Displaying 1-10 of 208 results found.
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Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1,x_2,...,x_k) = 1 (2<=k<=n).
+0
6
1, 2, 1, 2, 3, 1, 4, 6, 4, 1, 2, 9, 10, 5, 1, 6, 15, 20, 15, 6, 1, 4, 18, 34, 35, 21, 7, 1, 6, 27, 56, 70, 56, 28, 8, 1, 4, 30, 80, 125, 126, 84, 36, 9, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 4, 42, 154, 325, 461, 462, 330, 165, 55, 11, 1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1
COMMENTS
If instead we require that the individual parts (x_i,x_j) be relatively prime, we get A282748. This is the question studied by Shonhiwa (2006). - N. J. A. Sloane, Mar 05 2017.
FORMULA
T(n, k) = Sum_{d|n} binomial(d-1, k-1)*mobius(n/d).
EXAMPLE
T(6,3)=9 because we have 411,141,114 and the six permutations of 123 (222 does not qualify).
T(8,3)=18 because binomial(0,2)*mobius(8/1)+binomial(1,2)*mobius(8/2)+binomial(3,2)*mobius(8/4)+binomial(7,2)*mobius(8/8)=0+0+(-3)+21=18.
Triangle begins:
1,
2, 1,
2, 3, 1,
4, 6, 4, 1,
2, 9, 10, 5, 1,
6, 15, 20, 15, 6, 1,
4, 18, 34, 35, 21, 7, 1,
6, 27, 56, 70, 56, 28, 8, 1,
4, 30, 80, 125, 126, 84, 36, 9, 1,
10, 45, 120, 210, 252, 210, 120, 45, 10, 1,
4, 42, 154, 325, 461, 462, 330, 165, 55, 11, 1,
12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1,
...
Row n = 6 counts the following compositions:
(15) (114) (1113) (11112) (111111)
(51) (123) (1122) (11121)
(132) (1131) (11211)
(141) (1212) (12111)
(213) (1221) (21111)
(231) (1311)
(312) (2112)
(321) (2121)
(411) (2211)
(3111)
Missing are: (42), (24), (33), (222).
(End)
MAPLE
with(numtheory): T:=proc(n, k) local d, j, b: d:=divisors(n): for j from 1 to tau(n) do b[j]:=binomial(d[j]-1, k-1)*mobius(n/d[j]) od: sum(b[i], i=1..tau(n)) end: for n from 2 to 14 do seq(T(n, k), k=2..n) od; # yields the sequence in triangular form
MATHEMATICA
t[n_, k_] := Sum[Binomial[d-1, k-1]*MoebiusMu[n/d], {d, Divisors[n]}]; Table[t[n, k], {n, 2, 14}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jan 20 2014 *) Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {k}], GCD@@#==1&]], {n, 10}, {k, 2, n}] (* change {k, 2, n} to {k, 1, n} for the version with zeros. - Gus Wiseman, Oct 19 2020 *)
PROG
(PARI) T(n, k) = sumdiv(n, d, binomial(d-1, k-1)*moebius(n/d)); \\ Michel Marcus, Mar 09 2016
CROSSREFS
A000837 counts relatively prime partitions.
A135278 counts compositions by length.
A282748 is the pairwise coprime instead of relatively prime version.
A291166 ranks these compositions (evidently).
Powers of squarefree numbers.
+0
124
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97
COMMENTS
Heinz numbers of uniform partitions. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 16 2018
MATHEMATICA
Select[Range[100], Length[Union[FactorInteger[#][[All, 2]]]] == 1 &] (* Geoffrey Critzer, Mar 30 2015 *)
PROG
(Haskell)
import Data.Map (empty, findMin, deleteMin, insert)
import qualified Data.Map.Lazy as Map (null)
a072774 n = a072774_list !! (n-1)
(a072774_list, a072775_list, a072776_list) = unzip3 $
(1, 1, 1) : f (tail a005117_list) empty where
f vs'@(v:vs) m
| Map.null m || xx > v = (v, v, 1) :
f vs (insert (v^2) (v, 2) m)
| otherwise = (xx, bx, ex) :
f vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
where (xx, (bx, ex)) = findMin m
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
def f(x): return n-2+x-sum(g(integer_nthroot(x, k)[0]) for k in range(1, x.bit_length()))
kmin, kmax = 1, 2
while f(kmax) >= kmax:
kmax <<= 1
while True:
kmid = kmax+kmin>>1
if f(kmid) < kmid:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
CROSSREFS
Cf. A000009, A000837, A007916, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796, A302979.
Heinz numbers of integer partitions whose length is equal to the GCD of all the parts.
+0
10
1, 2, 9, 21, 39, 57, 87, 91, 111, 125, 129, 159, 183, 203, 213, 237, 247, 267, 301, 303, 321, 325, 339, 377, 393, 417, 427, 453, 489, 519, 543, 551, 553, 559, 575, 579, 597, 669, 687, 689, 707, 717, 753, 789, 791, 813, 817, 843, 845, 879, 923, 925, 933, 951, 973
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
2 is the only even term in the sequence. 3k is in the sequence if and only if k is in A031215. 5k is in the sequence if and only if k = pq with p and q in A031336.
EXAMPLE
Sequence of integer partitions whose length is equal to their GCD begins: (), (1), (2,2), (4,2), (6,2), (8,2), (10,2), (6,4), (12,2), (3,3,3), (14,2), (16,2), (18,2), (10,4), (20,2), (22,2), (8,6), (24,2), (14,4), (26,2), (28,2), (6,3,3).
MATHEMATICA
Select[Range[200], PrimeOmega[#]==GCD@@Cases[FactorInteger[#], {p_, k_}:>PrimePi[p]]&]
CROSSREFS
Cf. A056239, A067538, A074761, A143773, A289508, A289509, A296150, A316413, A316431, A316432, A316433, A031215, A031336.
Heinz numbers of integer partitions, whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts, in increasing order.
+0
1
2, 1495, 179417, 231133, 727531, 1378583, 1787387, 3744103, 4556993, 7566167, 18977519, 29629391, 30870587, 34174939, 39973571, 53508983, 70946617, 110779141, 138820187, 139681069, 170583017, 225817751, 409219217, 441317981, 493580417, 539462099, 544392433, 712797613, 802903541
COMMENTS
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
EXAMPLE
The corresponding sequence of partitions, whose length is equal to their GCD and whose sum is equal to their LCM: (1), (9,6,3), (20,8,8,4), (24,16,4,4), (16,16,12,4).
MATHEMATICA
Select[Range[2, 10000], With[{m=If[#==1, {}, Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]}, And[LCM@@m==Total[m], GCD@@m==Length[m]]]&]
CROSSREFS
Cf. A056239, A067538, A074761, A143773, A289508, A289509, A290103, A290104, A316431, A316432, A319328, A319330, A319333.
The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.
(Formerly M0472 N0173)
+0
2101
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
COMMENTS
For some authors, the terms "natural numbers" and "counting numbers" include 0, i.e., refer to the nonnegative integers A001477; the term "whole numbers" frequently also designates the whole set of (signed) integers A001057. a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n (cf. A007378). Inverse Euler transform of A000219. The rectangular array having A000027 as antidiagonals is the dispersion of the complement of the triangular numbers, A000217 (which triangularly form column 1 of this array). The array is also the transpose of A038722. - Clark Kimberling, Apr 05 2003 For nonzero x, define f(n) = floor(nx) - floor(n/x). Then f= A000027 if and only if x=tau or x=-tau. - Clark Kimberling, Jan 09 2005 If the offset were changed to 0, we would have the following pattern: a(n)=binomial(n,0) + binomial(n,1) for the present sequence (number of regions in 1-space defined by n points), A000124 (number of regions in 2-space defined by n straight lines), A000125 (number of regions in 3-space defined by n planes), A000127 (number of regions in 4-space defined by n hyperplanes), A006261, A008859, A008860, A008861, A008862 and A008863, where the last six sequences are interpreted analogously and in each "... by n ..." clause an offset of 0 has been assumed, resulting in a(0)=1 for all of them, which corresponds to the case of not cutting with a hyperplane at all and therefore having one region. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006 Define a number of points on a straight line to be in general arrangement when no two points coincide. Then these are the numbers of regions defined by n points in general arrangement on a straight line, when an offset of 0 is assumed. For instance, a(0)=1, since using no point at all leaves one region. The sequence satisfies the recursion a(n) = a(n-1) + 1. This has the following geometrical interpretation: Suppose there are already n-1 points in general arrangement, thus defining the maximal number of regions on a straight line obtainable by n-1 points, and now one more point is added in general arrangement. Then it will coincide with no other point and act as a dividing wall thereby creating one new region in addition to the a(n-1)=(n-1)+1=n regions already there, hence a(n)=a(n-1)+1. Cf. the comments on A000124 for an analogous interpretation. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006 The sequence a(n)=n (for n=1,2,3) and a(n)=n+1 (for n=4,5,...) gives to the rank (minimal cardinality of a generating set) for the semigroup I_n\S_n, where I_n and S_n denote the symmetric inverse semigroup and symmetric group on [n]. - James East, May 03 2007 The sequence a(n)=n (for n=1,2), a(n)=n+1 (for n=3) and a(n)=n+2 (for n=4,5,...) gives the rank (minimal cardinality of a generating set) for the semigroup PT_n\T_n, where PT_n and T_n denote the partial transformation semigroup and transformation semigroup on [n]. - James East, May 03 2007 "God made the integers; all else is the work of man." This famous quotation is a translation of "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk," spoken by Leopold Kronecker in a lecture at the Berliner Naturforscher-Versammlung in 1886. Possibly the first publication of the statement is in Heinrich Weber's "Leopold Kronecker," Jahresberichte D.M.V. 2 (1893) 5-31. - Clark Kimberling, Jul 07 2007 Writing A000027 as N, perhaps the simplest one-to-one correspondence between N X N and N is this: f(m,n) = ((m+n)^2 - m - 3n + 2)/2. Its inverse is given by I(k)=(g,h), where g = k - J(J-1)/2, h = J + 1 - g, J = floor((1 + sqrt(8k - 7))/2). Thus I(1)=(1,1), I(2)=(1,2), I(3)=(2,1) and so on; the mapping I fills the first-quadrant lattice by successive antidiagonals. - Clark Kimberling, Sep 11 2008 a(n) is also the mean of the first n odd integers. - Ian Kent, Dec 23 2008 Equals INVERTi transform of A001906, the even-indexed Fibonacci numbers starting (1, 3, 8, 21, 55, ...). - Gary W. Adamson, Jun 05 2009 These are also the 2-rough numbers: positive integers that have no prime factors less than 2. - Michael B. Porter, Oct 08 2009 Totally multiplicative sequence with a(p) = p for prime p. Totally multiplicative sequence with a(p) = a(p-1) + 1 for prime p. - Jaroslav Krizek, Oct 18 2009 Triangle T(k,j) of natural numbers, read by rows, with T(k,j) = binomial(k,2) + j = (k^2-k)/2 + j where 1 <= j <= k. In other words, a(n) = n = binomial(k,2) + j where k is the largest integer such that binomial(k,2) < n and j = n - binomial(k,2). For example, T(4,1)=7, T(4,2)=8, T(4,3)=9, and T(4,4)=10. Note that T(n,n)= A000217(n), the n-th triangular number. - Dennis P. Walsh, Nov 19 2009 Hofstadter-Conway-like sequence (see A004001): a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = 1, a(2) = 2. - Jaroslav Krizek, Dec 11 2009 a(n) is also the dimension of the irreducible representations of the Lie algebra sl(2). - Leonid Bedratyuk, Jan 04 2010 Generated from a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1), r = 2; in an infinite set, row 2 of the array shown in A178568. - Gary W. Adamson, May 29 2010 1/n = continued fraction [n]. Let barover[n] = [n,n,n,...] = 1/k. Then k - 1/k = n. Example: [2,2,2,...] = (sqrt(2) - 1) = 1/k, with k = (sqrt(2) + 1). Then 2 = k - 1/k. - Gary W. Adamson, Jul 15 2010 Number of n-digit numbers the binary expansion of which contains one run of 1's. - Vladimir Shevelev, Jul 30 2010 Let T denote the "natural number array A000027": 1 2 4 7 ...
3 5 8 12 ...
6 9 13 18 ...
10 14 19 25 ...
T(n,k) = n+(n+k-2)*(n+k-1)/2. See A185787 for a list of sequences based on T, such as rows, columns, diagonals, and sub-arrays. (End) The Stern polynomial B(n,x) evaluated at x=2. See A125184. - T. D. Noe, Feb 28 2011 The denominator in the Maclaurin series of log(2), which is 1 - 1/2 + 1/3 - 1/4 + .... - Mohammad K. Azarian, Oct 13 2011 As a function of Bernoulli numbers B_n (cf. A027641: (1, -1/2, 1/6, 0, -1/30, 0, 1/42, ...)): let V = a variant of B_n changing the (-1/2) to (1/2). Then triangle A074909 (the beheaded Pascal's triangle) * [1, 1/2, 1/6, 0, -1/30, ...] = the vector [1, 2, 3, 4, 5, ...]. - Gary W. Adamson, Mar 05 2012 Number of partitions of 2n+1 into exactly two parts. - Wesley Ivan Hurt, Jul 15 2013 Integers n dividing u(n) = 2u(n-1) - u(n-2); u(0)=0, u(1)=1 (Lucas sequence A001477). - Thomas M. Bridge, Nov 03 2013 For this sequence, the generalized continued fraction a(1)+a(1)/(a(2)+a(2)/(a(3)+a(3)/(a(4)+...))), evaluates to 1/(e-2) = A194807. - Stanislav Sykora, Jan 20 2014 a(n) is the number of permutations of length n simultaneously avoiding 213, 231 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014 a(n) is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl, Aug 07 2014 a(n) = least k such that 2*Pi - Sum_{h=1..k} 1/(h^2 - h + 3/16) < 1/n. - Clark Kimberling, Sep 28 2014 a(n) = least k such that Pi^2/6 - Sum_{h=1..k} 1/h^2 < 1/n. - Clark Kimberling, Oct 02 2014 Determinants of the spiral knots S(2,k,(1)). a(k) = det(S(2,k,(1))). These knots are also the torus knots T(2,k). - Ryan Stees, Dec 15 2014 As a function, the restriction of the identity map on the nonnegative integers {0,1,2,3...}, A001477, to the positive integers {1,2,3,...}. - M. F. Hasler, Jan 18 2015 See also A131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^k+ k) / k! takes integral values for all i>=0: For k=1, A131685(k)=1, which implies that this is a well defined integer sequence. - Alexander R. Povolotsky, Apr 24 2015 a(n) is the number of compositions of n+2 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016 Does not satisfy Benford's law [Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017 For n >= 3, a(n)=n is the least area that can be obtained for an irregular octagon drawn in a square of n units side, whose sides are parallel to the axes, with 4 vertices that coincide with the 4 vertices of the square, and the 4 remaining vertices having integer coordinates. See Affaire de Logique link. - Michel Marcus, Apr 28 2018 a(n+1) is the order of rowmotion on a poset defined by a disjoint union of chains of length n. - Nick Mayers, Jun 08 2018 Number of 1's in n-th generation of 1-D Cellular Automata using Rules 50, 58, 114, 122, 178, 186, 206, 220, 238, 242, 250 or 252 in the Wolfram numbering scheme, started with a single 1. - Frank Hollstein, Mar 25 2019 (1, 2, 3, 4, 5, ...) is the fourth INVERT transform of (1, -2, 3, -4, 5, ...). - Gary W. Adamson, Jul 15 2019
REFERENCES
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 1.
T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25.
W. Fulton and J. Harris, Representation theory: a first course, (1991), page 149. [From Leonid Bedratyuk, Jan 04 2010] I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
R. E. Schwartz, You Can Count on Monsters: The First 100 numbers and Their Characters, A. K. Peters and MAA, 2010.
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
N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, Spiral knots, Missouri J. of Math. Sci., 22 (2010). M. DeLong, M. Russell, and J. Schrock, Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m), Involve, Vol. 8 (2015), No. 3, 361-384. Eric Weisstein's World of Mathematics, Natural Number, Positive Integer, Counting Number Composition, Davenport-Schinzel Sequence, Idempotent Number, N, Smarandache Ceil Function, Whole Number, Engel Expansion, and Trinomial Coefficient
FORMULA
Another g.f.: Sum_{n>0} phi(n)*x^n/(1-x^n) (Apostol).
When seen as an array: T(k, n) = n+1 + (k+n)*(k+n+1)/2. Main diagonal is 2n*(n+1)+1 ( A001844), antidiagonal sums are n*(n^2+1)/2 ( A006003). - Ralf Stephan, Oct 17 2004 G.f.: x/(1-x)^2. E.g.f.: x*exp(x). a(n)=n. a(-n)=-a(n).
Series reversion of g.f. A(x) is x*C(-x)^2 where C(x) is the g.f. of A000108. - Michael Somos, Sep 04 2006 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 4*u*v. - Michael Somos, Oct 03 2006 a(n) = 2*a(n-1)-a(n-2); a(1)=1, a(2)=2. a(n) = 1+a(n-1). - Philippe Deléham, Nov 03 2008 G.f.: x * Product_{j>=0} (1+x^(2^j))^2 = x * (1+2*x+x^2) * (1+2*x^2+x^4) * (1+2*x^4+x^8) * ... = x + 2x^2 + 3x^3 + ... . - Gary W. Adamson, Jun 26 2012 a(n) = det(binomial(i+1,j), 1 <= i,j <= n). - Mircea Merca, Apr 06 2013 E.g.f.: x*E(0), where E(k) = 1 + 1/(x - x^3/(x^2 + (k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 03 2013 a(n) = Product_{k=1..n-1} 2*sin(Pi*k/n), n > 1.
a(n) = Product_{k=1..n-1} (2*sin(Pi*k/(2*n)))^2, n > 1.
These identities are used in the calculation of products of ratios of lengths of certain lines in a regular n-gon. For the first identity see the Gradstein-Ryshik reference, p. 62, 1.392 1., bringing the first factor there to the left hand side and taking the limit x -> 0 (L'Hôpital). The second line follows from the first one. Thanks to Seppo Mustonen who led me to consider n-gon lengths products. (End) a(n) = Sum_{j=0..k} (-1)^(j-1)*j*binomial(n,j)*binomial(n-1+k-j,k-j), k>=0. - Mircea Merca, Jan 25 2014 a(n) = Sum_{k=1..n^2+2*n} 1/(sqrt(k)+sqrt(k+1)). - Pierre CAMI, Apr 25 2014 a(n) = floor(1/sin(1/n)) = floor(cot(1/(n+1))) = ceiling(cot(1/n)). - Clark Kimberling, Oct 08 2014 a(n) = 1/(1/(n+1) + 1/(n+1)^2 + 1/(n+1)^3 + ...). - Pierre CAMI, Jan 22 2015 a(n) = Sum_{m=0..n-1} Stirling1(n-1,m)*Bell(m+1), for n >= 1. This corresponds to Bell(m+1) = Sum_{k=0..m} Stirling2(m, k)*(k+1), for m >= 0, from the fact that Stirling2*Stirling1 = identity matrix. See A048993, A048994 and A000110. - Wolfdieter Lang, Feb 03 2015 a(n) = Sum_{k=1..2n-1}(-1)^(k+1)*k*(2n-k). In addition, surprisingly, a(n) = Sum_{k=1..2n-1}(-1)^(k+1)*k^2*(2n-k)^2. - Charlie Marion, Jan 05 2016 G.f.: x/(1-x)^2 = (x * r(x) *r(x^3) * r(x^9) * r(x^27) * ...), where r(x) = (1 + x + x^2)^2 = (1 + 2x + 3x^2 + 2x^3 + x^4). - Gary W. Adamson, Jan 11 2017
PROG
(Magma) [ n : n in [1..100]];
(PARI) {a(n) = n};
(R) 1:100
(Shell) seq 1 100
(Haskell)
a000027 = id
(Maxima) makelist(n, n, 1, 30); /* Martin Ettl, Nov 07 2012 */ (Python)
(Julia) print([n for n in 1:280]) # Paul Muljadi, Apr 09 2024
CROSSREFS
Cf. A026081 = integers in reverse alphabetical order in U.S. English, A107322 = English name for number and its reverse have the same number of letters, A119796 = zero through ten in alphabetical order of English reverse spelling, A005589, etc. Cf. A185787 (includes a list of sequences based on the natural number array A000027). Cf. A038722 (mirrored when seen as triangle), A056011 (boustrophedon).
Number of strict integer partitions of n where all parts have neighbors.
+0
10
1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 2, 3, 2, 2, 5, 2, 4, 5, 5, 4, 8, 5, 7, 9, 8, 8, 13, 10, 11, 16, 13, 15, 20, 18, 18, 27, 21, 26, 31, 30, 30, 43, 34, 42, 49, 48, 48, 65, 56, 65, 76, 74, 77, 97, 88, 98, 117, 111, 119, 143, 137, 146, 175, 165, 182, 208
COMMENTS
A part x has a neighbor if either x - 1 or x + 1 is a part.
FORMULA
G.f.: 1 + Sum_{i>0} A(x,i), where A(x,i) = x^((2*i)+1) * G(x,i+1) for i > 0, is the g.f. for partitions of this kind with least part i, and G(x,k) = 1 + x^(k+1) * G(x,k+1) + Sum_{m>=0} x^(2*(k+m)+5) * G(x,m+k+3). - John Tyler Rascoe, Feb 16 2024
EXAMPLE
The a(n) partitions for n = 0, 1, 3, 9, 15, 18, 20, 24 (A = 10, B = 11):
() . (21) (54) (87) (765) (7643) (987)
(432) (654) (6543) (8732) (8754)
(54321) (7632) (9821) (9843)
(8721) (65432) (A932)
(65421) (BA21)
(87432)
(87621)
(765321)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Function[ptn, UnsameQ@@ptn&&And@@Table[MemberQ[ptn, x-1]||MemberQ[ptn, x+1], {x, Union[ptn]}]]]], {n, 0, 30}]
PROG
(Python) # see linked program
Number of integer partitions of n with a neighborless part.
+0
10
0, 1, 2, 2, 4, 4, 8, 9, 16, 20, 31, 40, 59, 76, 105, 138, 184, 238, 311, 400, 515, 656, 831, 1052, 1322, 1659, 2064, 2572, 3182, 3934, 4837, 5942, 7264, 8872, 10789, 13109, 15865, 19174, 23105, 27796, 33361, 39956, 47766, 56985, 67871, 80675, 95750, 113416
COMMENTS
A part x of a partition is neighborless if neither x - 1 nor x + 1 are parts.
EXAMPLE
The a(1) = 1 through a(8) = 9 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (111) (22) (41) (33) (52)
(31) (311) (42) (61)
(1111) (11111) (51) (331)
(222) (421)
(411) (511)
(3111) (4111)
(111111) (31111)
(1111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Function[ptn, Or@@Table[!MemberQ[ptn, x-1]&&!MemberQ[ptn, x+1], {x, Union[ptn]}]]]], {n, 0, 30}]
CROSSREFS
These partitions are ranked by the complement of A356736.
Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.
+0
11
1, 0, 0, 1, 1, 3, 3, 6, 6, 10, 11, 16, 18, 25, 30, 38, 47, 59, 74, 90, 112, 136, 171, 203, 253, 299, 372, 438, 536, 631, 767, 900, 1085, 1271, 1521, 1774, 2112, 2463, 2910, 3389, 3977, 4627, 5408, 6276, 7304, 8459, 9808, 11338, 13099, 15112, 17404, 20044, 23018, 26450, 30299, 34746, 39711, 45452, 51832
COMMENTS
These are partitions without a neighborless part, where a part x is neighborless if neither x - 1 nor x + 1 are parts. The first counted partition that does not cover an interval is (5,4,2,1).
EXAMPLE
The a(0) = 1 through a(9) = 11 partitions:
() . . (21) (211) (32) (321) (43) (332) (54)
(221) (2211) (322) (3221) (432)
(2111) (21111) (2221) (22211) (3222)
(3211) (32111) (3321)
(22111) (221111) (22221)
(211111) (2111111) (32211)
(222111)
(321111)
(2211111)
(21111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Function[ptn, !Or@@Table[!MemberQ[ptn, x-1]&&!MemberQ[ptn, x+1], {x, Union[ptn]}]]]], {n, 0, 30}]
CROSSREFS
These partitions are ranked by A356736.
Number of strict integer partitions of n with at least one neighborless part.
+0
10
0, 1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 13, 17, 20, 24, 30, 36, 41, 52, 60, 71, 84, 100, 114, 137, 158, 183, 214, 248, 283, 330, 379, 432, 499, 570, 648, 742, 846, 955, 1092, 1234, 1395, 1580, 1786, 2005, 2270, 2548, 2861, 3216, 3610, 4032, 4526, 5055, 5642, 6304, 7031, 7820, 8720, 9694
COMMENTS
A part x is neighborless if neither x - 1 nor x + 1 are parts.
EXAMPLE
The a(0) = 0 through a(9) = 6 partitions:
. (1) (2) (3) (4) (5) (6) (7) (8) (9)
(31) (41) (42) (52) (53) (63)
(51) (61) (62) (72)
(421) (71) (81)
(431) (531)
(521) (621)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Function[ptn, UnsameQ@@ptn&&Or@@Table[!MemberQ[ptn, x-1]&&!MemberQ[ptn, x+1], {x, Union[ptn]}]]]], {n, 0, 30}]
Number of partitions of n into 3 unordered relatively prime parts.
+0
22
1, 1, 2, 2, 4, 4, 6, 6, 10, 8, 14, 12, 16, 16, 24, 18, 30, 24, 32, 30, 44, 32, 50, 42, 54, 48, 70, 48, 80, 64, 80, 72, 96, 72, 114, 90, 112, 96, 140, 96, 154, 120, 144, 132, 184, 128, 196, 150, 192, 168, 234, 162, 240, 192, 240, 210, 290, 192, 310, 240, 288, 256, 336, 240, 374
FORMULA
G.f. for the number of partitions of n into m unordered relatively prime parts is Sum(moebius(k)*x^(m*k)/Product(1-x^(i*k), i=1..m), k=1..infinity). - Vladeta Jovovic, Dec 21 2004 a(n) = (n^2/12)*Product_{prime p|n} (1 - 1/p^2) = A007434(n)/12 for n > 3 (proved by Mohamed El Bachraoui). [ Jonathan Sondow, May 27 2009] a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} floor(1/gcd(i,k,n-i-k)). - Wesley Ivan Hurt, Jan 02 2021
EXAMPLE
The a(3) = 1 through a(13) = 14 triples (A = 10, B = 11):
111 211 221 321 322 332 432 433 443 543 544
311 411 331 431 441 532 533 552 553
421 521 522 541 542 651 643
511 611 531 631 551 732 652
621 721 632 741 661
711 811 641 831 733
722 921 742
731 A11 751
821 832
911 841
922
931
A21
B11
(End)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n, {3}], GCD@@#==1&]], {n, 3, 50}] (* Gus Wiseman, Oct 08 2020 *)
CROSSREFS
A000837 counts these partitions of any length.
A001399(n-3) does not require relative primality.
A284825 counts the case that is also pairwise non-coprime.
A307719 is the pairwise coprime instead of relatively prime version.
A337599 is the pairwise non-coprime instead of relative prime version.
A008284 counts partitions by sum and length.
A078374 counts relatively prime strict partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000010, A000217, A007434, A055684, A078374, A200976, A220377, A302698, A327516, A337563, A337600, A337605.
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