A169695 -id:A169695 - cp's OEIS frontend

A171369 Triangle read by rows, replace 2's with 3's in A169695.

1, 3, 3, 1, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Gary W. Adamson, Dec 06 2009

Given triangle A275437 (wrapped to contain n terms per row) or sequence A169695, replace 2's with 3's, other terms remain as is.

			First few rows of the triangle =
.
1;
3, 3;
1, 3, 3;
3, 3, 1, 3;
3, 3, 3, 3, 3;
1, 3, 3, 3, 3, 3;
3, 3, 3, 1, 3, 3, 3;
3, 3, 3, 3, 3, 3, 3, 1;
...

Cf. A169695, A275437.

T(n,k) = 1 if n*(n-1)/2+k is a square, otherwise T(n,k) = 3.

Description corrected by Robert Israel, Jan 03 2019

A207375 Irregular array read by rows in which row n lists the (one or two) central divisors of n in increasing order.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 2, 3, 1, 7, 2, 4, 3, 2, 5, 1, 11, 3, 4, 1, 13, 2, 7, 3, 5, 4, 1, 17, 3, 6, 1, 19, 4, 5, 3, 7, 2, 11, 1, 23, 4, 6, 5, 2, 13, 3, 9, 4, 7, 1, 29, 5, 6, 1, 31, 4, 8, 3, 11, 2, 17, 5, 7, 6, 1, 37, 2, 19, 3, 13, 5, 8, 1, 41, 6, 7, 1, 43

Offset: 1

Omar E. Pol, Feb 23 2012

If n is a square then row n lists only the square root of n because the squares (A000290) have only one central divisor.
If n is not a square then row n lists the pair (j, k) of divisors of n, nearest to the square root of n, such that j*k = n.
Conjecture 1: It appears that the n-th record in this sequence is the last member of row A008578(n).
Column 1 gives A033676. Right border gives A033677. - Omar E. Pol, Feb 26 2019
The conjecture 1 follows from Bertrand's Postulate. - Charles R Greathouse IV, Feb 11 2022
Row products give A097448. - Omar E. Pol, Feb 17 2022

			Array begins:
  1;
  1,  2;
  1,  3;
  2;
  1,  5;
  2,  3;
  1,  7;
  2,  4;
  3;
  2,  5;
  1, 11;
  3,  4;
  1, 13;
...

Alois P. Heinz, Rows n = 1..5000
Omar E. Pol, Illustration of the divisors of the first 12 positive integers

Row n has length A169695(n).
Row sums give A207376.
Cf. A000005, A000290, A008578, A027750, A033676, A033677, A097448, A161901, A161904.

A207375row[n_] := ArrayPad[#, -Floor[(Length[#] - 1)/2]] & [Divisors[n]];
Array[A207375row, 50] (* Paolo Xausa, Apr 07 2025 *)

A161840 Number of noncentral divisors of n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jun 21 2009

Noncentral divisors in the following sense: if we sort the divisors of n in natural order, there is one "central", median divisor if the number of divisors tau(n) = A000005(n) is odd, and there are two "central" divisors if tau(n) is even. a(n) is the number of divisors not counting the median or two central divisors.

			The divisors of 4 are 1, 2, 4 so the noncentral divisors of 4 are 1, 4 because its central divisor is 2.
The divisors of 12 are 1, 2, 3, 4, 6, 12 so the noncentral divisors of 12 are 1, 2, 6, 12 because its central divisors  are 3, 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A001620, A049240, A010052, A161841, A169695, A183002, A183003, A200213, A323643.

A000005 := proc(n) numtheory[tau](n) ; end: A010052 := proc(n) if issqr(n) then 1; else 0 ; fi; end: A161840 := proc(n) A000005(n)+A010052(n)-2 ; end: seq(A161840(n),n=1..100) ; # R. J. Mathar, Jul 04 2009

If[EvenQ[#],#-2,#-1]&/@DivisorSigma[0,Range[100]] (* Harvey P. Dale, Sep 22 2024 *)

A161840(n) = numdiv(n)+issquare(n)-2; \\ Antti Karttunen, Jul 07 2017

(define (A161840 n) (+ (A000005 n) (A010052 n) -2)) ;; Antti Karttunen, Jul 07 2017

a(n) = tau(n)-2 + (tau(n) mod 2), tau = A000005.
a(n) = A000005(n) - A049240(n) - 1.
a(n) = A000005(n) + A010052(n) - 2.
a(n) = A000005(n) - A169695(n).
For n >= 2, a(n) = A200213(n) + 2*A010052(n). - Antti Karttunen, Jul 07 2017
a(n) = 2*A072670(n-1). - Omar E. Pol, Jul 08 2017
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 14 2024

More terms from R. J. Mathar, Jul 04 2009

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

A275437 Triangle read by rows: T(n,k) is the number of 01-avoiding binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= floor(n/2)).

Original entry on oeis.org

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

Offset: 0

Emeric Deutsch, Aug 15 2016

The degree of asymmetry of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the degree of asymmetry of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
A sequence is palindromic if and only if its degree of asymmetry is 0.
Number of entries in row n is 1 + floor(n/2).
Sum of entries in row n is n+1.
Sum(k*T(n,k), k>=0) = A002620(n).

			Row 4 is [2,2,1] because the 01-avoiding binary words of length 4 are 0000, 1000, 1100, 1110, and 1111, having asymmetry degrees 0, 1, 2, 1, and 0, respectively.
Triangle starts:
  1;
  2;
  2, 1;
  2, 2;
  2, 2, 1;
  2, 2, 2.

Michael De Vlieger, Table of n, a(n) for n = 0..9999 (rows 0 <= n <= 199).

Cf. A002620, A169695.

T:= proc(n,k) if n = 2*k then 1 elif k <= floor((1/2)*n) then 2 else 0 end if end proc: for n from 0 to 20 do seq(T(n,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

Table[BinCounts[#, {0, Floor[n/2] + 1, 1}] &@ Map[Total@ BitXor[Take[#, Ceiling[Length[#]/2]], Reverse@ Take[#, -Ceiling[Length[#]/2]]] &, Select[PadLeft[IntegerDigits[#, 2], n] & /@ Range[0, 2^n - 1], Length@ SequenceCases[#, {0, 1}] == 0 &]], {n, 0, 15}] // Flatten (* Michael De Vlieger, Aug 15 2016, Version 10.1 *)
Table[If[k == n/2, 1, 2], {n, 15}, {k, Floor[n/2]}] (* Michael De Vlieger, Nov 05 2017 *)

T(2k,k)=1 (k >= 0); T(n,k)=2 if k <= floor(n/2); T(n,k)=0 if k > floor(n/2).

G.f.: G(t,z) = (1 + z)/((1 - z)(1 - tz^2)).

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).

A234398 Distribution of the natural numbers using the sequences family mentioned in the comments.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 6, 1, 2, 6, 2, 2, 6, 3, 2, 6, 4, 2, 6, 5, 2, 6, 6, 2, 6, 7, 2, 6, 8, 2, 6, 9, 2, 6, 10, 2, 6, 10, 1, 2, 6, 10, 2, 2, 6, 10, 3, 2, 6, 10, 4, 2, 6, 10, 5, 2, 6, 10, 6, 2, 6, 10, 7

Offset: 1

Paul Curtz, Dec 25 2013

Based on A016825=2,6,10,..., the family is
N16(1)=1, followed by 2's                  =A040000,
N16(2)=1,2,3,4,5, followed by 6's          =A101272,
N16(3)=1,2,3,4,5,6,7,8,9, followed by 10's, not in the OEIS,
N16(4)=1,2,3,4,5,6,7,8,9,10,11,12,13, followed by 14's, idem.
The N16(n) gives the successive columns beginning at row 1, 3, 9, 19, ... =A058331.
Sum of every row: n =A000027.
Note that with only N16(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, etc
is A169695(n+1).
A169695(n) corresponds to A028310.

			1,
2,
2, 1,
2, 2,
2, 3,
2, 4,
2, 5,
2, 6,
2, 6, 1,
2, 6, 2,
2, 6, 3,
2, 6, 4, etc.

Cf. A173642.

cp's OEIS Frontend

A171369 Triangle read by rows, replace 2's with 3's in A169695.

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A207375 Irregular array read by rows in which row n lists the (one or two) central divisors of n in increasing order.

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A161840 Number of noncentral divisors of n.

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A207376 Sum of central divisors of n.

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A275437 Triangle read by rows: T(n,k) is the number of 01-avoiding binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= floor(n/2)).

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

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A234398 Distribution of the natural numbers using the sequences family mentioned in the comments.

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