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 A274517 Number T(n,k) of integer partitions of n with exactly k distinct primes. 2
 1, 1, 1, 1, 1, 2, 2, 3, 2, 4, 1, 3, 7, 1, 3, 9, 3, 5, 12, 5, 6, 15, 9, 8, 22, 11, 1, 8, 28, 19, 1, 12, 38, 24, 3, 13, 46, 38, 4, 17, 62, 48, 8, 19, 77, 68, 12, 26, 98, 87, 20, 28, 117, 127, 24, 1, 37, 152, 154, 41, 1, 40, 183, 210, 55, 2, 52, 230, 260, 82, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Row lengths increase by 1 at row A007504(n). Columns k=0-1 give: A002095, A132381. Row sums give: A000041. LINKS Alois P. Heinz, Rows n = 0..1000, flattened FORMULA G.f.: Product_{k>=1} (1 - x^prime(k))/(1 - x^k)*(y/(1-x^prime(k)) - y + 1). EXAMPLE 1, 1, 1, 1, 1, 2, 2, 3, 2, 4,  1, 3, 7,  1, 3, 9,  3, 5, 12, 5, 6, 15, 9, 8, 22, 11, 1, T(6,1) = 7 because we have: 5+1, 4+2, 3+3, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1+1. MAPLE b:= proc(n, i) option remember; expand(       `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*       `if`(j>0 and isprime(i), x, 1), j=0..n/i))))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n\$2)): seq(T(n), n=0..30);  # Alois P. Heinz, Jun 26 2016 MATHEMATICA nn = 20; Map[Select[#, # > 0 &] &, CoefficientList[Series[Product[       1/(1 - z^k), {k, Select[Range[1000], PrimeQ[#] == False &]}] Product[       u/(1 - z^j) - u + 1, {j, Table[Prime[n], {n, 1, nn}]}], {z, 0,      nn}], {z, u}]] // Grid CROSSREFS Cf. A000041, A002095, A007504, A132381, A222656. Sequence in context: A242213 A035213 A083901 * A038148 A141829 A111336 Adjacent sequences:  A274514 A274515 A274516 * A274518 A274519 A274520 KEYWORD nonn,tabf AUTHOR Geoffrey Critzer, Jun 25 2016 STATUS approved

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Last modified October 23 17:19 EDT 2019. Contains 328373 sequences. (Running on oeis4.)