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 A073593 Number of cards needed to be drawn (with replacement) from a deck of n cards to have a 50% or greater chance of seeing each card at least once. 4
 1, 2, 5, 7, 10, 13, 17, 20, 23, 27, 31, 35, 38, 42, 46, 51, 55, 59, 63, 67, 72, 76, 81, 85, 90, 94, 99, 104, 108, 113, 118, 123, 128, 133, 137, 142, 147, 152, 157, 162, 167, 173, 178, 183, 188, 193, 198, 204, 209, 214, 219, 225, 230, 235, 241, 246, 251, 257, 262 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A version of the coupon collector's problem (A178923). REFERENCES W. Feller, An Introduction to Probability Theory and Its Applications: Volume 1. S. Ross, A First Course in Probability, Prentice-Hall, 3rd ed., Chapter 7, Example 3g. LINKS Jens Kruse Andersen, Table of n, a(n) for n = 1..1000 P. Erdős and A. Rényi, On a classical problem of probability theory, Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei, 1961 Sci.math.probability newsgroup, Collecting a deck of cards on the street, Aug 2002. Math.stackexchange, The smallest number of boxes to buy to have probability at least 1/2 of collecting all pictures, Oct 2014. FORMULA a(n) seems to be asymptotic to n*(log(n)+c) with c=0.3(6)...and maybe c=1/e. - Benoit Cloitre, Sep 07 2002 c likely to be closer to -log(log(2)) about 0.3665. - Henry Bottomley, Jun 01 2022 MATHEMATICA f[n_] := Block[{k = 1}, While[2StirlingS2[k, n]*n!/n^k < 1, k++ ]; k]; Table[ f[n], {n, 60}] PROG (PARI) S2(n, k) = if(k<1 || k>n, 0, if(n==1, 1, k*S2(n-1, k)+S2(n-1, k-1))); a(n)=if(n<0, 0, k=1; while( 2*S2(k, n)*n!/n^k<1, k++); k) (PARI) a(n)=v=vector(n+1); k=1; v[n]=1.0; while(v<0.5, k++; for(i=1, n, v[i]=v[i]*(n+1-i)/n+v[i+1]*i/n)); k \\ Faster program. Jens Kruse Andersen, Aug 03 2014 CROSSREFS Cf. A090582, A178923 (Coupon Collector's Problem). Sequence in context: A067008 A189757 A094065 * A241510 A088947 A288209 Adjacent sequences: A073590 A073591 A073592 * A073594 A073595 A073596 KEYWORD nonn AUTHOR Robert G. Wilson v, Aug 28 2002 STATUS approved

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Last modified March 31 15:18 EDT 2023. Contains 361668 sequences. (Running on oeis4.)