A249809 -id:A249809 - cp's OEIS frontend

A078898 Number of times the smallest prime factor of n is the smallest prime factor for numbers <= n; a(0)=0, a(1)=1.

0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 4, 11, 1, 12, 2, 13, 5, 14, 1, 15, 1, 16, 6, 17, 3, 18, 1, 19, 7, 20, 1, 21, 1, 22, 8, 23, 1, 24, 2, 25, 9, 26, 1, 27, 4, 28, 10, 29, 1, 30, 1, 31, 11, 32, 5, 33, 1, 34, 12, 35, 1, 36, 1, 37, 13, 38, 3, 39, 1, 40, 14, 41, 1, 42, 6, 43

Offset: 0

Reinhard Zumkeller, Dec 12 2002

nonn

From Antti Karttunen, Dec 06 2014: (Start)
For n >= 2, a(n) tells in which column of the sieve of Eratosthenes (see A083140, A083221) n occurs in. A055396 gives the corresponding row index.
(End)

Harvey P. Dale (terms 1 - 1000) & Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A002110, A008683, A008836, A020639, A032742, A054272, A055396, A078899, A078896, A083140, A083221, A243055, A246277, A249738, A249744, A249808, A249809, A249810, A249820, A249818, A250470, A250474, A250477, A250478, A251719, A251724, A251728.

Cf. A001248, A006094, A038110, A038111, A090076.

import Data.IntMap (empty, findWithDefault, insert)
a078898 n = a078898_list !! n
a078898_list = 0 : 1 : f empty 2 where
   f m x = y : f (insert p y m) (x + 1) where
           y = findWithDefault 0 p m + 1
           p = a020639 x
-- Reinhard Zumkeller, Apr 06 2015

N:= 1000: # to get a(0) to a(N)
Primes:= select(isprime, [2,seq(2*i+1,i=1..floor((N-1)/2))]):
A:= Vector(N):
for p in Primes do
  t:= 1:
  A[p]:= 1:
  for n from p^2 to N by p do
    if A[n] = 0 then
       t:= t+1:
       A[n]:= t
    fi
  od
od:
0,1,seq(A[i],i=2..N); # Robert Israel, Jan 04 2015

Module[{nn=90,spfs},spfs=Table[FactorInteger[n][[1,1]],{n,nn}];Table[ Count[ Take[spfs,i],spfs[[i]]],{i,nn}]] (* Harvey P. Dale, Sep 01 2014 *)

\\ Not practical for computing, but demonstrates the sum moebius formula:
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A055396(n) = { if(1==n,0,primepi(A020639(n))); };
A002110(n) = prod(i=1, n, prime(i));
A078898(n) = { my(k,p); if(1==n, n, k = A002110(A055396(n)-1); p = A020639(n); sumdiv(k, d, moebius(d)*(n\(p*d)))); };
\\ Antti Karttunen, Dec 05 2014

;; With memoizing definec-macro.
(definec (A078898 n) (if (< n 2) n (+ 1 (A078898 (A249744 n)))))
;; Much better for computing. Needs also code from A249738 and A249744. - Antti Karttunen, Dec 06 2014

Ordinal transform of A020639 (Lpf). - Franklin T. Adams-Watters, Aug 28 2006
From Antti Karttunen, Dec 05-08 2014: (Start)
a(0) = 0, a(1) = 1, a(n) = 1 + a(A249744(n)).
a(0) = 0, a(1) = 1, a(n) = sum_{d | A002110(A055396(n)-1)} moebius(d) * floor(n / (A020639(n)*d)).
a(0) = 0, a(1) = 1, a(n) = sum_{d | A002110(A055396(n)-1)} moebius(d) * floor(A032742(n) / d).
[Instead of Moebius mu (A008683) one could use Liouville's lambda (A008836) in the above formulas, because all primorials (A002110) are squarefree. A020639(n) gives the smallest prime dividing n, and A055396 gives its index].
a(0) = 0, a(1) = 1, a(2n) = n, a(2n+1) = a(A250470(2n+1)). [After a similar recursive formula for A246277. However, this cannot be used for computing the sequence, unless a definition for A250470(n) is found which doesn't require computing the value of A078898(n).]
For n > 1: a(n) = A249810(n) - A249820(n).
(End)
Other identities:
a(2*n) = n.
For n > 1: a(n)=1 if and only if n is prime.
For n > 1: a(n) = A249808(n, A055396(n)) = A249809(n, A055396(n)).
For n > 1: a(n) = A246277(A249818(n)).
From Antti Karttunen, Jan 04 2015: (Start)
a(n) = 2 if and only if n is a square of a prime.
For all n >= 1: a(A251728(n)) = A243055(A251728(n)) + 2. That is, if n is a semiprime of the form prime(i)*prime(j), prime(i) <= prime(j) < prime(i)^2, then a(n) = (j-i)+2.
(End)
a(A000040(n)^2) = 2; a(A000040(n)*A000040(n+1)) = 3. - Reinhard Zumkeller, Apr 06 2015
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum_{k>=1} (A038110(k)/A038111(k))^2 = 0.2847976823663... . - Amiram Eldar, Oct 26 2024

a(0) = 0 prepended for recurrence's sake by Antti Karttunen, Dec 06 2014

A249727 Start with a(1) = 1; then numbers 1 .. primepi(2), followed by numbers 1 .. primepi(3), and then numbers 1 .. primepi(4), ..., etc, where A000720 gives primepi.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

Can be used to construct the irregular table A249809.

This is a fractal sequence; i.e., the removal of the first occurrence of each term in A249727 leaves A249727, so that the sequence contains itself infinitely many times. The corresponding interspersion is A272616. - Clark Kimberling, May 12 2016

Antti Karttunen, Table of n, a(n) for n = 1..10016

Cf. A000720, A046992, A249728, A249809, A272616.

Flatten[Table[Range[PrimePi[n]], {n, 2, 100}]]
(* Clark Kimberling, May 14 2016 *)

(define (A249727 n) (if (= 1 n) 1 (- n (+ 1 (A046992 (- (A249728 n) 1))))))

A249728 After a(1) = 1 each n appears A000720(n) times.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10016

Cf. A000720, A046992, A249727, A249809.

Join[{1},Table[Table[n,PrimePi[n]],{n,20}]//Flatten] (* Harvey P. Dale, Nov 24 2018 *)

A249808 Triangular table read by rows, a lower right triangular region of square array A(n,k) is the number of times prime p_k has occurred as the smallest prime factor of numbers 1..n.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 1, 1, 0, 0, 3, 1, 1, 0, 0, 0, 3, 1, 1, 1, 0, 0, 0, 4, 1, 1, 1, 0, 0, 0, 0, 4, 2, 1, 1, 0, 0, 0, 0, 0, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 5, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 6, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 6, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 7, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 7, 3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Nov 06 2014

Table is read by taking n terms from the beginning of row n: A(1,1), A(2,1), A(2,2), A(3,1), A(3,2), A(3,3), ...

See also A249809 for a version with extra zeros removed.

			The first eleven rows of this triangular table:
  0;
  1, 0;
  1, 1, 0;
  2, 1, 0, 0;
  2, 1, 1, 0, 0;
  3, 1, 1, 0, 0, 0;
  3, 1, 1, 1, 0, 0, 0;
  4, 1, 1, 1, 0, 0, 0, 0;
  4, 2, 1, 1, 0, 0, 0, 0, 0;
  5, 2, 1, 1, 0, 0, 0, 0, 0, 0;
  5, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0;
  ...

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 rows of triangular table

A249809 is a more compact version.
A004526 gives the left edge, A001477 the row sums.
Cf. A000040, A020639, A055396, A078898, A002024, A002260.

FoldList[Append[MapAt[# + 1 &, #1, PrimePi@ FactorInteger[#2][[1, 1]]], 0] &, {0}, Range[2, 15]] // Flatten (* Michael De Vlieger, Nov 24 2017 *)

(define (A249808 n) (A249808bi (A002024 n) (A002260 n)))
(define (A249808bi row col) (if (= 1 row) 0 (+ (A249808bi (- row 1) col) (if (= (A055396 row) col) 1 0))))

If row n = 1, A(n,k) = 0, otherwise A(n,k) = A(n-1,k) + [A055396(n) = k], where the subexpression with the Iverson bracket is 1 if the index of the smallest prime dividing n is equal to k, and 0 otherwise. This is a formula for a full square array containing mostly zeros. The terms of this sequence are those collected from the lower right triangle of that square array.

For n > 1, A078898(n) = A(n, A055396(n)).

A248809 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 8, 7, 2, 1, 15, 18, 12, 2, 1, 48, 57, 30, 18, 2, 1, 105, 174, 141, 44, 25, 2, 1, 384, 561, 414, 285, 60, 33, 2, 1, 945, 1950, 1830, 810, 510, 78, 42, 2, 1, 3840, 6555, 6090, 4680, 1410, 840, 98, 52, 2, 1, 10395, 25290, 26685, 15000, 10290

Offset: 0

Clark Kimberling, Oct 23 2014

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + (n + 1)/f(n-1,x), where f(0,x) = 1.
(Sum of numbers in row n) = A000982(n+1) for n >= 0.
(Column 1) is essentially A006882 (double factorials).

			f(0,x) = 1/1, so that p(0,x) = 1.
f(1,x) = (2 + x)/1, so that p(1,x) = 2 + x.
f(2,x) = (3 + 2 x + x^2)/(2 + x), so that p(2,x) = 3 + 2 x + x^2.
First 6 rows of the triangle of coefficients:
1
2    1
3    2     1
8    7     2    1
15   18   12    2    1
48   57   30    18   2    1

Clark Kimberling, Rows 0..100, flattened.

Cf. A000982, A006882.

z = 15; f[x_, n_] := x + (n + 1)/f[x, n - 1]; f[x_, 0] = 1;
t = Table[Factor[f[x, n]], {n, 0, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (*A248809 array*)
Flatten[CoefficientList[u, x]] (*A249809 sequence*)

rown(n) = if (n==0, 1, x + (n+1)/rown(n-1));
tabl(nn) = for (n=0, nn, print(Vecrev(numerator(rown(n))))); \\ Michel Marcus, Oct 25 2014

cp's OEIS Frontend

A078898 Number of times the smallest prime factor of n is the smallest prime factor for numbers <= n; a(0)=0, a(1)=1.

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A249727 Start with a(1) = 1; then numbers 1 .. primepi(2), followed by numbers 1 .. primepi(3), and then numbers 1 .. primepi(4), ..., etc, where A000720 gives primepi.

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A249728 After a(1) = 1 each n appears A000720(n) times.

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A249808 Triangular table read by rows, a lower right triangular region of square array A(n,k) is the number of times prime p_k has occurred as the smallest prime factor of numbers 1..n.

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A248809 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

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