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 A146289 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001221(n)), giving the number of divisors of n with m distinct prime factors. 13
 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 3, 2, 1, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 3, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 4, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 3, 1, 1, 1, 3, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS The formula used in obtaining the n-th row (see below) also gives the number of divisors of the k-th power of n. Two numbers have identical rows in the table if and only if they have the same prime signature. T(n,0)=1. LINKS G. C. Greubel, Table of n, a(n) for the first 500 rows, flattened Anonymous?, Polynomial calculator Eric Weisstein's World of Mathematics, Distinct Prime Factors G. Xiao, WIMS server, Factoris (both expands and factors polynomials) FORMULA If the canonical factorization of n into prime powers is Product p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (1 + e(p) k). EXAMPLE Rows begin: 1; 1,1; 1,1; 1,2; 1,1; 1,2,1; 1,1; 1,3; 1,2; 1,2,1; ... 12 has 1 divisor with 0 distinct prime factors (1); 3 with 1 (2, 3 and 4); and 2 with 2 (6 and 12), for a total of 6. The 12th row of the table therefore reads (1, 3, 2). These are the positive coefficients of the polynomial equation 1 + 3k + 2k^2 = (1 + 2k)(1 + k), derived from the prime factorization of 12 (namely, 2^2*3^1). MAPLE f:= proc(n)    local F, G, f, t, k;    F:= ifactors(n)[2];    G:= mul(1+f[2]*t, f= F);    seq(coeff(G, t, k), k=0..nops(F)); end proc: seq(f(n), n=1..100); # Robert Israel, Feb 10 2015 MATHEMATICA Join[{{1}}, Table[nn = DivisorSigma[0, n]; CoefficientList[Series[Product[1 + i x, {i, FactorInteger[n][[All, 2]]}], {x, 0, nn}], x], {n, 2, 100}]] // Grid (* Geoffrey Critzer, Feb 09 2015 *) PROG (PARI) tabf(nn) = {for (n=1, nn, vd = divisors(n); vo = vector(#vd, k, omega(vd[k])); for (k=0, vecmax(vo), print1(#select(x->x==k, vo), ", "); ); print(); ); } \\ Michel Marcus, Apr 22 2017 CROSSREFS Row sums equal A000005(n). T(n, 1) = A001222(n) for n>1. T(n, A001221(n)) = A005361(n). Row n of A007318 is identical to row A002110(n) of this table and also identical to the row for any squarefree number with n prime factors. Cf. A146290. Also cf. A146291, A146292. Sequence in context: A101873 A336691 A177991 * A214575 A081418 A088951 Adjacent sequences:  A146286 A146287 A146288 * A146290 A146291 A146292 KEYWORD nonn,tabf AUTHOR Matthew Vandermast, Nov 11 2008 STATUS approved

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Last modified June 13 00:57 EDT 2021. Contains 344980 sequences. (Running on oeis4.)