A161840 -id:A161840 - cp's OEIS frontend

A072670 Number of ways to write n as i*j + i + j, 0 < i <= j.

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

Offset: 0

Reinhard Zumkeller, Jun 30 2002

nonn

a(n) is the number of partitions of n+1 with summands in arithmetic progression having common difference 2. For example a(29)=3 because there are 3 partitions of 30 that are in arithmetic progressions: 2+4+6+8+10, 8+10+12 and 14+16. - N-E. Fahssi, Feb 01 2008
From Daniel Forgues, Sep 20 2011: (Start)
a(n) is the number of nontrivial factorizations of n+1, in two factors.
a(n) is the number of ways to write n+1 as i*j + i + j + 1 = (i+1)(j+1), 0 < i <= j. (End)
a(n) is the number of ways to write n+1 as i*j, 1 < i <= j. - Arkadiusz Wesolowski, Nov 18 2012
For a generalization, see comment in A260804. - Vladimir Shevelev, Aug 04 2015
Number of partitions of n into 3 parts whose largest part is equal to the product of the other two. - Wesley Ivan Hurt, Jan 04 2022

			a(11)=2: 11 = 1*5 + 1 + 5 = 2*3 + 2 + 3.
From _Daniel Forgues_, Sep 20 2011 (Start)
Number of nontrivial factorizations of n+1 in two factors:
  0 for the unit 1 and prime numbers
  1 for a square: n^2 = n*n
  1 for 6 (2*3), 10 (2*5), 14 (2*7), 15 (3*5)
  1 for a cube: n^3 = n*n^2
  2 for 12 (2*6, 3*4), for 18 (2*9, 3*6) (End)

Robert Israel, Table of n, a(n) for n = 0..10000
Joseph W. Andrushkiw, Roman I. Andrushkiw and Clifton E. Corzatt, Representations of Positive Integers as Sums of Arithmetic Progressions, Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 245-248.
M. A. Nyblom and C. Evans, On the enumeration of partitions with summands in arithmetic progression, Australian Journal of Combinatorics, Vol. 28 (2003), pp. 149-159.
Vladimir Shevelev, Representation of positive integers by the form x1...xk+x1+...+xk, arXiv:1508.03970 [math.NT], 2015.

Cf. A001620, A067432, A066938, A072668, A006093, A072671, A161840, A260803, A260804.

0, seq(ceil(numtheory:-tau(n+1)/2)-1, n=1..100); # Robert Israel, Aug 04 2015

p2[n_] := 1/2 (Length[Divisors[n]] - 2 + ((-1)^(Length[Divisors[n]] + 1) + 1)/2); Table[p2[n + 1], {n, 0, 104}] (* N-E. Fahssi, Feb 01 2008 *)
Table[Ceiling[DivisorSigma[0, n + 1]/2] - 1, {n, 0, 104}] (* Arkadiusz Wesolowski, Nov 18 2012 *)

is_ok(k,i,j)=0=i&&k===i*j+i+j;
first(m)=my(v=vector(m,z,0));for(l=1,m,for(j=1,l,for(i=1,j,if(is_ok(l,i,j),v[l]++))));concat([0],v); /* Anders Hellström, Aug 04 2015 */

a(n)=(numdiv(n+1)+issquare(n+1))/2-1 \\ Charles R Greathouse IV, Jul 14 2017

a(n) = A038548(n+1) - 1.
From N-E. Fahssi, Feb 01 2008: (Start)
a(n) = p2(n+1), where p2(n) = (1/2)*(d(n) - 2 + ((-1)^(d(n)+1)+1)/2); d(n) is the number of divisors of n: A000005.
G.f.: Sum_{n>=1} a(n) x^n = 1/x Sum_{k>=2} x^(k^2)/(1-x^k). (End)
lim_{n->infinity} a(A002110(n)-1) = infinity. - Vladimir Shevelev, Aug 04 2015
a(n) = A161840(n+1)/2. - Omar E. Pol, Feb 27 2019
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3) / 2, where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 14 2024

A169695 a(n) = 1 if n is a square, otherwise a(n) = 2.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2

Offset: 0

N. J. A. Sloane, Apr 15 2010

Also number of central divisors of n, if n >= 1. - Omar E. Pol, Feb 10 2011.

Length of n-th row in the array of A207375, if n >= 1. - Omar E. Pol, Feb 27 2012

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A010052, A275437.

Cf. A000005, A049240, A161840.

Array[2 - Boole[IntegerQ@ Sqrt@ #] &, 105, 0] (* Michael De Vlieger, Nov 03 2017 *)

a(n) = 2 - issquare(n); \\ Amiram Eldar, Apr 17 2024

From Omar E. Pol, Feb 10 2011: (Start)
a(n) = 2 - A010052(n).
a(n) = 1 + A049240(n), n >=1.
a(n) = A000005(n)-A161840(n), n >= 1. (End)

A200213 Ordered factorizations of n with 2 distinct parts, both > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 2, 2, 0, 4, 0, 4, 2, 2, 0, 6, 0, 2, 2, 4, 0, 6, 0, 4, 2, 2, 2, 6, 0, 2, 2, 6, 0, 6, 0, 4, 4, 2, 0, 8, 0, 4, 2, 4, 0, 6, 2, 6, 2, 2, 0, 10, 0, 2, 4, 4, 2, 6, 0, 4, 2, 6, 0, 10, 0, 2, 4, 4, 2, 6, 0, 8, 2, 2, 0, 10, 2

Offset: 1

Peter Luschny, Nov 14 2011

			a(24) = 6 = card({{2,12},{3,8},{4,6},{6,4},{8,3},{12,2}}).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1).

Cf. A000005, A010052, A070824, A161840, A200214, A211159.

a := n -> `if`(n<2, 0, numtheory:-tau(n) - `if`(issqr(n), 3, 2)):
seq(a(n), n = 1..85); # Peter Luschny, Jul 10 2017

OrderedFactorizations[1] = {{}}; OrderedFactorizations[n_?PrimeQ] := {{n}}; OrderedFactorizations[n_] := OrderedFactorizations[n] = Flatten[Function[d, Prepend[#, d] & /@ OrderedFactorizations[n/d]] /@ Rest[Divisors[n]], 1]; a[n_] := With[{of2 = Sort /@ Select[OrderedFactorizations[n], Length[#] == 2 && Length[# // Union] == 2 &] // Union}, Length[Permutations /@ of2 // Flatten[#, 1] &]];  Table[a[n], {n, 1, 85}] (* Jean-François Alcover, Jul 02 2013, copied and adapted from The Mathematica Journal *)

A200213(n) = if(!n,n,sumdiv(n, d, (d<>(n/d))*(d>1)*(dAntti Karttunen, Jul 07 2017

a(n) = if (n==1, 0, numdiv(n) - issquare(n) - 2); \\ Michel Marcus, Jul 07 2017

(define (A200213 n) (if (<= n 1) 0 (- (A000005 n) 2 (A010052 n)))) ;; Antti Karttunen, Jul 07 2017

From Antti Karttunen, Jul 07 & Jul 09 2017: (Start)
a(1) = 0; for n > 1, a(n) = A000005(n) - A010052(n) - 2.
For n >= 2, a(n) = A161840(n) - 2*A010052(n). (End)

Description clarified and term a(0) removed by Antti Karttunen, Jul 09 2017

A207376 Sum of central divisors of n.

Original entry on oeis.org

1, 3, 4, 2, 6, 5, 8, 6, 3, 7, 12, 7, 14, 9, 8, 4, 18, 9, 20, 9, 10, 13, 24, 10, 5, 15, 12, 11, 30, 11, 32, 12, 14, 19, 12, 6, 38, 21, 16, 13, 42, 13, 44, 15, 14, 25, 48, 14, 7, 15, 20, 17, 54, 15, 16, 15, 22, 31, 60, 16, 62, 33, 16, 8, 18, 17, 68, 21, 26, 17

Offset: 1

Omar E. Pol, Feb 23 2012

If n is a square (A000290) then a(n) = sqrt(n) because the squares have only one central divisor. If n is a prime p then a(n) = 1 + p = A000203(n). For the number of central divisors of n see A169695.

			For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12. The central (or middle) divisors of 12 are 3 and 4, so a(12) = 3 + 4 = 7.

Alois P. Heinz, Table of n, a(n) for n = 1..5000
Omar E. Pol, Illustration of the divisors of the first 12 positive integers

Row sums of A207375. Where records occur give A008578.

Cf. A000005, A000040, A000203, A000290, A027750, A161840, A169695, A323643.

cdn[n_]:=Module[{dn=Divisors[n],len},len=Length[dn]; Which[ IntegerQ[ Sqrt[n]], Sqrt[n], PrimeQ[n],n+1, OddQ[len],dn[[Floor[len/2]+1]], EvenQ[len],dn[[len/2]]+dn[[len/2+1]]]]; Array[cdn,70] (* Harvey P. Dale, Nov 07 2012 *)

a(n) = A000203(n) - A323643(n). - Omar E. Pol, Feb 26 2019

A323643 a(n) is the sum of the noncentral divisors of n.

Original entry on oeis.org

0, 0, 0, 5, 0, 7, 0, 9, 10, 11, 0, 21, 0, 15, 16, 27, 0, 30, 0, 33, 22, 23, 0, 50, 26, 27, 28, 45, 0, 61, 0, 51, 34, 35, 36, 85, 0, 39, 40, 77, 0, 83, 0, 69, 64, 47, 0, 110, 50, 78, 52, 81, 0, 105, 56, 105, 58, 59, 0, 152, 0, 63, 88, 119, 66, 127, 0, 105, 70, 127

Offset: 1

Omar E. Pol, Feb 25 2019

a(n) = 0 iff n is 1 or a prime (A008578).

			For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12. The central divisors of 12 are both 3 and 4, therefore the noncentral divisors are 1, 2, 6, 12, and the sum of them is 1 + 2 + 6 + 12 = 21, so a(12) = 21.
For n = 16 the divisors of 16 are 1, 2, 4, 8, 16. The central divisor of 16 is 4, therefore the noncentral divisors of 16 are 1, 2, 8, 16, and the sum of them is 1 + 2 + 8 + 16 = 27, so a(16) = 27.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A008578, A027750, A161840, A169695, A207375, A207376.

sncd[n_]:=Module[{d=Divisors[n],len},len=Length[d];If[EvenQ[len],Total[ Drop[ d, {len/2,len/2+1}]],Total[Drop[d,{(len+1)/2}]]]]; Array[sncd,70] (* Harvey P. Dale, Apr 13 2019 *)

a(n) = A000203(n) - A207376(n).

A323644 Numbers with 3 or 4 divisors.

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 15, 21, 22, 25, 26, 27, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205

Offset: 1

Omar E. Pol, Feb 26 2019

Also numbers k such that the noncentral divisors of k are 1 and k.

Also numbers which are either semiprimes (A001358) or the cube of a prime (A030078). In other words: numbers which are either the product of two distinct primes (A006881) or the square of a prime (A001248) or the cube of a prime (A030078).

			4 is in the sequence because 4 has three divisors, they are 1, 2, 4. On the other hand, the noncentral divisors of 4 are 1 and 4, in accordance with the first comment.
6 is in the sequence because 6 has four divisors, they are 1, 2, 3, 6. On the other hand, the noncentral divisors of 6 are 1 and 6, in accordance with the first comment.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Union of A001248 and A030513.

Cf. A001358, A006881, A030078, A161840, A323643.

Select[Range[200], MemberQ[{3, 4}, DivisorSigma[0, #]] &] (* Amiram Eldar, Dec 03 2020 *)

isok(n) = my(nd=numdiv(n)); (nd==3) || (nd==4); \\ Michel Marcus, Feb 26 2019

A183002 a(n) is the total number of noncentral divisors in all positive integers <= n.

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 4, 6, 8, 10, 10, 14, 14, 16, 18, 22, 22, 26, 26, 30, 32, 34, 34, 40, 42, 44, 46, 50, 50, 56, 56, 60, 62, 64, 66, 74, 74, 76, 78, 84, 84, 90, 90, 94, 98, 100, 100, 108, 110, 114, 116, 120, 120, 126, 128, 134, 136, 138, 138, 148, 148, 150, 154

Offset: 1

Omar E. Pol, Jan 27 2011

The original name was: Partial sums of A161840.

Cf. A000005, A001620, A161840, A183003.

with(numtheory):
b:= n-> (x-> x-2+(x mod 2))(tau(n)):
a:= proc(n) option remember; b(n) +`if`(n=1, 0, a(n-1)) end:
seq(a(n), n=1..100);

Accumulate[Table[d = DivisorSigma[0, n]; If[OddQ[d], d - 1, d - 2], {n, 100}]]

lista(nmax) = {my(s = 0, d); for(n = 1, nmax, d = numdiv(n); s += (d + d%2 - 2); print1(s, ", ")); } \\ Amiram Eldar, Jan 19 2024

a(n) = Sum_{k=1..n} (tau(k)-2 + (tau(k) mod 2)), tau = A000005.

a(n) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 19 2024

New name from Omar E. Pol, Jan 04 2022

cp's OEIS Frontend

A072670 Number of ways to write n as i*j + i + j, 0 < i <= j.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A169695 a(n) = 1 if n is a square, otherwise a(n) = 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A200213 Ordered factorizations of n with 2 distinct parts, both > 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Scheme

Formula

Extensions

A207376 Sum of central divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A323643 a(n) is the sum of the noncentral divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A323644 Numbers with 3 or 4 divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A183002 a(n) is the total number of noncentral divisors in all positive integers <= n.

Views

Author

Keywords

Comments

Crossrefs