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A306353 Number of composites among the first n composite numbers whose least prime factor p is that of the n-th composite number. 1


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

%C Composites with least prime factor p are on that row of A083140 which begins with p

%C Sequence with similar values: A122005.

%C Sequence written as a jagged array A with new row when a(n) > a(n+1):

%C 1, 2, 3,

%C 1, 4, 5, 6,

%C 2, 7, 8, 9,

%C 3, 10, 11,

%C 1, 12,

%C 4, 13, 14, 15,

%C 5, 16,

%C 2, 17, 18,

%C 6, 19, 20, 21,

%C 7, 22, 23,

%C 1, 24,

%C 8, 25, 26,

%C 3, 27,

%C 9, 28, 29, 30.

%C A153196 is the list B of the first values in successive rows with length 4.

%C B is given by the formula for A002808(x)=A256388(n+3), an(x)=A153196(n+2)

%C For example: A002808(26)=A256388(3+3), an(26)=A153196(3+2).

%C A243811 is the list of the second values in successive rows with length 4.

%C 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 Sequence written as an irregular triangle C with new row when a(n)=1:

%C 1,2,3,

%C 1,4,5,6,2,7,8,9,3,10,11,

%C 1,12,4,13,14,15,5,16,2,17,18,6,19,20,21,7,22,23,

%C 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 A243887 is the last value in each row of C.

%C 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 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 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 The second row of the irregular triangle C is A117385(b) for 3 < b < 15.

%C The third row of the irregular triangle C has similar values as A117385 in different order.

%H Jamie Morken, <a href="/A306353/b306353.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) is approximately equal to A002808(n)*(A038110(x)/A038111(x)), with A000040(x)=A020639(A002808(n)).

%F For example if n=325, a(325)~= A002808(325)*(A038110(2)/A038111(2)) with A000040(2)=A020639(A002808(325)).

%F This gives an estimate of 67.499... and the actual value of a(n)=67.

%e First composite 4, least prime factor is 2, first case for 2 so a(1)=1.

%e Next composite 6, least prime factor is 2, second case for 2 so a(2)=2.

%e Next composite 8, least prime factor is 2, third case for 2 so a(3)=3.

%e Next composite 9, least prime factor is 3, first case for 3 so a(4)=1.

%e Next composite 10, least prime factor is 2, fourth case for 2 so a(5)=4.

%t counts = {}

%t values = {}

%t For[i = 2, i < 130, i = i + 1,

%t If[PrimeQ[i], ,

%t x = PrimePi[FactorInteger[i][[1, 1]]];

%t If[Length[counts] >= x,

%t counts[[x]] = counts[[x]] + 1;

%t AppendTo[values, counts[[x]]], AppendTo[counts, 1];

%t AppendTo[values, 1]]]]

%t (* Print[counts] *)

%t Print[values]

%o (PARI) c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808

%o 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 Cf. A002808, A256388, A056608, A083140, A122005, A153196, A243811, A047845, A104279, A243887, A117385, A216244, A084921, A066872, A053683, A038110, A038111, A020639.

%K nonn,hear

%O 1,2

%A _Jamie Morken_ and _Vincenzo Librandi_, Feb 09 2019

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Last modified January 23 19:12 EST 2020. Contains 331175 sequences. (Running on oeis4.)