cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A262025 a(n) = (A262024(n)-1)/2: a(n)*(a(n) + 1) = d(n)*Y(n)^2 with d(n) = A007969 and Y(n) = A261250(n).

Original entry on oeis.org

1, 4, 2, 9, 3, 324, 7, 16, 8, 4, 27, 98, 25, 63, 4900, 5, 11, 17, 36, 18, 12, 1024, 6, 99, 80, 12167, 49, 324, 33124, 242, 44, 7, 75, 9801, 15, 883159524, 31, 64, 32, 16, 3887, 125, 8, 1140624, 1849, 28899, 175, 26, 81, 27, 142884, 5202, 250000, 9, 575, 6075, 1071647, 19, 31404816, 49, 100, 50, 20, 16040025, 675, 79035335993124, 10, 147, 63, 602176, 512, 4900, 324, 153458
Offset: 1

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Author

Wolfdieter Lang, Sep 19 2015

Keywords

Comments

The positive fundamental solutions (x0(n), y0(n)) of the Pell equation x^2 - d(n) y^2 = +1, with d not a square, have only even y solutions for d(n) = A007969 (Conway's products of 1-happy couples). The proof is now given in the W. Lang link under A007969. The solutions x0 and y0 = 2*Y0 are given in A262024 and 2*A261250, respectively. The numbers X0(n) = (x0(n) - 1)/2 = a(n) satisfy a(n)*(a(n) + 1) = d(n)*Y0(n)^2. See the mentioned link.

Crossrefs

Cf. A007969, A007970, A262024, A261250.

A002378 Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).

Original entry on oeis.org

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

4*a(n) + 1 are the odd squares A016754(n).
The word "pronic" (used by Dickson) is incorrect. - Michael Somos
According to the 2nd edition of Webster, the correct word is "promic". - R. K. Guy
a(n) is the number of minimal vectors in the root lattice A_n (see Conway and Sloane, p. 109).
Let M_n denote the n X n matrix M_n(i, j) = (i + j); then the characteristic polynomial of M_n is x^(n-2) * (x^2 - a(n)*x - A002415(n)). - Benoit Cloitre, Nov 09 2002
The greatest LCM of all pairs (j, k) for j < k <= n for n > 1. - Robert G. Wilson v, Jun 19 2004
First differences are a(n+1) - a(n) = 2*n + 2 = 2, 4, 6, ... (while first differences of the squares are (n+1)^2 - n^2 = 2*n + 1 = 1, 3, 5, ...). - Alexandre Wajnberg, Dec 29 2005
25 appended to these numbers corresponds to squares of numbers ending in 5 (i.e., to squares of A017329). - Lekraj Beedassy, Mar 24 2006
A rapid (mental) multiplication/factorization technique -- a generalization of Lekraj Beedassy's comment: For all bases b >= 2 and positive integers n, c, d, k with c + d = b^k, we have (n*b^k + c)*(n*b^k + d) = a(n)*b^(2*k) + c*d. Thus the last 2*k base-b digits of the product are exactly those of c*d -- including leading 0(s) as necessary -- with the preceding base-b digit(s) the same as a(n)'s. Examples: In decimal, 113*117 = 13221 (as n = 11, b = 10 = 3 + 7, k = 1, 3*7 = 21, and a(11) = 132); in octal, 61*67 = 5207 (52 is a(6) in octal). In particular, for even b = 2*m (m > 0) and c = d = m, such a product is a square of this type. Decimal factoring: 5609 is immediately seen to be 71*79. Likewise, 120099 = 301*399 (k = 2 here) and 99990000001996 = 9999002*9999998 (k = 3). - Rick L. Shepherd, Jul 24 2021
Number of circular binary words of length n + 1 having exactly one occurrence of 01. Example: a(2) = 6 because we have 001, 010, 011, 100, 101 and 110. Column 1 of A119462. - Emeric Deutsch, May 21 2006
The sequence of iterated square roots sqrt(N + sqrt(N + ...)) has for N = 1, 2, ... the limit (1 + sqrt(1 + 4*N))/2. For N = a(n) this limit is n + 1, n = 1, 2, .... For all other numbers N, N >= 1, this limit is not a natural number. Examples: n = 1, a(1) = 2: sqrt(2 + sqrt(2 + ...)) = 1 + 1 = 2; n = 2, a(2) = 6: sqrt(6 + sqrt(6 + ...)) = 1 + 2 = 3. - Wolfdieter Lang, May 05 2006
Nonsquare integers m divisible by ceiling(sqrt(m)), except for m = 0. - Max Alekseyev, Nov 27 2006
The number of off-diagonal elements of an (n + 1) X (n + 1) matrix. - Artur Jasinski, Jan 11 2007
a(n) is equal to the number of functions f:{1, 2} -> {1, 2, ..., n + 1} such that for a fixed x in {1, 2} and a fixed y in {1, 2, ..., n + 1} we have f(x) <> y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007
Numbers m >= 0 such that round(sqrt(m+1)) - round(sqrt(m)) = 1. - Hieronymus Fischer, Aug 06 2007
Numbers m >= 0 such that ceiling(2*sqrt(m+1)) - 1 = 1 + floor(2*sqrt(m)). - Hieronymus Fischer, Aug 06 2007
Numbers m >= 0 such that fract(sqrt(m+1)) > 1/2 and fract(sqrt(m)) < 1/2 where fract(x) is the fractional part (fract(x) = x - floor(x), x >= 0). - Hieronymus Fischer, Aug 06 2007
X values of solutions to the equation 4*X^3 + X^2 = Y^2. To find Y values: b(n) = n(n+1)(2n+1). - Mohamed Bouhamida, Nov 06 2007
Nonvanishing diagonal of A132792, the infinitesimal Lah matrix, so "generalized factorials" composed of a(n) are given by the elements of the Lah matrix, unsigned A111596, e.g., a(1)*a(2)*a(3) / 3! = -A111596(4,1) = 24. - Tom Copeland, Nov 20 2007
If Y is a 2-subset of an n-set X then, for n >= 2, a(n-2) is the number of 2-subsets and 3-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
a(n) coincides with the vertex of a parabola of even width in the Redheffer matrix, directed toward zero. An integer p is prime if and only if for all integer k, the parabola y = kx - x^2 has no integer solution with 1 < x < k when y = p; a(n) corresponds to odd k. - Reikku Kulon, Nov 30 2008
The third differences of certain values of the hypergeometric function 3F2 lead to the squares of the oblong numbers i.e., 3F2([1, n + 1, n + 1], [n + 2, n + 2], z = 1) - 3*3F2([1, n + 2, n + 2], [n + 3, n + 3], z = 1) + 3*3F2([1, n + 3, n + 3], [n + 4, n + 4], z = 1) - 3F2([1, n + 4, n + 4], [n + 5, n + 5], z = 1) = (1/((n+2)*(n+3)))^2 for n = -1, 0, 1, 2, ... . See also A162990. - Johannes W. Meijer, Jul 21 2009
Generalized factorials, [a.(n!)] = a(n)*a(n-1)*...*a(0) = A010790(n), with a(0) = 1 are related to A001263. - Tom Copeland, Sep 21 2011
For n > 1, a(n) is the number of functions f:{1, 2} -> {1, ..., n + 2} where f(1) > 1 and f(2) > 2. Note that there are n + 1 possible values for f(1) and n possible values for f(2). For example, a(3) = 12 since there are 12 functions f from {1, 2} to {1, 2, 3, 4, 5} with f(1) > 1 and f(2) > 2. - Dennis P. Walsh, Dec 24 2011
a(n) gives the number of (n + 1) X (n + 1) symmetric (0, 1)-matrices containing two ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012
a(n) is the number of positions of a domino in a rectangled triangular board with both legs equal to n + 1. - César Eliud Lozada, Sep 26 2012
a(n) is the number of ordered pairs (x, y) in [n+2] X [n+2] with |x-y| > 1. - Dennis P. Walsh, Nov 27 2012
a(n) is the number of injective functions from {1, 2} into {1, 2, ..., n + 1}. - Dennis P. Walsh, Nov 27 2012
a(n) is the sum of the positive differences of the partition parts of 2n + 2 into exactly two parts (see example). - Wesley Ivan Hurt, Jun 02 2013
a(n)/a(n-1) is asymptotic to e^(2/n). - Richard R. Forberg, Jun 22 2013
Number of positive roots in the root system of type D_{n + 1} (for n > 2). - Tom Edgar, Nov 05 2013
Number of roots in the root system of type A_n (for n > 0). - Tom Edgar, Nov 05 2013
From Felix P. Muga II, Mar 18 2014: (Start)
a(m), for m >= 1, are the only positive integer values t for which the Binet-de Moivre formula for the recurrence b(n) = b(n-1) + t*b(n-2) with b(0) = 0 and b(1) = 1 has a root of a square. PROOF (as suggested by Wolfdieter Lang, Mar 26 2014): The sqrt(1 + 4t) appearing in the zeros r1 and r2 of the characteristic equation is (a positive) integer for positive integer t precisely if 4t + 1 = (2m + 1)^2, that is t = a(m), m >= 1. Thus, the characteristic roots are integers: r1 = m + 1 and r2 = -m.
Let m > 1 be an integer. If b(n) = b(n-1) + a(m)*b(n-2), n >= 2, b(0) = 0, b(1) = 1, then lim_{n->oo} b(n+1)/b(n) = m + 1. (End)
Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices (chromatic polynomial) of the complete graphs (here simply K_2). - Tom Copeland, Apr 05 2014
The set of integers k for which k + sqrt(k + sqrt(k + sqrt(k + sqrt(k + ...) ... is an integer. - Leslie Koller, Apr 11 2014
a(n-1) is the largest number k such that (n*k)/(n+k) is an integer. - Derek Orr, May 22 2014
Number of ways to place a domino and a singleton on a strip of length n - 2. - Ralf Stephan, Jun 09 2014
With offset 1, this appears to give the maximal number of crossings between n nonconcentric circles of equal radius. - Felix Fröhlich, Jul 14 2014
For n > 1, the harmonic mean of the n values a(1) to a(n) is n + 1. The lowest infinite sequence of increasing positive integers whose cumulative harmonic mean is integral. - Ian Duff, Feb 01 2015
a(n) is the maximum number of queens of one color that can coexist without attacking one queen of the opponent's color on an (n+2) X (n+2) chessboard. The lone queen can be placed in any position on the perimeter of the board. - Bob Selcoe, Feb 07 2015
With a(0) = 1, a(n-1) is the smallest positive number not in the sequence such that Sum_{i = 1..n} 1/a(i-1) has a denominator equal to n. - Derek Orr, Jun 17 2015
The positive members of this sequence are a proper subsequence of the so-called 1-happy couple products A007969. See the W. Lang link there, eq. (4), with Y_0 = 1, with a table at the end. - Wolfdieter Lang, Sep 19 2015
For n > 0, a(n) is the reciprocal of the area bounded above by y = x^(n-1) and below by y = x^n for x in the interval [0, 1]. Summing all such areas visually demonstrates the formula below giving Sum_{n >= 1} 1/a(n) = 1. - Rick L. Shepherd, Oct 26 2015
It appears that, except for a(0) = 0, this is the set of positive integers n such that x*floor(x) = n has no solution. (For example, to get 3, take x = -3/2.) - Melvin Peralta, Apr 14 2016
If two independent real random variables, x and y, are distributed according to the same exponential distribution: pdf(x) = lambda * exp(-lambda * x), lambda > 0, then the probability that n - 1 <= x/y < n is given by 1/a(n). - Andres Cicuttin, Dec 03 2016
a(n) is equal to the sum of all possible differences between n different pairs of consecutive odd numbers (see example). - Miquel Cerda, Dec 04 2016
a(n+1) is the dimension of the space of vector fields in the plane with polynomial coefficients up to order n. - Martin Licht, Dec 04 2016
It appears that a(n) + 3 is the area of the largest possible pond in a square (A268311). - Craig Knecht, May 04 2017
Also the number of 3-cycles in the (n+3)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
Also the Wiener index of the (n+2)-wheel graph. - Eric W. Weisstein, Sep 08 2017
The left edge of a Floyd's triangle that consists of even numbers: 0; 2, 4; 6, 8, 10; 12, 14, 16, 18; 20, 22, 24, 26, 28; ... giving 0, 2, 6, 12, 20, ... The right edge generates A028552. - Waldemar Puszkarz, Feb 02 2018
a(n+1) is the order of rowmotion on a poset obtained by adjoining a unique minimal (or maximal) element to a disjoint union of at least two chains of n elements. - Nick Mayers, Jun 01 2018
From Juhani Heino, Feb 05 2019: (Start)
For n > 0, 1/a(n) = n/(n+1) - (n-1)/n.
For example, 1/6 = 2/3 - 1/2; 1/12 = 3/4 - 2/3.
Corollary of this:
Take 1/2 pill.
Next day, take 1/6 pill. 1/2 + 1/6 = 2/3, so your daily average is 1/3.
Next day, take 1/12 pill. 2/3 + 1/12 = 3/4, so your daily average is 1/4.
And so on. (End)
From Bernard Schott, May 22 2020: (Start)
For an oblong number m >= 6 there exists a Euclidean division m = d*q + r with q < r < d which are in geometric progression, in this order, with a common integer ratio b. For b >= 2 and q >= 1, the Euclidean division is m = qb*(qb+1) = qb^2 * q + qb where (q, qb, qb^2) are in geometric progression.
Some examples with distinct ratios and quotients:
6 | 4 30 | 25 42 | 18
----- ----- -----
2 | 1 , 5 | 1 , 6 | 2 ,
and also:
42 | 12 420 | 100
----- -----
6 | 3 , 20 | 4 .
Some oblong numbers also satisfy a Euclidean division m = d*q + r with q < r < d that are in geometric progression in this order but with a common noninteger ratio b > 1 (see A335064). (End)
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {2, 2n}]. For n=1, this collapses to [1; {2}]. - Magus K. Chu, Sep 09 2022
a(n-2) is the maximum irregularity over all trees with n vertices. The extremal graphs are stars. (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023
For n > 0, number of diagonals in a regular 2*(n+1)-gon that are not parallel to any edge (cf. A367204). - Paolo Xausa, Mar 30 2024
a(n-1) is the maximum Zagreb index over all trees with n vertices. The extremal graphs are stars. (The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.) - Allan Bickle, Apr 11 2024
For n >= 1, a(n) is the determinant of the distance matrix of a cycle graph on 2*n + 1 vertices (if the length of the cycle is even such a determinant is zero). - Miquel A. Fiol, Aug 20 2024
For n > 1, the continued fraction expansion of sqrt(16*a(n)) is [2n+1; {1, 2n-1, 1, 8n+2}]. - Magus K. Chu, Nov 20 2024
For n>=2, a(n) is the number of faces on a n+1-zone rhombic zonohedron. Each pair of a collection of great circles on a sphere intersects at two points, so there are 2*binomial(n+1,2) intersections. The dual of the implied polyhedron is a rhombic zonohedron, its faces corresponding to the intersections. - Shel Kaphan, Aug 12 2025

Examples

			a(3) = 12, since 2(3)+2 = 8 has 4 partitions with exactly two parts: (7,1), (6,2), (5,3), (4,4). Taking the positive differences of the parts in each partition and adding, we get: 6 + 4 + 2 + 0 = 12. - _Wesley Ivan Hurt_, Jun 02 2013
G.f. = 2*x + 6*x^2 + 12*x^3 + 20*x^4 + 30*x^5 + 42*x^6 + 56*x^7 + ... - _Michael Somos_, May 22 2014
From _Miquel Cerda_, Dec 04 2016: (Start)
a(1) = 2, since 45-43 = 2;
a(2) = 6, since 47-45 = 2 and 47-43 = 4, then 2+4 = 6;
a(3) = 12, since 49-47 = 2, 49-45 = 4, and 49-43 = 6, then 2+4+6 = 12. (End)

References

  • W. W. Berman and D. E. Smith, A Brief History of Mathematics, 1910, Open Court, page 67.
  • J. H. Conway and R. K. Guy, The Book of Numbers, 1996, p. 34.
  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
  • L. E. Dickson, History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 357, 1952.
  • L. E. Dickson, History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 6, 232-233, 350 and 407, 1952.
  • H. Eves, An Introduction to the History of Mathematics, revised, Holt, Rinehart and Winston, 1964, page 72.
  • Nicomachus of Gerasa, Introduction to Arithmetic, translation by Martin Luther D'Ooge, Ann Arbor, University of Michigan Press, 1938, p. 254.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 291.
  • Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), pp. 980-981.
  • C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, pp. 61-62.
  • Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 54-55.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • F. J. Swetz, From Five Fingers to Infinity, Open Court, 1994, p. 219.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 2-6.

Links

Crossrefs

Partial sums of A005843 (even numbers). Twice triangular numbers (A000217).
1/beta(n, 2) in A061928.
Cf. A035106, A087811, A119462, A127235, A049598, A124080, A033996, A028896, A046092, A000217, A005563, A046092, A001082, A059300, A059297, A059298, A166373, A002943 (bisection), A002939 (bisection), A078358 (complement).
A036689 and A036690 are subsequences. Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488. - Bruno Berselli, Jun 10 2013
Row n=2 of A185651.
Cf. A007745, A169810, A213541, A005369 (characteristic function).
Cf. A281026. - Bruno Berselli, Jan 16 2017
Cf. A045943 (4-cycles in triangular honeycomb acute knight graph), A028896 (5-cycles), A152773 (6-cycles).
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
A335064 is a subsequence.
Second column of A003506.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).
Cf. A347213 (Dgf at s=4).
Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
Cf. A016742, A049450.

Programs

Formula

G.f.: 2*x/(1-x)^3. - Simon Plouffe in his 1992 dissertation.
a(n) = a(n-1) + 2*n, a(0) = 0.
Sum_{n >= 1} a(n) = n*(n+1)*(n+2)/3 (cf. A007290, partial sums).
Sum_{n >= 1} 1/a(n) = 1. (Cf. Tijdeman)
Sum_{n >= 1} (-1)^(n+1)/a(n) = log(4) - 1 = A016627 - 1 [Jolley eq (235)].
1 = 1/2 + Sum_{n >= 1} 1/(2*a(n)) = 1/2 + 1/4 + 1/12 + 1/24 + 1/40 + 1/60 + ... with partial sums: 1/2, 3/4, 5/6, 7/8, 9/10, 11/12, 13/14, ... - Gary W. Adamson, Jun 16 2003
a(n)*a(n+1) = a(n*(n+2)); e.g., a(3)*a(4) = 12*20 = 240 = a(3*5). - Charlie Marion, Dec 29 2003
Sum_{k = 1..n} 1/a(k) = n/(n+1). - Robert G. Wilson v, Feb 04 2005
a(n) = A046092(n)/2. - Zerinvary Lajos, Jan 08 2006
Log 2 = Sum_{n >= 0} 1/a(2n+1) = 1/2 + 1/12 + 1/30 + 1/56 + 1/90 + ... = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + ... = Sum_{n >= 0} (-1)^n/(n+1) = A002162. - Gary W. Adamson, Jun 22 2003
a(n) = A110660(2*n). - N. J. A. Sloane, Sep 21 2005
a(n-1) = n^2 - n = A000290(n) - A000027(n) for n >= 1. a(n) is the inverse (frequency distribution) sequence of A000194(n). - Mohammad K. Azarian, Jul 26 2007
(2, 6, 12, 20, 30, ...) = binomial transform of (2, 4, 2). - Gary W. Adamson, Nov 28 2007
a(n) = 2*Sum_{i=0..n} i = 2*A000217(n). - Artur Jasinski, Jan 09 2007, and Omar E. Pol, May 14 2008
a(n) = A006503(n) - A000292(n). - Reinhard Zumkeller, Sep 24 2008
a(n) = A061037(4*n) = (n+1/2)^2 - 1/4 = ((2n+1)^2 - 1)/4 = (A005408(n)^2 - 1)/4. - Paul Curtz, Oct 03 2008 and Klaus Purath, Jan 13 2022
a(0) = 0, a(n) = a(n-1) + 1 + floor(x), where x is the minimal positive solution to fract(sqrt(a(n-1) + 1 + x)) = 1/2. - Hieronymus Fischer, Dec 31 2008
E.g.f.: (x+2)*x*exp(x). - Geoffrey Critzer, Feb 06 2009
Product_{i >= 2} (1-1/a(i)) = -2*sin(Pi*A001622)/Pi = -2*sin(A094886)/A000796 = 2*A146481. - R. J. Mathar, Mar 12 2009, Mar 15 2009
E.g.f.: ((-x+1)*log(-x+1)+x)/x^2 also Integral_{x = 0..1} ((-x+1)*log(-x+1) + x)/x^2 = zeta(2) - 1. - Stephen Crowley, Jul 11 2009
a(A007018(n)) = A007018(n+1), i.e., A007018(n+1) = A007018(n)-th oblong numbers. - Jaroslav Krizek, Sep 13 2009
a(n) = floor((n + 1/2)^2). a(n) = A035608(n) + A004526(n+1). - Reinhard Zumkeller, Jan 27 2010
a(n) = 2*(2*A006578(n) - A035608(n)). - Reinhard Zumkeller, Feb 07 2010
a(n-1) = floor(n^5/(n^3 + n^2 + 1)). - Gary Detlefs, Feb 11 2010
For n > 1: a(n) = A173333(n+1, n-1). - Reinhard Zumkeller, Feb 19 2010
a(n) = A004202(A000217(n)). - Reinhard Zumkeller, Feb 12 2011
a(n) = A188652(2*n+1) + 1. - Reinhard Zumkeller, Apr 13 2011
For n > 0 a(n) = 1/(Integral_{x=0..Pi/2} 2*(sin(x))^(2*n-1)*(cos(x))^3). - Francesco Daddi, Aug 02 2011
a(n) = A002061(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(0) = 0, a(n) = A005408(A034856(n)) - A005408(n-1). - Ivan N. Ianakiev, Dec 06 2012
a(n) = A005408(A000096(n)) - A005408(n). - Ivan N. Ianakiev, Dec 07 2012
a(n) = A001318(n) + A085787(n). - Omar E. Pol, Jan 11 2013
Sum_{n >= 1} 1/(a(n))^(2s) = Sum_{t = 1..2*s} binomial(4*s - t - 1, 2*s - 1) * ( (1 + (-1)^t)*zeta(t) - 1). See Arxiv:1301.6293. - R. J. Mathar, Feb 03 2013
a(n)^2 + a(n+1)^2 = 2 * a((n+1)^2), for n > 0. - Ivan N. Ianakiev, Apr 08 2013
a(n) = floor(n^2 * e^(1/n)) and a(n-1) = floor(n^2 / e^(1/n)). - Richard R. Forberg, Jun 22 2013
a(n) = 2*C(n+1, 2), for n >= 0. - Felix P. Muga II, Mar 11 2014
A005369(a(n)) = 1. - Reinhard Zumkeller, Jul 05 2014
Binomial transform of [0, 2, 2, 0, 0, 0, ...]. - Alois P. Heinz, Mar 10 2015
a(2n) = A002943(n) for n >= 0, a(2n-1) = A002939(n) for n >= 1. - M. F. Hasler, Oct 11 2015
For n > 0, a(n) = 1/(Integral_{x=0..1} (x^(n-1) - x^n) dx). - Rick L. Shepherd, Oct 26 2015
a(n) = A005902(n) - A007588(n). - Peter M. Chema, Jan 09 2016
For n > 0, a(n) = lim_{m -> oo} (1/m)*1/(Sum_{i=m*n..m*(n+1)} 1/i^2), with error of ~1/m. - Richard R. Forberg, Jul 27 2016
From Ilya Gutkovskiy, Jul 28 2016: (Start)
Dirichlet g.f.: zeta(s-2) + zeta(s-1).
Convolution of nonnegative integers (A001477) and constant sequence (A007395).
Sum_{n >= 0} a(n)/n! = 3*exp(1). (End)
From Charlie Marion, Mar 06 2020: (Start)
a(n)*a(n+2k-1) + (n+k)^2 = ((2n+1)*k + n^2)^2.
a(n)*a(n+2k) + k^2 = ((2n+1)*k + a(n))^2. (End)
Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)/Pi. - Amiram Eldar, Jan 20 2021
A generalization of the Dec 29 2003 formula, a(n)*a(n+1) = a(n*(n+2)), follows. a(n)*a(n+k) = a(n*(n+k+1)) + (k-1)*n*(n+k+1). - Charlie Marion, Jan 02 2023
a(n) = A016742(n) - A049450(n). - Leo Tavares, Mar 15 2025

Extensions

Additional comments from Michael Somos
Comment and cross-reference added by Christopher Hunt Gribble, Oct 13 2009

A002144 Pythagorean primes: primes of the form 4*k + 1.

Original entry on oeis.org

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Rational primes that decompose in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017
These are the prime terms of A009003.
-1 is a quadratic residue mod a prime p if and only if p is in this sequence.
Sin(a(n)*Pi/2) = 1 with Pi = 3.1415..., see A070750. - Reinhard Zumkeller, May 04 2002
If at least one of the odd primes p, q belongs to the sequence, then either both or neither of the congruences x^2 = p (mod q), x^2 = q (mod p) are solvable, according to Gauss reciprocity law. - Lekraj Beedassy, Jul 17 2003
Odd primes such that binomial(p-1, (p-1)/2) == 1 (mod p). - Benoit Cloitre, Feb 07 2004
Primes that are the hypotenuse of a right triangle with integer sides. The Pythagorean triple is {A002365(n), A002366(n), a(n)}.
Also, primes of the form a^k + b^k, k > 1. - Amarnath Murthy, Nov 17 2003
The square of a(n) is the average of two other squares. This fact gives rise to a class of monic polynomials x^2 + bx + c with b = a(n) that will factor over the integers regardless of the sign of c. See A114200. - Owen Mertens (owenmertens(AT)missouristate.edu), Nov 16 2005
Also such primes p that the last digit is always 1 for the Nexus numbers of form n^p - (n-1)^p. - Alexander Adamchuk, Aug 10 2006
The set of Pythagorean primes is a proper subset of the set of positive fundamental discriminants (A003658). - Paul Muljadi, Mar 28 2008
A079260(a(n)) = 1; complement of A137409. - Reinhard Zumkeller, Oct 11 2008
From Artur Jasinski, Dec 10 2008: (Start)
If we take 4 numbers: 1, A002314(n), A152676(n), A152680(n) then multiplication table modulo a(n) is isomorphic to the Latin square:
1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1
and isomorphic to the multiplication table of {1, i, -i, -1} where i is sqrt(-1), A152680(n) is isomorphic to -1, A002314(n) with i or -i and A152676(n) vice versa -i or i. 1, A002314(n), A152676(n), A152680(n) are subfield of Galois field [a(n)]. (End)
Primes p such that the arithmetic mean of divisors of p^3 is an integer. There are 2 sequences of such primes: this one and A002145. - Ctibor O. Zizka, Oct 20 2009
Equivalently, the primes p for which the smallest extension of F_p containing the square roots of unity (necessarily F_p) contains the 4th roots of unity. In this respect, the n = 2 case of a family of sequences: see n=3 (A129805) and n=5 (A172469). - Katherine E. Stange, Feb 03 2010
Subsequence of A007969. - Reinhard Zumkeller, Jun 18 2011
A151763(a(n)) = 1.
k^k - 1 is divisible by 4*k + 1 if 4*k + 1 is a prime (see Dickson reference). - Gary Detlefs, May 22 2013
Not only are the squares of these primes the sum of two nonzero squares, but the primes themselves are also. 2 is the only prime equal to the sum of two nonzero squares and whose square is not. 2 is therefore not a Pythagorean prime. - Jean-Christophe Hervé, Nov 10 2013
The statement that these primes are the sum of two nonzero squares follows from Fermat's theorem on the sum of two squares. - Jerzy R Borysowicz, Jan 02 2019
The decompositions of the prime and its square into two nonzero squares are unique. - Jean-Christophe Hervé, Nov 11 2013. See the Dickson reference, Vol. II, (B) on p. 227. - Wolfdieter Lang, Jan 13 2015
p^e for p prime of the form 4*k+1 and e >= 1 is the sum of 2 nonzero squares. - Jon Perry, Nov 23 2014
Primes p such that the area of the isosceles triangle of sides (p, p, q) for some integer q is an integer. - Michel Lagneau, Dec 31 2014
This is the set of all primes that are the average of two squares. - Richard R. Forberg, Mar 01 2015
Numbers k such that ((k-3)!!)^2 == -1 (mod k). - Thomas Ordowski, Jul 28 2016
This is a subsequence of primes of A004431 and also of A016813. - Bernard Schott, Apr 30 2022
In addition to the comment from Jean-Christophe Hervé, Nov 10 2013: All powers as well as the products of any of these primes are the sum of two nonzero squares. They are terms of A001481, which is closed under multiplication. - Klaus Purath, Nov 19 2023

Examples

			The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2 + d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2 - a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
  ---------------------------------
   p  a  b  t_1  c   d t_2 t_3  t_4
  ---------------------------------
   5  1  2   1   3   4   4   3    6
  13  2  3   3   5  12  12   5   30
  17  1  4   2   8  15   8  15   60
  29  2  5   5  20  21  20  21  210
  37  1  6   3  12  35  12  35  210
  41  4  5  10   9  40  40   9  180
  53  2  7   7  28  45  28  45  630
  ...
a(7) = 53 = A002972(7)^2 + (2*A002973(7))^2 = 7^2 + (2*1)^2 = 49 + 4, and this is the only way. - _Wolfdieter Lang_, Jan 13 2015

References

  • David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
  • L. E. Dickson, "History of the Theory of Numbers", Chelsea Publishing Company, 1919, Vol I, page 386
  • L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
  • M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 76.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 241, 243.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 90.

Links

Crossrefs

Cf. A002145, A002314, A002476, A002972, A002973, A003658, A004431, A007519, A010051, A016813, A076339, A094407.
Cf. A114200, A133870, A142925, A152676, A152680, A173330, A173331, A208177, A208178, A334912.
Cf. A004613 (multiplicative closure).
Apart from initial term, same as A002313.
For values of n see A005098.
Primes in A020668.

Programs

  • Haskell
    a002144 n = a002144_list !! (n-1)
a002144_list = filter ((== 1) . a010051) [1,5..]
-- Reinhard Zumkeller, Mar 06 2012, Feb 22 2011
  • Magma
    [a: n in [0..200] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Nov 23 2014
  • Maple
    a := []; for n from 1 to 500 do if isprime(4*n+1) then a := [op(a),4*n+1]; fi; od: A002144 := n->a[n];
# alternative
A002144 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        5;
    else
        for a from procname(n-1)+4 by 4 do
            if isprime(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A002144(n),n=1..100) ; # R. J. Mathar, Jan 31 2024
  • Mathematica
    Select[4*Range[140] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 16 2006 *)
Select[Prime[Range[150]],Mod[#,4]==1&] (* Harvey P. Dale, Jan 28 2021 *)
  • PARI
    select(p->p%4==1,primes(1000))
  • PARI
    A002144_next(p=A2144[#A2144])={until(isprime(p+=4),);p} /* NB: p must be of the form 4k+1. Beyond primelimit, this is *much* faster than forprime(p=...,, p%4==1 && return(p)). */
A2144=List(5); A002144(n)={while(#A2144A002144_next())); A2144[n]}
\\ M. F. Hasler, Jul 06 2024
  • Python
    from sympy import prime
A002144 = [n for n in (prime(x) for x in range(1,10**3)) if not (n-1) % 4]
# Chai Wah Wu, Sep 01 2014
  • Python
    from sympy import isprime
print(list(filter(isprime, range(1, 618, 4)))) # Michael S. Branicky, May 13 2021
  • SageMath
    def A002144_list(n): # returns all Pythagorean primes <= n
    return [x for x in prime_range(5,n+1) if x % 4 == 1]
A002144_list(617) # Peter Luschny, Sep 12 2012

Formula

Odd primes of form x^2 + y^2, (x=A002331, y=A002330, with x < y) or of form u^2 + 4*v^2, (u = A002972, v = A002973, with u odd). - Lekraj Beedassy, Jul 16 2004
p^2 - 1 = 12*Sum_{i = 0..floor(p/4)} floor(sqrt(i*p)) where p = a(n) = 4*n + 1. [Shirali]
a(n) = A000290(A002972(n)) + A000290(2*A002973(n)) = A000290(A002331(n+1)) + A000290(A002330(n+1)). - Reinhard Zumkeller, Feb 16 2010
a(n) = A002972(n)^2 + (2*A002973(n))^2, n >= 1. See the Jean-Christophe Hervé Nov 11 2013 comment. - Wolfdieter Lang, Jan 13 2015
a(n) = 4*A005098(n) + 1. - Zak Seidov, Sep 16 2018
From Vaclav Kotesovec, Apr 30 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = A088539.
Product_{k>=1} (1 + 1/a(k)^2) = A243380.
Product_{k>=1} (1 - 1/a(k)^3) = A334425.
Product_{k>=1} (1 + 1/a(k)^3) = A334424.
Product_{k>=1} (1 - 1/a(k)^4) = A334446.
Product_{k>=1} (1 + 1/a(k)^4) = A334445.
Product_{k>=1} (1 - 1/a(k)^5) = A334450.
Product_{k>=1} (1 + 1/a(k)^5) = A334449. (End)
From Vaclav Kotesovec, May 05 2020: (Start)
Product_{k>=1} (1 + 1/A002145(k)) / (1 + 1/a(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962...
Product_{k>=1} (1 - 1/A002145(k)) / (1 - 1/a(k)) = Pi/(8*A064533^2) = 0.6723864219124347812758324971213817835333659546161816055555481... (End)
Sum_{k >= 1} 1/a(k)^s = (1/2) * Sum_{n >= 1 odd numbers} moebius(n) * log((2*n*s)! * zeta(n*s) * abs(EulerE(n*s - 1)) / (Pi^(n*s) * 2^(2*n*s) * BernoulliB(2*n*s) * (2^(n*s) + 1) * (n*s - 1)!))/n, s >= 3 odd number. - Dimitris Valianatos, May 21 2020
Legendre symbol (-1, a(n)) = +1, for n >= 1. - Wolfdieter Lang, Mar 03 2021

A000037 Numbers that are not squares (or, the nonsquares).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99
Offset: 1

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

Note the remarkable formula for the n-th term (see the FORMULA section)!
These are the natural numbers with an even number of divisors. The number of divisors is odd for the complementary sequence, the squares (sequence A000290) and the numbers for which the number of divisors is divisible by 3 is sequence A059269. - Ola Veshta (olaveshta(AT)my-deja.com), Apr 04 2001
a(n) is the largest integer m not equal to n such that n = (floor(n^2/m) + m)/2. - Alexander R. Povolotsky, Feb 10 2008
Union of A007969 and A007970; A007968(a(n)) > 0. - Reinhard Zumkeller, Jun 18 2011
Terms of even numbered rows in the triangle A199332. - Reinhard Zumkeller, Nov 23 2011
If a(n) and a(n+1) are of the same parity then (a(n)+a(n+1))/2 is a square. - Zak Seidov, Aug 13 2012
Theaetetus of Athens proved the irrationality of the square roots of these numbers in the 4th century BC. - Charles R Greathouse IV, Apr 18 2013
4*a(n) are the even members of A079896, the discriminants of indefinite binary quadratic forms. - Wolfdieter Lang, Jun 14 2013

Examples

			For example note that the squares 0, 1, 4, 9, 16 are not included.

References

  • Titu Andreescu, Dorin Andrica, and Zuming Feng, 104 Number Theory Problems, Birkhäuser, 2006, 58-60.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A007412, A000005, A000290, A059269, A134986, A087153, A172151, A000196, A049068 (subsequence).
Cf. A242401 (subsequence).
Cf. A086849 (partial sums), A048395.

Programs

  • Haskell
    a000037 n = n + a000196 (n + a000196 n)
-- Reinhard Zumkeller, Nov 23 2011
  • Magma
    [n : n in [1..1000] | not IsSquare(n) ];
  • Magma
    at:=0; for n in [1..10000] do if not IsSquare(n) then at:=at+1; print at, n; end if; end for;
  • Maple
    A000037 := n->n+floor(1/2+sqrt(n));
  • Mathematica
    a[n_] := (n + Floor[Sqrt[n + Floor[Sqrt[n]]]]); Table[a[n], {n, 71}] (* Robert G. Wilson v, Sep 24 2004 *)
With[{upto=100},Complement[Range[upto],Range[Floor[Sqrt[upto]]]^2]] (* Harvey P. Dale, Dec 02 2011 *)
a[ n_] :=  If[ n < 0, 0, n + Round @ Sqrt @ n]; (* Michael Somos, May 28 2014 *)
  • Maxima
    A000037(n):=n + floor(1/2 + sqrt(n))$ makelist(A000037(n),n,1,50); /* Martin Ettl, Nov 15 2012 */
  • PARI
    {a(n) = if( n<0, 0, n + (1 + sqrtint(4*n)) \ 2)};
  • Python
    from math import isqrt
def A000037(n): return n+isqrt(n+isqrt(n)) # Chai Wah Wu, Mar 31 2022
  • Python
    from math import isqrt
def A000037(n): return n+(k:=isqrt(n))+int(n>=k*(k+1)+1) # Chai Wah Wu, Jun 17 2024

Formula

a(n) = n + floor(1/2 + sqrt(n)).
a(n) = n + floor(sqrt( n + floor(sqrt n))).
A010052(a(n)) = 0. - Reinhard Zumkeller, Jan 26 2010
A173517(a(n)) = n; a(n)^2 = A030140(n). - Reinhard Zumkeller, Feb 20 2010
a(n) = A000194(n) + n. - Jaroslav Krizek, Jun 14 2009
a(A002061(n)) = a(n^2-n+1) = A002522(n) = n^2 + 1. - Jaroslav Krizek, Jun 21 2009

Extensions

Edited by Charles R Greathouse IV, Oct 30 2009

A003814 Numbers k such that the continued fraction for sqrt(k) has odd period length.

Original entry on oeis.org

2, 5, 10, 13, 17, 26, 29, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 125, 130, 137, 145, 149, 157, 170, 173, 181, 185, 193, 197, 202, 218, 226, 229, 233, 241, 250, 257, 265, 269, 274, 277, 281, 290, 293, 298, 313, 314, 317
Offset: 1

Views

Author

N. J. A. Sloane, Walter Gilbert

Keywords

Comments

All primes of the form 4m + 1 are here. - T. D. Noe, Mar 19 2012
These numbers have no prime factors of the form 4m + 3. - Thomas Ordowski, Jul 01 2013
This sequence is a proper subsequence of the so-called 1-happy number products A007969. See the W. Lang link there, eq. (1), with B = 1, C = a(n), also with a table at the end. This is due to the soluble Pell equation R^2 - C*S^2 = -1 for C = a(n). See e.g., Perron, Satz 3.18. on p. 93, and the table on p. 91 with the numbers D of the first column that do not have a number in brackets in the second column (Teilnenner von sqrt(D)). - Wolfdieter Lang, Sep 19 2015

References

  • W. Paulsen, Calkin-Wilf sequences for irrational numbers, Fib. Q., 61:1 (2023), 51-59.
  • O. Perron, Die Lehre von den Kettenbrüchen, Band I, Teubner Verlagsgesellschaft, Stuttgart, 1954.
  • Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).

Links

Crossrefs

Cf. A010333, A003654, A007969.
Cf. A031396.
Cf. A206586 (period has positive even length).

Programs

  • Maple
    isA003814 := proc(n)
    local cf,p ;
    if issqr(n) then
        return false;
    end if;
    for p in numtheory[factorset](n) do
        if modp(p,4) = 3 then
            return false;
        end if;
    end do:
    cf := numtheory[cfrac](sqrt(n),'periodic','quotients') ;
    type( nops(op(2,cf)),'odd') ;
end proc:
A003814 := proc(n)
    option remember;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA003814(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A003814(n),n=1..40) ; # R. J. Mathar, Oct 19 2014
  • Mathematica
    Select[Range[100], ! IntegerQ[Sqrt[#]] && OddQ[Length[ContinuedFraction[Sqrt[#]][[2]]]] &] (* T. D. Noe, Mar 19 2012 *)
  • PARI
    cyc(cf) = {
  if(#cf==1, return([])); \\ There is no cycle
  my(s=[]);
  for(k=2, #cf,
    s=concat(s, cf[k]);
    if(cf[k]==2*cf[1], return(s)) \\ Cycle found
  );
  0 \\ Cycle not found
}
select(n->#cyc(contfrac(sqrt(n)))%2==1, vector(400, n, n)) \\ Colin Barker, Oct 19 2014

A007970 Rhombic numbers.

Original entry on oeis.org

3, 7, 8, 11, 15, 19, 23, 24, 27, 31, 32, 35, 40, 43, 47, 48, 51, 59, 63, 67, 71, 75, 79, 80, 83, 87, 88, 91, 96, 99, 103, 104, 107, 115, 119, 120, 123, 127, 128, 131, 135, 136, 139, 143, 151, 152, 159, 160, 163, 167, 168, 171, 175, 176, 179
Offset: 1

Views

Author

J. H. Conway

Keywords

Comments

A191856(n) = A007966(a(n)); A191857(n) = A007967(a(n)). - Reinhard Zumkeller, Jun 18 2011
This sequence gives the values d of the Pell equation x^2 - d*y^2 = +1 that have positive fundamental solutions (x0, y0) with odd y0. This was first conjectured and is proved provided Conway's theorem in the link is assumed and the proof of the conjecture stated in A007869, given there in a W. Lang link, is used. - Wolfdieter Lang, Sep 19 2015
For a proof of Conway's theorem on the happy number factorization see the W. Lang link (together with the link given under A007969). - Wolfdieter Lang, Oct 04 2015

Links

Crossrefs

Every number belongs to exactly one of A000290, A007969, A007970.
Cf. A007968.
Subsequence of A000037, A002145 is a subsequence.
A263008 (T numbers), A263009 (U numbers).

Programs

  • Haskell
    a007970 n = a007970_list !! (n-1)
a007970_list = filter ((== 2) . a007968) [0..]
-- Reinhard Zumkeller, Oct 11 2015
  • Mathematica
    r[b_, c_] := (red = Reduce[x > 0 && y > 0 && b*x^2 + 2 == c*y^2, {x, y}, Integers] /. C[1] -> 1 // Simplify; If[Head[red] === Or, First[red], red]);
f[n_] := f[n] = If[! IntegerQ[Sqrt[n]], Catch[Do[{b, c} = bc; If[ (r0 = r[b, c]) =!= False, {x0, y0} = {x, y} /. ToRules[r0]; If[OddQ[x0] && OddQ[y0], Throw[n]]]; If[ (r0 = r[c, b]) =!= False, {x0, y0} = {x, y} /. ToRules[r0]; If[OddQ[x0] && OddQ[y0], Throw[n]]], {bc, Union[Sort[{#, n/#}] & /@ Divisors[n]]} ]]];
A007970 = Reap[ Table[ If[f[n] =!= Null, Print[f[n]]; Sow[f[n]]], {n, 1, 180}] ][[2, 1]](* Jean-François Alcover, Jun 26 2012 *)

Formula

a(n) = A191856(n)*A191857(n); A007968(a(n))=2. - Reinhard Zumkeller, Jun 18 2011
a(n) is in the sequence if a(n) = D*E with positive integers D and E, such that E*U^2 - D*T^2 = 2 has an integer solution with U*T odd (without loss of generality one may take U and T positive). See the Conway link. D and E are given in A191856 and A191857, respectively. - Wolfdieter Lang, Oct 05 2015

Extensions

159 and 175 inserted by Jean-François Alcover, Jun 26 2012

A007968 Type of happy factorization of n.

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 1, 2, 2, 0, 1, 2, 1, 1, 1, 2, 0, 1, 1, 2, 1, 1, 1, 2, 2, 0, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 0, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 2, 2
Offset: 0

Views

Author

J. H. Conway

Keywords

Links

Crossrefs

Cf. A000290, A007969, A007970.

Programs

  • Haskell
    a007968 = (\(hType,,,_,_) -> hType) . h
h 0 = (0, 0, 0, 0, 0)
h x = if a > 0 then (0, a, a, a, a) else h' 1 divs
      where a = a037213 x
            divs = a027750_row x
            h' r []                                = h' (r + 1) divs
            h' r (d:ds)
             | d' > 1 && rest1 == 0 && ss == s ^ 2 = (1, d, d', r, s)
             | rest2 == 0 && odd u && uu == u ^ 2  = (2, d, d', t, u)
             | otherwise                           = h' r ds
             where (ss, rest1) = divMod (d * r ^ 2 + 1) d'
                   (uu, rest2) = divMod (d * t ^ 2 + 2) d'
                   s = a000196 ss; u = a000196 uu; t = 2 * r - 1
                   d' = div x d
hs = map h [0..]
hCouples = map (\(, factor1, factor2, , _) -> (factor1, factor2)) hs
sqrtPair n = genericIndex sqrtPairs (n - 1)
sqrtPairs = map (\(, , _, sqrt1, sqrt2) -> (sqrt1, sqrt2)) hs
-- Reinhard Zumkeller, Oct 11 2015

Formula

a(A000290(n)) = 0; a(A007969(n)) = 1; a(A007970(n)) = 2.

A007966 First factor in happy factorization of n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 7, 2, 3, 1, 1, 3, 1, 7, 3, 4, 1, 2, 1, 4, 3, 2, 23, 4, 5, 1, 1, 7, 1, 5, 31, 16, 11, 17, 5, 6, 1, 2, 3, 2, 1, 6, 1, 11, 5, 23, 47, 6, 7, 1, 1, 4, 1, 2, 11, 7, 3, 1, 1, 15, 1, 31, 7, 8, 1, 2, 1, 4, 23, 5, 71, 8, 1, 1, 25, 19, 7, 26, 79, 8, 9, 1, 1, 3, 1, 2, 3, 4, 1, 9
Offset: 0

Views

Author

J. H. Conway

Keywords

Comments

a(n) = n / A007967(n);
a(A007969(n)) = A191854(n); a(A007970(n)) = A191856(n). - Reinhard Zumkeller, Jun 18 2011

Links

Crossrefs

Cf. A191914.
Cf. A007968, A007969, A007970, A191854, A191856.

Programs

  • Haskell
    import Data.List (genericIndex)
a007966 n = genericIndex a007966_list n
a007966_list = map fst hCouples
-- Pairs hCouples are defined in A007968.
-- Reinhard Zumkeller, Oct 11 2015
  • Mathematica
    r[b_, c_, d_] := (red = Reduce[x > 0 && y > 0 && b*x^2 + d == c*y^2, {x, y}, Integers] /. C[1] -> 1 // Simplify; If[Head[red] === Or, red[[1]], red]); f[n_] := f[n] = If[IntegerQ[rn = Sqrt[n]], {0, rn, rn, rn, rn}, Catch[Do[b = bc[[1]]; c = bc[[2]]; If[ c > 1 && (r0 = r[b, c, 1]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; Throw[{1, b, c, x0, y0}]]; If[ b > 1 && (r0 = r[c, b, 1]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; Throw[{1, c, b, x0, y0}]]; If[ (r0 = r[b, c, 2]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; If[OddQ[x0] && OddQ[y0], Throw[{2, b, c, x0, y0}]]]; If[ (r0 = r[c, b, 2]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; If[OddQ[x0] && OddQ[y0], Throw[{2, c, b, x0, y0}]]]; , {bc, Union[Sort[{#, n/#}] & /@ Divisors[n]]} ]]]; a[n_] := f[n][[2]]; A007966 = Table[Print[a[n]]; a[n], {n, 0, 90}] (* Jean-François Alcover, Jun 25 2012 *)

A007967 Second factor in happy factorization of n.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 3, 1, 4, 3, 10, 11, 4, 13, 2, 5, 4, 17, 9, 19, 5, 7, 11, 1, 6, 5, 26, 27, 4, 29, 6, 1, 2, 3, 2, 7, 6, 37, 19, 13, 20, 41, 7, 43, 4, 9, 2, 1, 8, 7, 50, 51, 13, 53, 27, 5, 8, 19, 58, 59, 4, 61, 2, 9, 8, 65, 33, 67, 17, 3, 14, 1, 9, 73, 74, 3, 4, 11, 3, 1, 10, 9, 82, 83
Offset: 0

Views

Author

J. H. Conway

Keywords

Comments

a(n) = n / A007966(n);
a(A007969(n)) = A191855(n); a(A007970(n)) = A191857(n). - Reinhard Zumkeller, Jun 18 2011

Links

Crossrefs

Cf. A191914.
Cf. A007968, A007969, A007970, A191855, A191857.

Programs

  • Haskell
    import Data.List (genericIndex)
a007967 n = genericIndex a007967_list n
a007967_list = map snd hCouples
-- Pairs hCouples are defined in A007968.
-- Reinhard Zumkeller, Oct 11 2015
  • Mathematica
    r[b_, c_,  d_] := (red = Reduce[x > 0 && y > 0 && b*x^2 + d == c*y^2, {x, y}, Integers] /. C[1] -> 1 // Simplify; If[Head[red] === Or, red[[1]], red]); f[n_] := f[n] =  If[IntegerQ[rn = Sqrt[n]], {0, rn, rn, rn, rn},  Catch[Do[b = bc[[1]]; c = bc[[2]]; If[c > 1 && (r0 = r[b, c, 1]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; Throw[{1, b, c, x0, y0}]]; If[b > 1 && (r0 = r[c, b, 1]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; Throw[{1, c, b, x0, y0}]]; If[(r0 = r[b, c, 2]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; If[OddQ[x0] && OddQ[y0], Throw[{2, b, c, x0, y0}]]]; If[(r0 = r[c, b, 2]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; If[OddQ[x0] && OddQ[y0], Throw[{2, c, b, x0, y0}]]];, {bc, Union[Sort[{#, n/#}] & /@ Divisors[n]]}]]];a[n_] := f[n][[3]]; A007967 = Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 90}] (* Jean-François Alcover, Sep 18 2015 *)

A191854 First factor in happy factorization of n-th rectangular number.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 7, 1, 2, 4, 3, 2, 1, 7, 1, 5, 11, 17, 1, 2, 3, 1, 6, 11, 5, 23, 1, 4, 1, 2, 11, 7, 3, 1, 15, 1, 31, 1, 2, 4, 23, 5, 8, 1, 1, 19, 7, 26, 1, 3, 1, 2, 1, 9, 23, 3, 47, 19, 1, 49, 1, 2, 5, 1, 27, 1, 10, 3, 7, 1, 2, 4, 9, 2, 1, 31, 1, 14, 3, 1
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 18 2011

Keywords

Comments

a(n) = A007966(A007969(n)) = A007969(n) / A191855(n);
(a(n), A191855(n)) is a 1-happy couple;
notation: B in the Conway link.

Links

Crossrefs

Cf. A007966, A007969, A191855.

Programs

  • Haskell
    a191854 = a007966 . a007969  -- Reinhard Zumkeller, Oct 11 2015
  • Mathematica
    r[b_, c_] := (red = Reduce[x>0 && y>0 && b*x^2 + 1 == c*y^2, {x, y}, Integers] /. C[1] -> 1 // Simplify; If[Head[red] === Or, First[red], red]); f[128] = {}(* to speed up *); f[n_] := f[n] = If[IntegerQ[Sqrt[n]], {}, Do[c = n/b; If[(r0 = r[b, c]) =!= False, {x0, y0} = {x, y} /. ToRules[r0]; Return[{b, c, x0, y0}]], {b, Divisors[n] // Most}]]; A191854 = Reap[Table[Print[n, " ", f[n]]; If[f[n] != {} && f[n] =!= Null, Sow[f[n][[1]]]], {n, 1, 130}]][[2, 1]] (* Jean-François Alcover, Sep 18 2015 *)

Extensions

Wrong formula removed (thanks to Wolfdieter Lang, who pointed this out) by Reinhard Zumkeller, Oct 11 2015
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