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A287352 Irregular triangle T(n,k) = A112798(n,1) followed by first differences of A112798(n). 6
0, 1, 2, 1, 0, 3, 1, 1, 4, 1, 0, 0, 2, 0, 1, 2, 5, 1, 0, 1, 6, 1, 3, 2, 1, 1, 0, 0, 0, 7, 1, 1, 0, 8, 1, 0, 2, 2, 2, 1, 4, 9, 1, 0, 0, 1, 3, 0, 1, 5, 2, 0, 0, 1, 0, 3, 10, 1, 1, 1, 11, 1, 0, 0, 0, 0, 2, 3, 1, 6, 3, 1, 1, 0, 1, 0, 12, 1, 7, 2, 4, 1, 0, 0, 2, 13 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Irregular triangle T(n,k) = first differences of indices of prime divisors p of n.

Row lengths = (big) Omega(n) = A001222(n).

Row sums = A061395(n).

Row maxima = A286469(n).

We can concatenate the rows 1 <= n <= 28 as none of the values of k in this range exceed 9: {0, 1, 2, 10, 3, 11, 4, 100, 20, 12, 5, 101, 6, 13, 21, 1000, 7, 110, 8, 102, 22, 14, 9, 1001, 30, 15, 200, 103}; a(29) = {10}, which would require a digit greater than 9.

a(1) = 0 by convention.

a(0) is not defined (i.e., null set). a(n) is defined for positive nonzero n.

a(p) = A000720(p) for p prime.

a(p^e) = A000720(p) followed by (e - 1) zeros.

a(Product(p^e)) is the concatenation of the a(p^e) of the unitary prime power divisors p^e of n, sorted by the prime p (i.e. the function a(n) mapped across the terms of row n of A141809).

a(A002110(n)) = an array of n 1s.

T(n,k) could be used to furnish A054841(n). We read data in row n of T(n,k). If T(n,1) = 0, then write 0. If T(n,1) > 0, then increment the k-th place from the right. For k > 1, increment the k-th place to the right of the last-incremented place.

T(n,k) can be used to render n in decimal. If T(n,1) = 0, then write 1. If T(n,1) > 0, then multiply 1 by A000720(T(n,1)). For k > 1, multiply the previous product by pi(x) = A000720(x) of the running total of T(n,k) for each k.

Ignoring zeros in row n > 1 and decoding the remaining values of T(n,k) as immediately above yields the squarefree kernel of n = A007947(n).

Leading zeros of a(n) are trimmed, but as in decimal notation numbers that include leading zeros symbolize the same n as without them. Zeros that precede nonzero values merely multiply implicit 1 by itself until we encounter nonzero values. Thus, {0,0,2} = 1*1*pi(2) = 3, as {2} = pi(2) = 3. Because of this no row n > 1 has 0 for k = 1 of T(n,k).

Interpreting n written in binary as a row of a(n) yields A057335(n).

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..15568 (rows 1 <= n <= 5000).

FORMULA

T(n,1) = A117798(n,1); T(n,k) = A117798(n,k) - A117798(n, k - 1) for 2 <= k <= A001222(n).

EXAMPLE

a(1) = {0} by convention.

a(2) = {pi(2)} = {1}.

a(4) = {pi(2), pi(2) - pi(2)}, = {1, 0} since 4 = 2 * 2.

a(6) = {pi(2), pi(3) - pi(2)} = {1, 1} since 6 = 2 * 3.

a(12) = {pi(2), pi(2) - pi(2), pi(3) - pi(2) - pi(2)} = {1, 0, 1}, since 12 = 2 * 2 * 3.

The triangle starts:

   1:  0;

   2:  1;

   3:  2;

   4:  1, 0;

   5:  3;

   6:  1, 1;

   7:  4;

   8:  1, 0, 0;

   9:  2, 0;

  10:  1, 2;

  11:  5;

  12:  1, 0, 1;

  13:  6;

  14:  1, 3;

  15:  2, 1;

  16:  1, 0, 0, 0;

  17:  7;

  18:  1, 1, 0;

  19:  8;

  20:  1, 0, 2;

       ...

MATHEMATICA

Table[Prepend[Differences@ #, First@ #] & Flatten[FactorInteger[n] /. {p_, e_} /; p > 0 :> ConstantArray[PrimePi@ p, e]], {n, 41}] // Flatten (* Michael De Vlieger, May 23 2017 *)

CROSSREFS

Cf. A000720, A001222, A007947, A054841, A057335, A061395, A112798, A141809, A286469.

Sequence in context: A262114 A320780 A029312 * A243715 A333661 A143256

Adjacent sequences:  A287349 A287350 A287351 * A287353 A287354 A287355

KEYWORD

nonn,tabf,easy

AUTHOR

Michael De Vlieger, May 23 2017

STATUS

approved

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Last modified October 25 17:13 EDT 2021. Contains 348255 sequences. (Running on oeis4.)