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A231347 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the odd numbers interleaved with k-1 zeros but T(n,1) = n - 1 and the first element of column k is in row k(k+1)/2. 13
0, 1, 2, 1, 3, 0, 4, 3, 5, 0, 1, 6, 5, 0, 7, 0, 0, 8, 7, 3, 9, 0, 0, 1, 10, 9, 0, 0, 11, 0, 5, 0, 12, 11, 0, 0, 13, 0, 0, 3, 14, 13, 7, 0, 1, 15, 0, 0, 0, 0, 16, 15, 0, 0, 0, 17, 0, 9, 5, 0, 18, 17, 0, 0, 0, 19, 0, 0, 0, 3, 20, 19, 11, 0, 0, 1, 21, 0, 0, 7, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Alternating sum of row n equals the sum of aliquot divisors of n, i.e., sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A001065(n).

Row n has length A003056(n).

Column k starts in row A000217(k).

The number of positive terms in row n is A001227(n), for n >= 2.

If n = 2^j then the only positive integer in row n is T(n,1) = n - 1, for j >= 1.

If n is an odd prime then the only two positive integers in row n are T(n,1) = n - 1 and T(n,2) = n - 2.

LINKS

Table of n, a(n) for n=1..82.

FORMULA

T(n,1) = n - 1.

T(n,k) = A196020(n,k), for k >= 2.

EXAMPLE

Triangle begins:

  0;

  1;

  2,   1;

  3,   0;

  4,   3;

  5,   0,  1;

  6,   5,  0;

  7,   0,  0;

  8,   7,  3;

  9,   0,  0,  1;

  10,  9,  0,  0;

  11,  0,  5,  0;

  12, 11,  0,  0;

  13,  0,  0,  3;

  14, 13,  7,  0,  1;

  15,  0,  0,  0,  0;

  16, 15,  0,  0,  0;

  17,  0,  9,  5,  0;

  18, 17,  0,  0,  0;

  19,  0,  0,  0,  3;

  20, 19, 11,  0,  0,  1;

  21,  0,  0,  7,  0,  0;

  22, 21,  0,  0,  0,  0;

  23,  0, 13,  0,  0,  0;

  ...

For n = 15 the aliquot divisors of 15 are 1, 3, 5, therefore the sum of aliquot divisors of 15 is 1 + 3 + 5 = 9. On the other hand the 15th row of triangle is 14, 13, 7, 0, 1, hence the alternating row sum is 14 - 13 + 7 - 0 + 1 = 9, equalling the sum of aliquot divisors of 15.

If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of aliquot divisors of n. Example: the sum of aliquot divisors of 24 is 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36, and the alternating sum of the 24th row of triangle is 23 - 0 + 13 - 0 + 0 - 0 = 36.

CROSSREFS

Columns 1-2: A001477, A193356.

Cf. A000217, A001065, A001227, A000203, A003056, A196020, A211343, A212119, A228813, A231345, A235791, A235794, A236104, A236106, A236112.

Sequence in context: A264520 A058208 A070817 * A180988 A127474 A326400

Adjacent sequences:  A231344 A231345 A231346 * A231348 A231349 A231350

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Dec 28 2013

STATUS

approved

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Last modified October 19 20:55 EDT 2019. Contains 328224 sequences. (Running on oeis4.)