cp's OEIS Frontend

A306353 Number of composites among the first n composite numbers whose least prime factor p is that of the n-th composite number.

%I A306353 #99 Apr 09 2019 05:12:01
%S A306353 1,2,3,1,4,5,6,2,7,8,9,3,10,11,1,12,4,13,14,15,5,16,2,17,18,6,19,20,
%T A306353 21,7,22,23,1,24,8,25,26,3,27,9,28,29,30,10,31,4,32,33,11,34,35,36,12,
%U A306353 37,2,38,39,13,40,41,5,42,14,43,44,3,45,15,46,6,47,48,16,49,50,51,17,52,53,54,18,55,56,7
%N A306353 Number of composites among the first n composite numbers whose least prime factor p is that of the n-th composite number.
%C A306353 Composites with least prime factor p are on that row of A083140 which begins with p
%C A306353 Sequence with similar values: A122005.
%C A306353 Sequence written as a jagged array A with new row when a(n) > a(n+1):
%C A306353   1,  2,  3,
%C A306353   1,  4,  5,  6,
%C A306353   2,  7,  8,  9,
%C A306353   3, 10, 11,
%C A306353   1, 12,
%C A306353   4, 13, 14, 15,
%C A306353   5, 16,
%C A306353   2, 17, 18,
%C A306353   6, 19, 20, 21,
%C A306353   7, 22, 23,
%C A306353   1, 24,
%C A306353   8, 25, 26,
%C A306353   3, 27,
%C A306353   9, 28, 29, 30.
%C A306353 A153196 is the list B of the first values in successive rows with length 4.
%C A306353   B is given by the formula for A002808(x)=A256388(n+3), an(x)=A153196(n+2)
%C A306353 For example: A002808(26)=A256388(3+3), an(26)=A153196(3+2).
%C A306353 A243811 is the list of the second values in successive rows with length 4.
%C A306353 A047845 is the list of values in the second column and A104279 is the list of values in the third column of the jagged array starting on the second row.
%C A306353 Sequence written as an irregular triangle C with new row when a(n)=1:
%C A306353   1,2,3,
%C A306353   1,4,5,6,2,7,8,9,3,10,11,
%C A306353   1,12,4,13,14,15,5,16,2,17,18,6,19,20,21,7,22,23,
%C A306353   1,24,8,25,26,3,27,9,28,29,30,10,31,4,32,33,11,34,35,36,12,37,2,38,39,13,40,41,5,42,14,43,44,3,45,15,46,6,47,48,16,49,50,51,17,52,53,54,18,55,56,7,57,19,58,4,59.
%C A306353 A243887 is the last value in each row of C.
%C A306353 The second value D on the row n > 1 of the irregular triangle C is a(A053683(n)) or equivalently A084921(n). For example for row 3 of the irregular triangle:
%C A306353   D = a(A053683(3)) = a(16) = 12 or D = A084921(3) = 12. This is the number of composites < A066872(3) with the same least prime factor p as the A053683(3) = 16th composite, A066872(3) = 26.
%C A306353 The number of values in each row of the irregular triangle C begins: 3,11,18,57,39,98,61,141,265,104,351,268,...
%C A306353 The second row of the irregular triangle C is A117385(b) for 3 < b < 15.
%C A306353 The third row of the irregular triangle C has similar values as A117385 in different order.
%H A306353 Jamie Morken, <a href="/A306353/b306353.txt">Table of n, a(n) for n = 1..10000</a>
%F A306353 a(n) is approximately equal to A002808(n)*(A038110(x)/A038111(x)), with A000040(x)=A020639(A002808(n)).
%F A306353 For example if n=325, a(325)~= A002808(325)*(A038110(2)/A038111(2)) with A000040(2)=A020639(A002808(325)).
%F A306353 This gives an estimate of 67.499... and the actual value of a(n)=67.
%e A306353 First composite 4, least prime factor is 2, first case for 2 so a(1)=1.
%e A306353 Next composite 6, least prime factor is 2, second case for 2 so a(2)=2.
%e A306353 Next composite 8, least prime factor is 2, third case for 2 so a(3)=3.
%e A306353 Next composite 9, least prime factor is 3, first case for 3 so a(4)=1.
%e A306353 Next composite 10, least prime factor is 2, fourth case for 2 so a(5)=4.
%t A306353 counts = {}
%t A306353 values = {}
%t A306353 For[i = 2, i < 130, i = i + 1,
%t A306353 If[PrimeQ[i], ,
%t A306353 x = PrimePi[FactorInteger[i][[1, 1]]];
%t A306353   If[Length[counts] >= x,
%t A306353    counts[[x]] = counts[[x]] + 1;
%t A306353    AppendTo[values, counts[[x]]], AppendTo[counts, 1];
%t A306353    AppendTo[values, 1]]]]
%t A306353    (* Print[counts] *)
%t A306353    Print[values]
%o A306353 (PARI) c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808
%o A306353 a(n) = my(c=c(n), lpf = vecmin(factor(c)[,1]), nb=0); for(k=2, c, if (!isprime(k) && vecmin(factor(k)[,1])==lpf, nb++)); nb; \\ _Michel Marcus_, Feb 10 2019
%Y A306353 Cf. A002808, A256388, A056608, A083140, A122005, A153196, A243811, A047845, A104279, A243887, A117385, A216244, A084921, A066872, A053683, A038110, A038111, A020639.
%K A306353 nonn,hear
%O A306353 1,2
%A A306353 _Jamie Morken_ and _Vincenzo Librandi_, Feb 09 2019