cp's OEIS Frontend

A070824 Number of divisors of n which are > 1 and < n (nontrivial divisors).

%I A070824 #81 Oct 14 2024 02:55:11
%S A070824 0,0,0,1,0,2,0,2,1,2,0,4,0,2,2,3,0,4,0,4,2,2,0,6,1,2,2,4,0,6,0,4,2,2,
%T A070824 2,7,0,2,2,6,0,6,0,4,4,2,0,8,1,4,2,4,0,6,2,6,2,2,0,10,0,2,4,5,2,6,0,4,
%U A070824 2,6,0,10,0,2,4,4,2,6,0,8,3,2,0,10,2,2
%N A070824 Number of divisors of n which are > 1 and < n (nontrivial divisors).
%C A070824 These are sometimes called the proper divisors, but see A032741 for the usual meaning of that term.
%C A070824 a(n) = number of ordered factorizations of n into two factors, n = 2, 3, ... If n has the prime factorization n=Product p^e(j), j=1..r, the number of compositions of the vector (e(1), ..., e(r)) equals the number of ordered factorizations of n. Andrews (1998, page 59) gives a formula for the number of m-compositions of (e(1), ..., e(r)) which equals the number f(n,m) of ordered m-factorizations of n. But with m=2 the formula reduces to f(n,2) = d(n)-2 = a(n). - _Augustine O. Munagi_, Mar 31 2005
%C A070824 a(n) = 0 if and only if n is 1 or prime. - _Jon Perry_, Nov 08 2008
%C A070824 For n > 2: number of zeros in n-th row of triangle A051778. - _Reinhard Zumkeller_, Dec 03 2014
%C A070824 a(n) = number of partitions of n in which largest and least parts occur exactly once and their difference is 2. Example: a(12) = 4 because we have [7,5], [5,4,3], [4,3,3,2], and [3,2,2,2,2,1]. In general, if d is a nontrivial divisor of n, then [d+1,{d}^(n/d-2),d-1] is a partition of n of the prescribed type. - _Emeric Deutsch_, Nov 03 2015
%C A070824 Absolute values of the inverse Möbius transform of (-1)^prime(n), n >= 2. - _Wesley Ivan Hurt_, Jun 22 2024
%D A070824 George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading 1976; reprinted, Cambridge University Press, Cambridge, 1984, 1998.
%H A070824 T. D. Noe, <a href="/A070824/b070824.txt">Table of n, a(n) for n = 1..10000</a>
%H A070824 Rahul Chhimpa and Avinash Chand Yadav, <a href="https://arxiv.org/abs/2410.05699">Scaling behavior in the number theoretic division model of self-organized criticality</a>, arXiv:2410.05699 [cond-mat.stat-mech], 2024. See p. 2.
%H A070824 Arnold Knopfmacher and Michael Mays, <a href="https://web.archive.org/web/20140119155532/http://www.mathematica-journal.com/issue/v10i1/contents/Factorizations/Factorizations_3.html">Ordered and Unordered Factorizations of Integers</a>, The Mathematica Journal, Vol 10 (1), 2006 (Wayback Machine link); <a href="https://www.researchgate.net/publication/255662882_Ordered_and_Unordered_Factorizations_of_Integers">ResearchGate link</a>.
%F A070824 a(n) = A000005(n)-2, n>=2 (with the number-of-divisors function d(n) = A000005(n)).
%F A070824 a(n) = d(n)-2, for n>=2, where d(n) is the number-of-divisors function. E.g., a(12) = 4 because 12 has 4 ordered factorizations into two factors: 2*6, 6*2, 3*4, 4*3. - _Augustine O. Munagi_, Mar 31 2005
%F A070824 G.f.: Sum_{k>=2} x^(2k)/(1-x^k). - _Jon Perry_, Nov 08 2008
%F A070824 Dirichlet generating function: (zeta(s)-1)^2. - _Mats Granvik_ May 25 2013
%F A070824 Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma - 3)*n, where gamma is Euler's constant (A001620). - _Amiram Eldar_, Nov 27 2022
%F A070824 a(n) = abs( Sum_{d|n} (-1)^prime(d) ), n >= 2 with a(1) = 0. - _Wesley Ivan Hurt_, Jun 22 2024
%F A070824 a(n) = Sum_{k=2..n-1} floor(n/k) - floor((n-1)/k), see Chhimpa and Yadav. - _Stefano Spezia_, Oct 13 2024
%e A070824 a(12) = 4 with the nontrivial divisors 2,3,4,6.
%e A070824 a(24) = 6 = card({{2,12},{3,8},{4,6},{6,4},{8,3},{12,2}}). - _Peter Luschny_, Nov 14 2011
%p A070824 0, seq(numtheory[tau](n)-2,n=2..100); # _Augustine O. Munagi_, Mar 31 2005
%t A070824 Join[{0},Rest[DivisorSigma[0,Range[90]]-2]] (* _Harvey P. Dale_, Jun 23 2012 *)
%t A070824 a[ n_] := SeriesCoefficient[ Sum[x^(2 k)/(1 - x^k), {k, 2, n/2}], {x, 0, n}]; (* _Michael Somos_, Jun 24 2019 *)
%o A070824 (Haskell) a070824 n = if n == 1 then 0 else length $ tail $ a027751_row n -- _Reinhard Zumkeller_, Dec 03 2014
%o A070824 (PARI) {a(n) = if( n<1, 0, my(v = vector(n, i, i>1)); dirmul(v, v)[n])}; /* _Michael Somos_, Jun 24 2019 */
%o A070824 (PARI) apply( A070824(n)=numdiv(n+(n<2))-2, [1..90]) \\ _M. F. Hasler_, Oct 11 2019
%o A070824 (Python)
%o A070824 from sympy import divisor_count
%o A070824 def A070824(n): return 0 if n == 1 else divisor_count(n)-2 # _Chai Wah Wu_, Jun 03 2022
%Y A070824 Cf. A000005, A001620, A074206, A032741, A200213.
%Y A070824 First column in the matrix power A175992^2.
%Y A070824 Row sums of A175992 starting from the second column.
%Y A070824 Cf. A027751, A051778.
%Y A070824 Column k=2 of A251683.
%K A070824 nonn,easy
%O A070824 1,6
%A A070824 _Wolfdieter Lang_, May 08 2002
%E A070824 a(1)=0 added by _Peter Luschny_, Nov 14 2011
%E A070824 Several minor edits by _M. F. Hasler_, Oct 14 2019