Proposed fix for A072916, plz review

[David Wilson]

Note 1: The second %S in the modified sequence should be %U, so the fixed sequence would be

%I A072916
%S A072916 3,8,19,41,117,254,616,1642,3766,9461,24183,60252,151368,385600,979844,
%T A072916 2507393,6428977,16513542,42642649,110283280,285776799,742428731,
%U A072916 1932223170,5038580446,13159683245,34423463648,90173540312
%N A072916 Number of m such that Floor[Prime[m]/m] = n.
%e A072916 There are 8 values of m giving Floor[Prime[m]/m] = 2, namely m = m = 1,5,6,7,8,9,10,11, so a(2) = 8.
%t A072916 a(n_) := Length[Cases[Table[Floor[Prime[m]/m], {m, 1, 1000000}], n]]
%K A072916 base,easy,nonn
%O A072916 1,1.
%e A072916 a(16) through a(27) from Farideh Firoozbakht Sep 13 2005
%A A072916 Zakir F. Seidov (zakseidov(AT)yahoo.com) Aug 11 2002

Note 2: The %t program as it stands is not sufficient to find the larger extended values.

Note 3: Once A072916 is fixed, A112547 may be removed as a duplicate of it.  A112547 is a very new sequence and unlikely to have been referenced.

----- Original Message ----- 
  From: David Wilson 
  To: Sequence Fans 
  Sent: Saturday, September 17, 2005 2:55 AM
  Subject: Proposed fix for A072916, plz review


  Currently A072916 looks like

  %I A072916
  %S A072916 3,7,19,41,117,254,616,1642,3766,9461,24183,60252,151368,385600,979844
  %N A072916 Number of m such that Floor[Prime[m]/m] = n.
  %C A072916 First 12 primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 First 12 Floor[Prime[m]/m]'s are: 2, 1, 1, 1\
    , 2, 2, 2, 2, 2, 2, 2, 3 Remove first term and count ones, twos etc and get the sequence: 3,7,19...
  %e A072916 a(2)=7, as seven primes 11, 13, 17, 19, 23, 29, 31 divided by their order give numbers between 2 and 3.
  %t A072916 a(n_) := Length[Cases[Table[Floor[Prime[m]/m], {m, 2, 1000000}], n]]
  %Y A072916 Adjacent sequences: A072913 A072914 A072915 this_sequence A072917 A072918 A072919
  %Y A072916 Sequence in context: A077313 A049490 A047025 this_sequence A096447 A086519 A090689
  %K A072916 base,easy,nonn
  %O A072916 1,1
  %A A072916 Zakir F. Seidov (zakseidov(AT)yahoo.com) Aug 11 2002

  According to the description a(n) counts the number of m with Floor[Prime[m]/m] = n.  The domain of Prime[m] is set of positive integers, so the natural assumption would be that a(n) counts positive m with Floor[Prime[m]/m] = n.  There are eight positive values of m satisfying Floor[Prime[m]/m] = 2, namely, m = 1,5,6,7,8,9,10,11.  We would therefore expect a(2) = 8, however, for some inexplicable reason, m = 1 is dropped from consideration (see "Remove first term" in comment) and so a(2) = 7 in the current sequence. Note that in the program given, m starts at 2 instead of 1 for no specified reason.

  My proposed version of this sequence counts m = 1 and extends the sequence.

  %I A072916
  %S A072916 3,8,19,41,117,254,616,1642,3766,9461,24183,60252,151368,385600,979844,
  %T A072916 2507393,6428977,16513542,42642649,110283280,285776799,742428731,
  %S A072916 1932223170,5038580446,13159683245,34423463648,90173540312
  %N A072916 Number of m such that Floor[Prime[m]/m] = n.
  %e A072916 There are 8 values of m giving Floor[Prime[m]/m] = 2, namely m = m = 1,5,6,7,8,9,10,11, so a(2) = 8.
  %t A072916 a(n_) := Length[Cases[Table[Floor[Prime[m]/m], {m, 1, 1000000}], n]]
  %K A072916 base,easy,nonn
  %O A072916 1,1.
  %e A072916 a(16) through a(27) from Farideh Firoozbakht Sep 13 2005
  %A A072916 Zakir F. Seidov (zakseidov(AT)yahoo.com) Aug 11 2002

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