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 A001227 Number of odd divisors of n. 282
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Also (1) number of ways to write n as difference of two triangular numbers (A000217); (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.) - Henry Bottomley, Apr 13 2000 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 for n = A000079. - Lekraj Beedassy, Apr 12 2005 Number of occurrences of n in A049777. - Philippe Deléham, Jun 19 2005 For n odd, n is prime iff the n-th term of the sequence is 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; 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 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 iff 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 REFERENCES 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). Graham, Knuth and 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. LINKS N. J. A. Sloane, Table of n, a(n) for n = 1..10000 K. S. Brown's Mathpages, Partitions into Consecutive Integers 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). 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, #99.1.6. Eric Weisstein's World of Mathematics, Binomial Number and Odd Divisor Function. Eric Weisstein's World of Mathematics, q-Polygamma Function. FORMULA Dirichlet g.f.: zeta(s)^2*(1-1/2^s). 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 G.f.: Sum_{n>=1} x^n/(1-x^(2*n)). - Vladeta Jovovic, Oct 16 2002 a(n) = A000005(A000265(n)). - Lekraj Beedassy, Jan 07 2005 G.f.: Sum_{k>0} x^(2k-1)/(1-x^(2k-1)) = Sum_{k>0} x^(k*(k+1)/2)/(1-x^k). - Michael Somos, Oct 30 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 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) - A183063(n). a(n) = d(n) if n is odd, else d(n) - d(n/2). (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 G.f.: Sum_{k>0} x^(k*(k+1)/2) / (1 - x^k). [Ramanujan, 2nd notebook, p. 355.] - Michael Somos, Oct 25 2014 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). - 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 EXAMPLE 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 + ... MAPLE 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)     a := 1 ;     for d in ifactors(n) 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 MATHEMATICA 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 (* 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]]>valuation(n, 2)) \\ Charles R Greathouse IV, Mar 16 2011 (PARI) 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 (PARI) a(n)=sumdivmult(n, d, d%2) \\ Charles R Greathouse IV, Aug 29 2013 (Haskell) a001227 = sum . a247795_row -- Reinhard Zumkeller, Sep 28 2014, May 01 2012, Jul 25 2011 (Sage) def A001227(n) : return len(filter(is_odd, divisors(n))) [A001227(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012 (MAGMA) [NumberOfDivisors(n)/Valuation(2*n, 2): n in [1..100]]; // Vincenzo Librandi, Jun 02 2019 CROSSREFS Cf. A000005, A000593, A050999, A051000, A051001, A051002, A051731, A054844, A069283, A069288, A109814, A115369, A118235, A118236, A125911, A136655, A183063, A183064, A247795, A272887, A273401, A279387. Cf. A000203, A000593, A053866. Sequence in context: A225843 A301957 A318874 * A060764 A105149 A295894 Adjacent sequences:  A001224 A001225 A001226 * A001228 A001229 A001230 KEYWORD nonn,easy,nice,mult,core AUTHOR STATUS approved

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Last modified August 24 18:52 EDT 2019. Contains 326295 sequences. (Running on oeis4.)