A137921 -id:A137921 - cp's OEIS frontend

A173569 Least number k such that the function described in A137921 equals n.

1, 3, 8, 15, 24, 36, 48, 72, 96, 140, 120, 180, 280, 240, 336, 420, 360, 480, 672, 864, 900, 720, 840, 1008, 1080, 1260, 1980, 1440, 2340, 1680, 3744, 2160, 3024, 2880, 4032, 2520, 3780, 3360, 4680, 4320, 6048, 9450, 9072, 6300, 7200, 6720, 5040, 8640

Offset: 1

Robert G. Wilson v, Feb 22 2010

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..2665 .

Cf. A137921, A173570.

Cf. A137921: a(n) = number of "divisor islands" of n. A divisor island is any set of consecutive divisors of a number where no pairs of consecutive divisors in the set are separated by 2 or more. [From Robert G. Wilson v, May 20 2010]

f[n_] := Length@ Split[ Divisors@n, #2 - #1 == 1 &] (* f(n) from Dr. Bobby R. Treat *); t = Table[0, {100}]; k = 1; While[k < 10^9, a = f@k; If[a < 101 && t[[a]] == 0, t[[a]] = k; Print[{k, a}]]; k++ ]; t

A174199 Bisection of A137921.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 3, 5, 3, 5, 5, 5, 3, 6, 3, 6, 5, 5, 3, 7, 5, 5, 6, 6, 3, 7, 3, 6, 6, 5, 7, 8, 3, 5, 6, 8, 3, 8, 3, 7, 8, 5, 3, 9, 5, 7, 6, 7, 3, 9, 6, 8, 6, 5, 3, 11, 3, 5, 9, 7, 7, 8, 3, 7, 6, 10, 3, 11, 3, 5, 9, 7, 7, 8, 3, 10, 8, 5, 3, 11, 7, 5, 6, 9, 3, 12, 6, 7, 6, 5, 7, 11, 3, 8, 10, 10, 3, 9, 3, 9, 11, 5, 3, 12, 3, 9, 6, 10, 3, 9, 7, 7, 10, 5, 7, 14

Offset: 1

N. J. A. Sloane, Nov 26 2010

nonn

Since the other bisection (A099774) is known - it is the bisection of A000005 - the obvious problem is to find a formula for this sequence.

N. J. A. Sloane, Table of n, a(n) for n = 1..5000

Cf. A000005, A001620, A099774, A137921.

a(n) = sumdiv(2*n, d, ((2*n)%(d+1) != 0)); \\ Amiram Eldar, Jan 18 2024

From Amiram Eldar, Jan 18 2024: (Start)
a(n) = A137921(2*n).
Sum_{k=1..n} a(k) ~ (n/2) * (3*log(n) + 6*gamma - 7 + log(2)), where gamma is Euler's constant (A001620). (End)

A001227 Number of odd divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 1, 4, 2, 4, 3, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 2, 3, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 6, 1, 4, 4, 2, 2, 4, 4, 2, 3, 2, 2, 6, 2, 4, 4, 2, 2, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4, 2, 2, 3, 6, 3, 2, 4, 2, 2, 8

Offset: 1

N. J. A. Sloane

Also (1) number of ways to write n as difference of two triangular numbers (A000217), see A136107; (2) number of ways to arrange n identical objects in a trapezoid. - Tom Verhoeff
Also number of partitions of n into consecutive positive integers including the trivial partition of length 1 (e.g., 9 = 2+3+4 or 4+5 or 9 so a(9)=3). (Useful for cribbage players.) See A069283. - Henry Bottomley, Apr 13 2000
This has been described as Sylvester's theorem, but to reduce ambiguity I suggest calling it Sylvester's enumeration. - Gus Wiseman, Oct 04 2022
a(n) is also the number of factors in the factorization of the Chebyshev polynomial of the first kind T_n(x). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
Number of factors in the factorization of the polynomial x^n+1 over the integers. See also A000005. - T. D. Noe, Apr 16 2003
a(n) = 1 if and only if n is a power of 2 (see A000079). - Lekraj Beedassy, Apr 12 2005
Number of occurrences of n in A049777. - Philippe Deléham, Jun 19 2005
For n odd, n is prime if and only if a(n) = 2. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 10 2005
Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly once. Example: a(9)=3 because we have [3,3,2,1],[2,2,2,2,1] and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 07 2006
Also the number of factors of the n-th Lucas polynomial. - T. D. Noe, Mar 09 2006
Lengths of rows of triangle A182469;
Denoted by Delta_0(n) in Glaisher 1907. - Michael Somos, May 17 2013
Also the number of partitions p of n into distinct parts such that max(p) - min(p) < length(p). - Clark Kimberling, Apr 18 2014
Row sums of triangle A247795. - Reinhard Zumkeller, Sep 28 2014
Row sums of triangle A237048. - Omar E. Pol, Oct 24 2014
A069288(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015
A000203, A000593 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
a(n) is equal to the number of ways to write 2*n-1 as (4*x + 2)*y + 4*x + 1 where x and y are nonnegative integers. Also a(n) is equal to the number of distinct values of k such that k/(2*n-1) + k divides (k/(2*n-1))^(k/(2*n-1)) + k, (k/(2*n-1))^k + k/(2*n-1) and k^(k/(2*n-1)) + k/(2*n-1). - Juri-Stepan Gerasimov, May 23 2016, Jul 15 2016
Also the number of odd divisors of n*2^m for m >= 0. - Juri-Stepan Gerasimov, Jul 15 2016
a(n) is odd if and only if n is a square or twice a square. - Juri-Stepan Gerasimov, Jul 17 2016
a(n) is also the number of subparts in the symmetric representation of sigma(n). For more information see A279387 and A237593. - Omar E. Pol, Nov 05 2016
a(n) is also the number of partitions of n into an odd number of equal parts. - Omar E. Pol, May 14 2017 [This follows from the g.f. Sum_{k >= 1} x^k/(1-x^(2*k)). - N. J. A. Sloane, Dec 03 2020]

			G.f. = q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + 2*q^10 + ...
From _Omar E. Pol_, Nov 30 2020: (Start)
For n = 9 there are three odd divisors of 9; they are [1, 3, 9]. On the other hand there are three partitions of 9 into consecutive parts: they are [9], [5, 4] and [4, 3, 2], so a(9) = 3.
Illustration of initial terms:
                              Diagram
   n   a(n)                         _
   1     1                        _|1|
   2     1                      _|1 _|
   3     2                    _|1  |1|
   4     1                  _|1   _| |
   5     2                _|1    |1 _|
   6     2              _|1     _| |1|
   7     2            _|1      |1  | |
   8     1          _|1       _|  _| |
   9     3        _|1        |1  |1 _|
  10     2      _|1         _|   | |1|
  11     2    _|1          |1   _| | |
  12     2   |1            |   |1  | |
...
a(n) is the number of horizontal line segments in the n-th level of the diagram. For more information see A286001. (End)

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 487 Entry 47.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 306.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4).
Ronald. L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
K. S. Brown's Mathpages, Partitions into Consecutive Integers.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 18.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
A. Heiligenbrunner, Sum of adjacent numbers (in German).
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 11.
Gerzson Kéri, The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_o(n).
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976.
N. J. A. Sloane, Transforms.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.
Eric Weisstein's World of Mathematics, Binomial Number and Odd Divisor Function.
Eric Weisstein's World of Mathematics, q-Polygamma Function.
Index entries for "core" sequences.
Index entries for sequences mentioned by Glaisher.

Cf. A000005, A000079, A000593, A010054 (char. func.), A038547 (positions of first appearances), A050999, A051000, A051001, A051002, A051731, A054844, A069283, A069288, A109814, A115369, A118235, A118236, A125911, A136655, A183063, A183064, A237593, A247795, A272887, A273401, A279387, A286001.
Cf. A000203, A001620, A002162, A053866, A060831.
If this sequence counts gapless sets by sum (by Sylvester's enumeration), these sets are ranked by A073485 and A356956. See also A055932, A066311, A073491, A107428, A137921, A333217, A356224, A356841, A356845.
Dirichlet inverse is A327276.

a001227 = sum . a247795_row
-- Reinhard Zumkeller, Sep 28 2014, May 01 2012, Jul 25 2011

[NumberOfDivisors(n)/Valuation(2*n, 2): n in [1..100]]; // Vincenzo Librandi, Jun 02 2019

for n from 1 by 1 to 100 do s := 0: for d from 1 by 2 to n do if n mod d = 0 then s := s+1: fi: od: print(s); od:
A001227 := proc(n) local a,d;
    a := 1 ;
    for d in ifactors(n)[2] do
        if op(1,d) > 2 then
            a := a*(op(2,d)+1) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jun 18 2015

f[n_] := Block[{d = Divisors[n]}, Count[ OddQ[d], True]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 27 2004 *)
Table[Total[Mod[Divisors[n], 2]],{n,105}] (* Zak Seidov, Apr 16 2010 *)
f[n_] := Block[{d = DivisorSigma[0, n]}, If[ OddQ@ n, d, d - DivisorSigma[0, n/2]]]; Array[f, 105] (* Robert G. Wilson v *)
a[ n_] := Sum[  Mod[ d, 2], { d, Divisors[ n]}]; (* Michael Somos, May 17 2013 *)
a[ n_] := DivisorSum[ n, Mod[ #, 2] &]; (* Michael Somos, May 17 2013 *)
Count[Divisors[#],?OddQ]&/@Range[110] (* _Harvey P. Dale, Feb 15 2015 *)
(* using a262045 from A262045 to compute a(n) = number of subparts in the symmetric representation of sigma(n) *)
(* cl = current level, cs = current subparts count *)
a001227[n_] := Module[{cs=0, cl=0, i, wL, k}, wL=a262045[n]; k=Length[wL]; For[i=1, i<=k, i++, If[wL[[i]]>cl, cs++; cl++]; If[wL[[i]]Hartmut F. W. Hoft, Dec 16 2016 *)
a[n_] := DivisorSigma[0, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

{a(n) = sumdiv(n, d, d%2)}; /* Michael Somos, Oct 06 2007 */

{a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( 4, p) * X))[n]}; /* Michael Somos, Oct 06 2007 */

a(n)=numdiv(n>>valuation(n,2)) \\ Charles R Greathouse IV, Mar 16 2011

a(n)=sum(k=1,round(solve(x=1,n,x*(x+1)/2-n)),(k^2-k+2*n)%(2*k)==0) \\ Charles R Greathouse IV, May 31 2013

a(n)=sumdivmult(n,d,d%2) \\ Charles R Greathouse IV, Aug 29 2013

from functools import reduce
from operator import mul
from sympy import factorint
def A001227(n): return reduce(mul,(q+1 for p, q in factorint(n).items() if p > 2),1) # Chai Wah Wu, Mar 08 2021

def A001227(n): return len([1 for d in divisors(n) if is_odd(d)])
[A001227(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012

Dirichlet g.f.: zeta(s)^2*(1-1/2^s).
Comment from N. J. A. Sloane, Dec 02 2020: (Start)
By counting the odd divisors f n in different ways, we get three different ways of writing the ordinary generating function.  It is:
A(x) =  x + x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + x^8 + 3*x^9 + 2*x^10 + ...
  = Sum_{k >= 1} x^(2*k-1)/(1-x^(2*k-1))
  = Sum_{k >= 1} x^k/(1-x^(2*k))
  = Sum_{k >= 1} x^(k*(k+1)/2)/(1-x^k)  [Ramanujan, 2nd notebook, p. 355.].
(This incorporates comments from Vladeta Jovovic, Oct 16 2002 and  Michael Somos, Oct 30 2005.) (End)
G.f.: x/(1-x) + Sum_{n>=1} x^(3*n)/(1-x^(2*n)), also L(x)-L(x^2) where L(x) = Sum_{n>=1} x^n/(1-x^n). - Joerg Arndt, Nov 06 2010
a(n) = A000005(n)/(A007814(n)+1) = A000005(n)/A001511(n).
Multiplicative with a(p^e) = 1 if p = 2; e+1 if p > 2. - David W. Wilson, Aug 01 2001
a(n) = A000005(A000265(n)). - Lekraj Beedassy, Jan 07 2005
Moebius transform is period 2 sequence [1, 0, ...] = A000035, which means a(n) is the Dirichlet convolution of A000035 and A057427.
a(n) = A113414(2*n). - N. J. A. Sloane, Jan 24 2006 (corrected Nov 10 2007)
a(n) = A001826(n) + A001842(n). - Reinhard Zumkeller, Apr 18 2006
Sequence = M*V = A115369 * A000005, where M = an infinite lower triangular matrix and V = A000005, d(n); as a vector: [1, 2, 2, 3, 2, 4, ...]. - Gary W. Adamson, Apr 15 2007
Equals A051731 * [1,0,1,0,1,...]; where A051731 is the inverse Mobius transform. - Gary W. Adamson, Nov 06 2007
a(n) = A000005(n) - A183063(n).
a(n) = d(n) if n is odd, or d(n) - d(n/2) if n is even, where d(n) is the number of divisors of n (A000005). (See the Weisstein page.) - Gary W. Adamson, Mar 15 2011
Dirichlet convolution of A000005 and A154955 (interpreted as a flat sequence). - R. J. Mathar, Jun 28 2011
a(A000079(n)) = 1; a(A057716(n)) > 1; a(A093641(n)) <= 2; a(A038550(n)) = 2; a(A105441(n)) > 2; a(A072502(n)) = 3. - Reinhard Zumkeller, May 01 2012
a(n) = 1 + A069283(n). - R. J. Mathar, Jun 18 2015
a(A002110(n)/2) = n, n >= 1. - Altug Alkan, Sep 29 2015
a(n*2^m) = a(n*2^i), a((2*j+1)^n) = n+1 for m >= 0, i >= 0 and j >= 0. a((2*x+1)^n) = a((2*y+1)^n) for positive x and y. - Juri-Stepan Gerasimov, Jul 17 2016
Conjectures: a(n) = A067742(n) + 2*A131576(n) = A082647(n) + A131576(n). - Omar E. Pol, Feb 15 2017
a(n) = A000005(2n) - A000005(n) = A099777(n)-A000005(n). - Danny Rorabaugh, Oct 03 2017
L.g.f.: -log(Product_{k>=1} (1 - x^(2*k-1))^(1/(2*k-1))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
G.f.: (psi_{q^2}(1/2) + log(1-q^2))/log(q), where psi_q(z) is the q-digamma function. - Michael Somos, Jun 01 2019
a(n) = A003056(n) - A238005(n). - Omar E. Pol, Sep 12 2021
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma + log(2)/2 - 1/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = log(2) (A002162). - Amiram Eldar, Mar 01 2023
a(n) = Sum_{i=1..n} (-1)^(i+1)*A135539(n,i). - Ridouane Oudra, Apr 13 2023

A287170 a(n) = number of runs of consecutive prime numbers among the prime divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2

Offset: 1

Rémy Sigrist, Jun 04 2017

nonn

a(n) = 0 iff n = 1.
a(n) = 1 iff n belongs to A073491.
a(p) = 1 for any prime p.
a(A002110(n)) = 1 for any n > 0.
a(n!) = 1 for any n > 1.
a(A066205(n)) = n for any n > 0.
a(n) = a(A007947(n)) for any n > 0.
a(n) = a(A003961(n)) for any n > 0.
a(n*m) <= a(n) + a(m) for any n > 0 and m > 0.
Each number n can be uniquely represented as a product of a(n) distinct terms from A073491; this representation is minimal relative to the number of terms.

			See illustration of the first terms in the Links section.
The prime indices of 18564 are {1,1,2,4,6,7}, which separate into maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 3; this corresponds to the ordered factorization 18564 = 12 * 7 * 221. - _Gus Wiseman_, Sep 03 2022

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, Scatterplot of the ordinal transform of the first 1000000 terms

Cf. A002110, A003961, A007947, A069010, A087207.
Positions of first appearances are A066205.
These are the row-lengths of A356226 and A356234. Other statistics are:
- length: A287170 (this sequence)
- minimum: A356227
- maximum: A356228
- bisected length: A356229
- standard composition: A356230
- Heinz number: A356231
- positions of first appearances: A356603 or A356232 (sorted)
A001222 counts prime factors, distinct A001221.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, complement A073492.
Cf. A000005, A053251, A055932, A061395, A073493, A107428, A132747, A137921, A193829, A286470.

Table[Length[Select[First/@If[n==1,{},FactorInteger[n]],!Divisible[n,NextPrime[#]]&]],{n,30}] (* Gus Wiseman, Sep 03 2022 *)

a(n) = my (f=factor(n)); if (#f~==0, return (0), return (#f~ - sum(i=1, #f~-1, if (primepi(f[i,1])+1 == primepi(f[i+1,1]), 1, 0))))

from sympy import factorint, primepi
def a087207(n):
    f=factorint(n)
    return sum([2**primepi(i - 1) for i in f])
def a069010(n): return sum(1 for d in bin(n)[2:].split('0') if len(d)) # this function from Chai Wah Wu
def a(n): return a069010(a087207(n)) # Indranil Ghosh, Jun 06 2017

a(n) = A069010(A087207(n))

A129308 a(n) is the number of positive integers k such that k*(k+1) divides n.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 5, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 1, 0

Offset: 1

Leroy Quet, May 26 2007

nonn

The usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception.

In other words, a(n) is the number of oblong numbers (A002378) dividing n. - Bernard Schott, Jul 29 2022

			The divisors of 20 are 1,2,4,5,10,20. Of these there are two that are of the form k(k+1): 2 = 1*2 and 20 = 4*5. So a(2) = 2.

Ray Chandler, Table of n, a(n) for n = 1..10000
P. Erdős and R. R. Hall, On some unconventional problems on the divisors of integers, J. Austral. Math. Soc., Ser. A, 25, 479-485 (1978).
MathOverflow, On the number of consecutive divisors of an integer.

Positions of 0's and 1's are A088725, whose characteristic function is A360128.
First appearance of n is A287142(n), with sorted version A328450.
The longest run of divisors of n has length A055874(n).
One less than A195155.
Cf. A000005, A002378, A003601, A007862, A027750, A033676, A060680, A060681, A072627, A088722, A181063, A199970, A328026, A328165, A328166.
Cf. A005369, A059841, A287142, A344005.

a = {}; For[n = 1, n < 90, n++, k = 1; co = 0; While[k < Sqrt[n], If[IntegerQ[ n/(k*(k + 1))], co++ ]; k++ ]; AppendTo[a, co]]; a (* Stefan Steinerberger, May 27 2007 *)
Table[Count[Differences[Divisors[n]],1],{n,30}] (* Gus Wiseman, Oct 15 2019 *)

a(n)=sumdiv(n, d, n%(d+1)==0); \\ Michel Marcus, Jan 06 2015

from itertools import pairwise
from sympy import divisors
def A129308(n): return 0 if n&1 else sum(1 for a, b in pairwise(divisors(n)) if a+1==b) # Chai Wah Wu, Jun 09 2025

a(2n-1) = 0; a(2n) = A007862(n). - Ray Chandler, Jun 24 2008
G.f.: Sum_{n>=1} x^(n*(n+1))/(1-x^(n*(n+1))). - Joerg Arndt, Jan 30 2011 [modified by Ilya Gutkovskiy, Apr 14 2021]
a(n) = A000005(n) - A137921(n), where A137921(n) is the number of maximal runs of successive divisors of n. - Gus Wiseman, Oct 15 2019
a(n) = Sum_{d|n} A005369(d). - Ridouane Oudra, Jan 22 2021
a(n) = A195155(n)-1. - Antti Karttunen, Feb 21 2023
From Amiram Eldar, Dec 31 2023: (Start)
a(n) = A088722(n) + A059841(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. (End)

More terms from Stefan Steinerberger, May 27 2007

Extended by Ray Chandler, Jun 24 2008

A356224 Number of divisors of n whose prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 1, 4, 1, 2, 1, 5, 1, 2, 1, 5, 1, 4, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 4, 1, 6, 1, 2, 1, 7, 1, 2, 1, 4, 1, 3, 1, 3, 1, 2, 1, 9, 1, 2, 1, 3, 1, 5, 1, 4, 1, 2, 1, 7, 1, 2, 1, 7, 1, 3, 1, 3, 1, 2, 1, 10, 1, 2, 1, 3, 1, 3, 1, 5, 1, 2, 1, 5, 1, 2, 1

Offset: 1

Gus Wiseman, Aug 04 2022

nonn

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.

			The a(n) gapless divisors of n = 1..24:
  1  2  1  4  1  6  1  8  1  2  1  12  1  2  1  16  1  18  1  4  1  2  1  24
     1     2     2     4     1     6      1     8      6      2     1     12
           1     1     2           4            4      2      1           8
                       1           2            2      1                  6
                                   1            1                         4
                                                                          2
                                                                          1
For example, the divisors of 12 are {1,2,3,4,6,12}, of which {1,2,4,6,12} belong to A055932, so a(12) = 5.

These divisors belong to A055932, a subset of A073491 (complement A073492).
The complement is A356225.
A001223 lists the prime gaps.
A328338 has third-largest divisor prime.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A000005, A001222, A028334, A029709, A055874, A056239, A070824, A112798, A119313, A137921, A287170, A289509, A356223.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
Table[Length[Select[Divisors[n],normQ[primeMS[#]]&]],{n,100}]

A107428 Number of gap-free compositions of n.

Original entry on oeis.org

1, 2, 4, 6, 11, 21, 39, 71, 141, 276, 542, 1070, 2110, 4189, 8351, 16618, 33134, 66129, 131937, 263483, 526453, 1051984, 2102582, 4203177, 8403116, 16800894, 33593742, 67174863, 134328816, 268624026, 537192064, 1074288649, 2148414285, 4296543181, 8592585289

Offset: 1

N. J. A. Sloane, May 26 2005

nonn

A gap-free composition contains all the parts between its smallest and largest part. a(5)=11 because we have: 5, 3+2, 2+3, 2+2+1, 2+1+2, 1+2+2, 2+1+1+1, 1+2+1+1, 1+1+2+1, 1+1+1+2, 1+1+1+1+1. - Geoffrey Critzer, Apr 13 2014

			From _Gus Wiseman_, Oct 04 2022: (Start)
The a(0) = 1 through a(5) = 11 gap-free compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (22)    (23)
                 (21)   (112)   (32)
                 (111)  (121)   (122)
                        (211)   (212)
                        (1111)  (221)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Alois P. Heinz, Plot of (a(n)-2^(n-2))/2^(n-2) for n = 42..1000
P. Hitczenko and A. Knopfmacher, Gap-free compositions and gap-free samples of geometric random variables, Discrete Math., 294 (2005), 225-239.

The unordered version (partitions) is A034296, ranked by A073491.
The initial case is A107429, unordered A000009, ranked by A333217.
The unordered complement is counted by A239955, ranked by A073492.
These compositions are ranked by A356841.
The complement is counted by A356846, ranked by A356842
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
Cf. A055932, A073493, A137921, A251729, A356224, A356843, A356845.

b:= proc(n, i, t) option remember; `if`(n=0, t!,
      `if`(i<1 or n add(b(n, i, 0), i=1..n):
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 14 2014

Table[Length[Select[Level[Map[Permutations,IntegerPartitions[n]],{2}],Length[Union[#]]==Max[#]-Min[#]+1&]],{n,1,20}] (* Geoffrey Critzer, Apr 13 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, t!, If[i < 1 || n < i, 0, Sum[b[n - i*j, i - 1, t + j]/j!, {j, 1, n/i}]]]; a[n_] := Sum[b[n, i, 0], {i, 1, n}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

a(n) ~ 2^(n-2). - Alois P. Heinz, Dec 07 2014

G.f.: Sum_{j>0} Sum_{k>=j} C({j..k},x) where C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/(1 - Sum_{i in {s}} (x^i)) is the g.f. for compositions such that the set of parts equals {s} with C({},x) = 1. - John Tyler Rascoe, Jun 01 2024

More terms from Vladeta Jovovic, May 26 2005

A356226 Irregular triangle giving the lengths of maximal gapless submultisets of the prime indices of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 4, 2, 1, 1, 3, 2, 1, 1, 3, 1, 5, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 5, 2, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 3, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 6

Offset: 1

Gus Wiseman, Aug 10 2022

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.

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.

			Triangle  begins: {}, {1}, {1}, {2}, {1}, {2}, {1}, {3}, {2}, {1,1}, {1}, {3}, {1}, {1,1}, {2}, {4}, {1}, {3}, {1}, {2,1}, ... For example, the prime indices of 20 are {1,1,3}, which separates into maximal gapless submultisets {{1,1},{3}}, so row 20 is (2,1).
The prime indices of 18564 are {1,1,2,4,6,7}, which separates into {1,1,2}, {4}, {6,7}, so row 18564 is (3,1,2). This corresponds to the factorization 18564 = 12 * 7 * 221.

Row sums are A001222.
Singleton row positions are A073491, complement A073492.
Length-2,3,4 row positions are A073493-A073495.
Row lengths are A287170, firsts A066205.
Row minima are A356227.
Row maxima are A356228.
Bisected run-lengths are A356229.
Standard composition numbers of rows are A356230.
Heinz numbers of rows are A356231.
Positions of first appearances are A356232.
A001221 counts distinct prime factors, with sum A001414.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A055874, A060680-A060683, A137921, A193829, A286470, A328166.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length/@Split[primeMS[n],#1>=#2-1&],{n,100}]

A356225 Number of divisors of n whose prime indices do not cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 1, 2, 3, 0, 1, 2, 1, 3, 3, 2, 1, 1, 2, 2, 3, 3, 1, 4, 1, 0, 3, 2, 3, 2, 1, 2, 3, 4, 1, 5, 1, 3, 5, 2, 1, 1, 2, 4, 3, 3, 1, 3, 3, 4, 3, 2, 1, 5, 1, 2, 5, 0, 3, 5, 1, 3, 3, 6, 1, 2, 1, 2, 5, 3, 3, 5, 1, 5, 4, 2, 1, 7, 3, 2, 3

Offset: 1

Gus Wiseman, Aug 13 2022

nonn

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.

			The a(70) = 6 divisors: 5, 7, 10, 14, 35, 70.

These divisors belong to the complement of A055932, a subset of A073491.
These divisors belong to A080259, a superset of A073492.
The complement is counted by A356224.
A001223 lists the prime gaps.
A328338 has third-largest divisor prime, smallest A119313.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A000005, A001222, A028334, A055874, A056239, A070824, A112798, A137921, A287170, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
Table[Length[Select[Divisors[n],!normQ[primeMS[#]]&]],{n,100}]

a(n) = A000005(n) - A356224(n).

A356606 Number of strict integer partitions of n where all parts have neighbors.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 24 2022

nonn

A part x has a neighbor if either x - 1 or x + 1 is a part.

			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)

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 301 terms from John Tyler Rascoe)
John Tyler Rascoe, Python program

This is the strict case of A355393 and A355394.
The complement is counted by A356607, non-strict A356235 and A356236.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A137921, A325160, A328171, A328172, A328187, A328220, A328221, A356237.

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}]

Python
```
# see linked program
```

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

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A356224 Number of divisors of n whose prime indices cover an initial interval of positive integers.

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