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A077592
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Table by antidiagonals of tau_k(n) the k-th Piltz function (see A007425), or n-th term of sequence resulting from applying inverse Moebius transform (k-1) times to all ones sequence.
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3
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1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 2, 1, 1, 6, 5, 10, 3, 4, 1, 1, 7, 6, 15, 4, 9, 2, 1, 1, 8, 7, 21, 5, 16, 3, 4, 1, 1, 9, 8, 28, 6, 25, 4, 10, 3, 1, 1, 10, 9, 36, 7, 36, 5, 20, 6, 4, 1, 1, 11, 10, 45, 8, 49, 6, 35, 10, 9, 2, 1, 1, 12, 11, 55, 9, 64, 7, 56, 15, 16, 3, 6, 1, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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FORMULA
| If n=Sum_i p_i^e_i, then T(n, k)=Product_i C(k+e_i-1, e_i). T(n, k) =sum_d{d|n}T(n-1, d) =A077593(n, k)-A077593(n-1, k).
Columns are multiplicative.
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EXAMPLE
| Rows start: 1,1,1,1,1,1,1,...; 1,2,3,4,5,6,7,...; 1,2,3,4,5,6,7,...; 1,3,6,10,15,21,28,...; 1,2,3,4,5,6,7,...; 1,4,9,16,25,36,49,...; etc.
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MATHEMATICA
| tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[tau[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten (* Robert G. Wilson v *)
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CROSSREFS
| Columns include A000012, A000005, A007425, A007426, A061200, A034695. Rows include (with multiplicity and some offsets) A000012, A000027, A000027, A000217, A000027, A000290, A000027, A000292, A000217, A000290, A000027, A002411, A000027, A000290, A000290, A000332 etc. Cf. A077593.
Sequence in context: A025474 A136575 A193592 * A194005 A055794 A092905
Adjacent sequences: A077589 A077590 A077591 * A077593 A077594 A077595
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KEYWORD
| mult,nonn,tabl
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Nov 08 2002
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