24 2016

Also the number of 3-cycles in the (n+4)-triangular honeycomb obtuse knight graph. - Eric W. Weisstein, Jul 29 2017

Also the Wiener index of the n-cocktail party graph for n > 1. - Eric W. Weisstein, Sep 07 2017

Numbers represented as the palindrome 242 in number base B including B=2 (binary), 3 (ternary) and 4: 242(2)=18, 242(3)=32, 242(4)=50, ... 242(9)=200, 242(10)=242, ... - Ron Knott, Nov 14 2017

a(n) is the square of the hypotenuse of an isosceles right triangle whose sides are equal to n. - Thomas M. Green, Aug 20 2019

The sequence contains all odd powers of 2 (A004171) but no even power of 2 (A000302). - Torlach Rush, Oct 10 2019

From Bernard Schott, Aug 31 2021 and Sep 16 2021: (Start)

Apart from 0, integers such that the number of even divisors (A183063) is odd.

Proof: every n = 2^q * (2k+1), q, k >= 0, then 2*n^2 = 2^(2q+1) * (2k+1)^2; now, gcd(2, 2k+1) = 1, tau(2^(2q+1)) = 2q+2 and tau((2k+1)^2) = 2u+1 because (2k+1)^2 is square, so, tau(2*n^2) = (2q+2) * (2u+1).

The 2q+2 divisors of 2^(2q+1) are {1, 2, 2^2, 2^3, ..., 2^(2q+1)}, so 2^(2q+1) has 2q+1 even divisors {2^1, 2^2, 2^3, ..., 2^(2q+1)}.

Conclusion: these 2q+1 even divisors create with the 2u+1 odd divisors of (2k+1)^2 exactly (2q+1)*(2u+1) even divisors of 2*n^2, and (2q+1)*(2u+1) is odd. (End)

a(n) with n>0 are the numbers with period length 2 for Bulgarian and Mancala solitaire. - Paul Weisenhorn, Jan 29 2022

Number of points at L1 distance = 2 from any given point in Z^n. - Shel Kaphan, Feb 25 2023

Integer that multiplies (h^2)/(m*L^2) to give the energy of a 1-D quantum mechanical particle in a box whenever it is an integer multiple of (h^2)/(m*L^2), where h = Planck's constant, m = mass of particle, and L = length of box. - A. Timothy Royappa, Mar 14 2025

Numbers n such that sum of divisors of n (A000203) is odd.

Also the numbers with an odd number of run sums (trapezoidal arrangements, number of ways of being written as the difference of two triangular numbers). - Ron Knott, Jan 27 2003

Pell(n)*Sum_{k|n} 1/Pell(k) is odd, where Pell(n) is A000129(n). - Paul Barry, Oct 12 2005

Number of odd divisors of n (A001227) is odd. - Vladeta Jovovic, Aug 28 2007

A071324(a(n)) is odd. - Reinhard Zumkeller, Jul 03 2008

Sigma(a(n)) = A000203(a(n)) = A152677(n). - Jaroslav Krizek, Oct 06 2009

Numbers n such that sum of odd divisors of n (A000593) is odd. - Omar E. Pol, Jul 05 2016

A187793(a(n)) is odd. - Timothy L. Tiffin, Jul 18 2016

If k is odd (k = 2m+1 for m >= 0), then 2^k = 2^(2m+1) = 2*(2^m)^2. If k is even (k = 2m for m >= 0), then 2^k = 2^(2m) = (2^m)^2. So, the powers of 2 sequence (A000079) is a subsequence of this one. - Timothy L. Tiffin, Jul 18 2016

Numbers n such that A175317(n) = Sum_{d|n} pod(d) is odd, where pod(m) = the product of divisors of m (A007955). - Jaroslav Krizek, Dec 28 2016

Positions of zeros in A292377 and A292383, positions of ones in A286357 and A292583. (See A292583 for why.) - Antti Karttunen, Sep 25 2017

Numbers of the form A000079(i)*A016754(j), i,j>=0. - R. J. Mathar, May 30 2020

Equivalently, numbers whose odd part is square. Cf. A042968. - Peter Munn, Jul 14 2020

These are the Heinz numbers of the partitions counted by A119620. - Gus Wiseman, Oct 29 2021

Numbers m whose abundance, A033880(m), is odd. - Peter Munn, May 23 2022

Numbers with an odd number of middle divisors (cf. A067742). - Omar E. Pol, Aug 02 2022

Also the rows of both triangles A237270 and A237593 interleaved.

Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).

The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.

The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.

Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.

Odd-indexed rows are also the rows of the irregular triangle A237270.

Even-indexed rows are also the rows of the triangle A237593.

Lengths of the odd-indexed rows are in A237271.

Lengths of the even-indexed rows give 2*A003056.

Row sums of the odd-indexed rows gives A000203, the sum of divisors function.

Row sums of the even-indexed rows give the positive even numbers (see A005843).

Row sums give A245092.

From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.

The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

The 229026 Zumkeller numbers less than 10^6 have a maximum difference of 12. This leads to the conjecture that any 12 consecutive numbers include at least one Zumkeller number. There are 1989 odd Zumkeller numbers less than 10^6; they are exactly the odd abundant numbers that have even abundance, A174865. - T. D. Noe, Mar 31 2010

For k >= 0, numbers of the form 18k + 6 and 18k + 12 are terms (see Remark 2.3. in Somu et al., 2023). Corollary: The maximum difference between any two consecutive terms is at most 12. - Ivan N. Ianakiev, Jan 02 2024

All 205283 odd abundant numbers less than 10^8 that have even abundance (see A174865) are Zumkeller numbers. - T. D. Noe, Nov 14 2010

Except for 1 and 2, all primorials (A002110) are Zumkeller numbers (follows from Fact 6 in the Rao/Peng paper). - Ivan N. Ianakiev, Mar 23 2016

Supersequence of A111592 (follows from Fact 3 in the Rao/Peng paper). - Ivan N. Ianakiev, Mar 20 2017

Conjecture: Any 4 consecutive terms include at least one number k such that sigma(k)/2 is also a Zumkeller number (verified for the first 10^5 Zumkeller numbers). - Ivan N. Ianakiev, Apr 03 2017

LeVan studied these numbers using the equivalent definition of numbers n such that n = Sum_{d|n, dA180332) "minimal integer-perfect numbers". - Amiram Eldar, Dec 20 2018

The numbers 3 * 2^k for k > 0 are all Zumkeller numbers: half of one such partition is {3*2^k, 3*2^(k-2), ...}, replacing 3 with 2 if it appears. With this and the lemma that the product of a Zumkeller number and a number coprime to it is again a Zumkeller number (see A179527), we have that all numbers divisible by 6 but not 9 (or numbers congruent to 6 or 12 modulo 18) are Zumkeller numbers, proving that the difference between consecutive Zumkeller numbers is at most 12. - Charlie Neder, Jan 15 2019

Improvements on the previous comment: 1) For every integer q > 0, every odd integer r > 0 and every integer s > 0 relatively prime to 6, the integer 2^q*3^r*s is a Zumkeller number, and therefore 2) there exist Zumkeller numbers divisible by 9 (such as 54, 90, 108, 126, etc.). - Ivan N. Ianakiev, Jan 16 2020

Conjecture: If d > 1, d|k and tau(d)*sigma(d) = k, then k is a Zumkeller number (cf. A331668). - Ivan N. Ianakiev, Apr 24 2020

This sequence contains A378541, the intersection of the practical numbers (A005153) with numbers with even sum of divisors (A028983). - David A. Corneth, Nov 03 2024

Sequence gives the positions of even terms in A119347, and correspondingly, of odd terms in A308605. - Antti Karttunen, Nov 29 2024

If s = sigma(m) is odd and p > s then m*p is not in the sequence. - David A. Corneth, Dec 07 2024

This is a multiplicative self-inverse permutation of the integers.

A225547 gives the fixed points.

From Antti Karttunen and Peter Munn, Feb 02 2020: (Start)

This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, in A329050, this sequence reflects these factors about the main diagonal of the array, substituting A329050[j,i] for A329050[i,j], and this results in many relationships including significant homomorphisms.

This sequence provides a relationship between the operations of squaring and prime shift (A003961) because each successive column of the A329050 array is the square of the previous column, and each successive row is the prime shift of the previous row.

A329050 gives examples of how significant sets of numbers can be formed by choosing their factors in relation to rows and/or columns. This sequence therefore maps equivalent derived sets by exchanging rows and columns. Thus odd numbers are exchanged for squares, squarefree numbers for powers of 2 etc.

Alternative construction: For n > 1, form a vector v of length A299090(n), where each element v[i] for i=1..A299090(n) is a product of those distinct prime factors p(i) of n whose exponent e(i) has the bit (i-1) "on", or 1 (as an empty product) if no such exponents are present. a(n) is then Product_{i=1..A299090(n)} A000040(i)^A048675(v[i]). Note that because each element of vector v is squarefree, it means that each exponent A048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary string A087207(n).

This permutation effects the following mappings:

A000035(a(n)) = A010052(n), A010052(a(n)) = A000035(n). [Odd numbers <-> Squares]

A008966(a(n)) = A209229(n), A209229(a(n)) = A008966(n). [Squarefree numbers <-> Powers of 2]

(End)

From Antti Karttunen, Jul 08 2020: (Start)

Moreover, we see also that this sequence maps between A016825 (Numbers of the form 4k+2) and A001105 (2*squares) as well as between A008586 (Multiples of 4) and A028983 (Numbers with even sum of the divisors).

(End)

Comment from N. J. A. Sloane, Jan 03 2021: (Start)

Theorem 1: (i) a(n) = (number of odd divisors of n <= sqrt(2*n)) - (number of odd divisors of n > sqrt(2*n)).

(ii) Let r(n) = A003056(n). Then a(n) = (number of odd divisors of n <= r(n)) - (number of odd divisors of n > r(n)).

(iii) a(n) = Sum_{k=1..r(n)} (-1)^(k+1)*A237048(n,k).

(iv) a(n) is odd iff n is a square or twice a square (cf. A053866). Indices of odd terms give A028982. Indices of even terms give A028983.

The proofs are straightforward. These results were conjectured by Omar E. Pol in 2017. (End)

Theorem 2: a(n) is equal to the difference between the number of partitions of n into an odd number of consecutive parts and the number of partitions of n into an even number of consecutive parts. [Chapman et al., 2001; Hirschhorn and Hirschhorn, 2005]. - Omar E. Pol, Feb 06 2017

From Omar E. Pol, Feb 06 2017: (Start)

Conjecture 1: This is the central column of the isosceles triangle of A249351.

Conjecture 2: a(n) is also the width of the terrace at the n-th level in the main diagonal of the pyramid described in A245092.

Conjecture 3: a(n) is also the number of central subparts of the symmetric representation of sigma(n). For more information see A279387.

Conjectures 2 and 3 were proposed after Michel Marcus's conjecture in A237593. (End)

Conjectures 1, 2, and 3 are all true. - N. J. A. Sloane, Feb 11 2021

The bisection {1,1,2,3,5,7,11,15,22,...} agrees with the initial terms of A008641, Number of partitions of n into at most 12 parts and also A008635, Molien series for A_12.

a(2n+1)=a(2n) for all n>0. If the partition {...,1} is a member of a(2n) then the partition {...,1,1} is a member of a(2n+1). - Robert G. Wilson v, Jun 09 2006

Number of partitions of n where all parts (except for possibly the first part) are even; see example. - Joerg Arndt, Apr 22 2013

For n >= 2, a(n) = number of partitions p of n such that floor(n/2) is a part of p. For n >= 1, a(n) = number of partitions p of n such that ceiling(n/2) is a part of p. - Clark Kimberling, Feb 28 2014

From Gus Wiseman, Oct 28 2021: (Start)

If we insert zeros every three terms, this counts partitions of n such that n = floor(3*k/2), where k is the number of parts. This counts by sum rather than length. These partitions are ranked by A347452.

Also the number of integer partitions of n with alternating product 1, where the alternating product of a sequence (y_1,...,y_k) is Product_i y_i^((-1)^(i-1)). These are the conjugates of the partitions (ranked by A336119) described in Arndt's comment above. For example, the a(2) = 1 through a(10) = 7 partitions are:

11 111 22 221 33 331 44 441 55

1111 11111 2211 22111 2222 22221 3322

111111 1111111 3311 33111 4411

221111 2211111 222211

11111111 111111111 331111

22111111

1111111111

These partitions are ranked by A028982. The odd-length case is A035363 (shifted), which is also the version for sum instead of product. The multiplicative version (factorizations) is A347438.

(End)

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

Also the number of factorizations of n with alternating sum >= 0.

Alternating row sums of A056538. - Omar E. Pol, Feb 17 2024

Does a constant analogous to the median abundancy index (see A353617) exist for this function? Particularly, does a constant exist such that the numbers having the value a(n)/n greater than this constant have natural density exactly 1/2? Using the first 10^7 values, one observes that if it exists, this constant appears to converge around 0.726. - Shreyansh Jaiswal, Apr 16 2025