A207376 -id:A207376 - cp's OEIS frontend

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

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 *)

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).

A272025 Irregular triangle read by rows, n >= 1, 1 <= k <= A038548(n), in which T(n,k) is the sum of the k-th pair of conjugate divisors of n, or T(n,k) is the central divisor of n if such a pair does not exist.

Original entry on oeis.org

1, 3, 4, 5, 2, 6, 7, 5, 8, 9, 6, 10, 3, 11, 7, 12, 13, 8, 7, 14, 15, 9, 16, 8, 17, 10, 4, 18, 19, 11, 9, 20, 21, 12, 9, 22, 10, 23, 13, 24, 25, 14, 11, 10, 26, 5, 27, 15, 28, 12, 29, 16, 11, 30, 31, 17, 13, 11, 32, 33, 18, 12, 34, 14, 35, 19, 36, 12, 37, 20, 15, 13, 6, 38, 39, 21, 40, 16, 41, 22, 14, 13, 42, 43, 23, 17, 13

Offset: 1

Omar E. Pol, Apr 21 2016

			Triangle begins:
  1;
  3;
  4;
  5, 2;
  6;
  7, 5;
  8;
  9, 6;
  10, 3;
  11, 7;
  12;
  13, 8, 7;
  ...
For n = 9 the divisors of 9 are [1, 3, 9]. There is only one pair of conjugate divisors: [1, 9], and the central divisor is 3, so the 9th row of the triangle is [10, 3].
For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12]. There are three pairs of conjugate divisors, they are [1, 12], [2, 6], [3, 4], so the 12th row of the triangle is [13, 8, 7].

Row sums give A000203.
Row lengths give A038548.
Right border gives A207376.
Column 1 is A065475.
Cf. A027750, A210959, A228814.

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

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A272025 Irregular triangle read by rows, n >= 1, 1 <= k <= A038548(n), in which T(n,k) is the sum of the k-th pair of conjugate divisors of n, or T(n,k) is the central divisor of n if such a pair does not exist.

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