A080681 -id:A080681 - cp's OEIS frontend

A080786 Triangle T(n,k) = number of k-smooth numbers <= n, read by rows.

1, 1, 2, 1, 2, 3, 1, 3, 4, 4, 1, 3, 4, 4, 5, 1, 3, 5, 5, 6, 6, 1, 3, 5, 5, 6, 6, 7, 1, 4, 6, 6, 7, 7, 8, 8, 1, 4, 7, 7, 8, 8, 9, 9, 9, 1, 4, 7, 7, 9, 9, 10, 10, 10, 10, 1, 4, 7, 7, 9, 9, 10, 10, 10, 10, 11, 1, 4, 8, 8, 10, 10, 11, 11, 11, 11, 12, 12, 1, 4, 8, 8, 10, 10, 11, 11, 11, 11, 12, 12, 13, 1, 4

Offset: 1

Reinhard Zumkeller, Mar 12 2003

T(n,n-1) = A014684(n) for n>1;
T(n,2) = A029837(n) for n>1; T(n,3) = A071521(n) for n>2; T(n,5) = A071520(n) for n>4.
A036234(n) = number of distinct terms in n-th row. - Reinhard Zumkeller, Sep 17 2013

			Triangle begins:
.................. 1
................ 1...2
.............. 1...2...3
............ 1...3...4...4
.......... 1...3...4...4...5
........ 1...3...5...5...6...6
...... 1...3...5...5...6...6...7
.... 1...4...6...6...7...7...8...8
.. 1...4...7...7...8...8...9...9...9.

Reinhard Zumkeller, Rows n=1..120 of triangle, flattened
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A000079, A002473, A003586, A006530, A014684, A029837, A036234, A051037, A051038, A071520, A071521, A080197, A080681, A080682, A080683.

a080786 n k = a080786_tabl !! (n-1) !! (k-1)
a080786_row n = a080786_tabl !! (n-1)
a080786_tabl = map reverse $ iterate f [1] where
   f xs@(x:_) = (x + 1) :
                (zipWith (+) xs (map (fromEnum . (lpf <=)) [x, x-1 ..]))
        where lpf = fromInteger $ a006530 $ fromIntegral (x + 1)
-- Reinhard Zumkeller, Sep 17 2013

A080786 := proc(x,y)
    local a,n ;
    a := 0 ;
    for n from 1 to x do
        if A006530(n) <= y then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Aug 31 2013

P[n_] := FactorInteger[n][[-1, 1]]; P[1]=1; T[n_, k_] := (For[j=0; m=1, m <= n, m++, If[P[m] <= k, j++]]; j); Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2015 *)

from itertools import count, islice
from sympy import prevprime, integer_log
def A080786_T(n,k):
    if k==1: return 1
    def g(x,m): return x.bit_length() if m==2 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    return g(n,prevprime(k+1))
def A080786_gen(): # generator of terms
    return (A080786_T(n,k) for n in count(1) for k in range(1,n+1))
A080786_list = list(islice(A080786_gen(),100)) # Chai Wah Wu, Oct 22 2024

A108275 Number of positive integers <= 10^n that are divisible by no prime exceeding 17.

Original entry on oeis.org

1, 10, 67, 287, 945, 2579, 6179, 13389, 26809, 50351, 89679, 152751, 250420, 397227, 612185, 919814, 1351182, 1945205, 2750000, 3824456, 5239923, 7082118, 9453184, 12473898, 16286197, 21055745, 26974877, 34265658, 43183191, 54019174

Offset: 0

Robert G. Wilson v, May 31 2005

nonn

Row 7 of A253635.

Cf. A011557, A080681.

n = 10; t = Select[ Flatten[ Table[17^g* Select[ Flatten[ Table[13^f* Select[ Flatten[ Table[11^e* Select[ Flatten[ Table[7^d* Select[ Flatten[ Table[5^c* Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &], {d, 0, n*Log[7, 10]}]], # <= 10^n &], {e, 0, n*Log[11, 10]}]], # <= 10^n &], {f, 0, n*Log[13, 10]}]], # <= 10^n &], {g, 0, n*Log[17, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 10}]

a(11)-a(18) from Donovan Johnson, Sep 16 2009

a(19)-a(29) from Max Alekseyev, Apr 27 2010

A184677 Number of numbers <= p^2 with largest prime factor <= p, where p is the n-th prime; a(0) = 1.

Original entry on oeis.org

1, 3, 7, 16, 30, 61, 88, 138, 177, 248, 361, 423, 569, 690, 777, 924, 1137, 1370, 1495, 1765, 1979, 2129, 2452, 2711, 3075, 3563, 3871, 4078, 4412, 4639, 4996, 6027, 6427, 6988, 7272, 8181, 8494, 9135, 9803, 10320, 11031, 11768, 12140, 13315, 13713, 14330

Offset: 0

Reinhard Zumkeller, Jun 27 2011

nonn

a(n) = #{m: m<=A001248(n) and A006530(m)<=A000040(n)} for n > 0.

			a(1) = #{1,2,4} = 3 = number of binary powers <= 4;
a(2) = #{1,2,3,4,6,8,9} = 7 = number of 3-smooth numbers <= 9;
a(3) = #{1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25} = 16 = number of 5-smooth numbers <= 25.

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Smooth numbers: A000079, A003586, A051037, A002473, A051038, A080197, A080681, A080682, A080683.

Cf. A027424.

Block[{nn = 45, w}, w = Array[FactorInteger[#][[All, 1]] &, Prime[nn]^2]; {1}~Join~Table[Count[w[[1 ;; p^2]], ?(AllTrue[#, # <= p &] &)], {p, Prime@ Range@ nn}]] (* _Michael De Vlieger, Mar 13 2021 *)

a(n)=if(n==0, return(1)); my(p=prime(n),s=p); forfactored(k=p+1,p^2, if(vecmax(k[2][,1])<=p, s++)); s \\ Charles R Greathouse IV, Nov 27 2017

A080785 Least p-smooth number not less n, where p is the smallest prime factor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 8, 9, 16, 11, 16, 13, 16, 16, 16, 17, 32, 19, 32, 24, 32, 23, 32, 25, 32, 27, 32, 29, 32, 31, 32, 36, 64, 36, 64, 37, 64, 48, 64, 41, 64, 43, 64, 48, 64, 47, 64, 49, 64, 54, 64, 53, 64, 60, 64, 64, 64, 59, 64, 61, 64, 64, 64, 72, 128, 67, 128, 72, 128, 71

Offset: 1

Reinhard Zumkeller, Mar 12 2003

nonn

a(n)<=2^k for n<=2^k.

a(n)=n for n in A000961. - Ivan Neretin, Apr 30 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A020639, A006530.

Cf. A000079, A003586, A051037, A002473, A051038, A080197, A080681, A080682, A080683.

a[n_] := Module[{p, k}, p = FactorInteger[n][[1, 1]]; For[k = n, True, k++, If[FactorInteger[k][[-1, 1]] <= p, Return[k]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 15 2021 *)

A373944 Lexicographically earliest sequence of distinct positive integers such that for A(k) <= n < A(k+1); rad(Product_{i = 1..n} a(i)) = A002110(k), where A = A002110, rad = A007947, k >= 0, n >= 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 3, 6, 9, 12, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 5, 10, 15, 20, 25, 30, 40, 45, 50, 60, 75, 80, 90, 100, 120, 125, 135, 150, 160, 180, 200, 225, 240, 250, 270, 300, 320, 324, 360, 375, 384, 400

Offset: 1

David James Sycamore, Jun 23 2024

Sequence is computed piecewise in blocks of A002110(k+1) - A002110(k) terms, for indices n in the range A002110(k) <= n < A002110(k+1), k = 0,1,2,... in which all terms are the ordered earliest prime(k)-smooth numbers not already recorded in earlier blocks. Since a(0) = 1, and for all k >= 1, all prime(k)-smooth numbers eventually appear in the sequence, this is a permutation of the positive integers, A000027.
From Michael De Vlieger, Jun 25 2024: (Start)
Let P(i) = A002110(i) be the product of i smallest primes.
Let rad = A007947 and let gpf = A006530.
Let S(i) = {k : rad(k) | P(i)}, the prime(i)-smooth numbers.
The notation S(i,j) denotes the j-th smallest term in i, i.e., the j-th term when S(i) is sorted.
This sequence can be seen as a table with row r = 0 {1}, r = 1 {2, 4, 8, 16}, etc.
Then row r contains k in S(r, 1..P(r+1)-1) such that terms k <= S(r-1, P(r)-1) such that gpf(k) < prime(r) are removed.
As a consequence, the sorted union of rows 0..r reconstructs S(r, 1..P(r+1)-1).
For example, A003586(1..29) is given by the sorted union of rows r = 0..2 of the sequence.
The sorted union of rows r = 0..3 gives A051037(1..209), etc.
For r > 1, P(r) is the P(r-1)-th term in row r. (End)

			k = 0 --> A(0) <= n < A(1) --> 1 <= n < 2 --> n = 1 --> a(1) = 1 since rad(1) = 1 = A(0).
k = 1 --> A(1) <= n < A(2) --> 2 <= n < 6 --> n = 2,3,4,5 --> a(2,3,4,5) = 2,4,8,16 (the first 4 terms of A000079, excluding 1).
k = 2 --> 6 <= n < 30 --> n = 6,7,8,9,...,29 --> a(6,7,8,9...,29) = 3,6,9,12,...,288 (the first 24 terms of A003586 excluding all above).
k = 3 --> 30 <= n < 210 --> n = 30,31,32,...,209 --> a(30,31,32,...,209) = 5,10,15,...,19200 (the first 180 terms of A051037 excluding all above).
Sequence can be presented as an irregular table T(n,k), in which the n-th row commences A008578(n); n >= 1, and T(n,k) is the k-th prime(n)-smooth number which has not appeared earlier.
Table starts:
  1;
  2,4,8,16;
  3,6,9,12,18,24,27,32,...,288;
  5,10,15,20,25,30,40,45,50,60,...,19200;
  7,14,21,28,...,13829760;

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n) n = 1..510509.

Cf. A000040, A002110, A002473, A003586, A007947, A051037, A051038, A080197, A080681, A080682, A080683.

(* First, load function f from A162306 *)
P = m = 1; Flatten@ Join[{{1}}, Reap[Do[P *= Prime[i]; (Sow@ Select[#, Nand[# <= m, FactorInteger[#][[-1, 1]] < Prime[i]] &]; m = #[[-1]]) &@ f[P, P^4][[;; P*Prime[i + 1] - 1]], {i, 3}] ][[-1, 1]]] (* Michael De Vlieger, Jun 24 2024 *)

a(A002110(n)) = A000040(n), n >= 1.

A363794 a(n) = smallest prime(n)-smooth number k such that r(k) >= r(P(n+1)), where r(n) = A010846(n) and P(n) = A002110(n).

Original entry on oeis.org

16, 72, 540, 6300, 92400, 1681680, 36756720, 921470550, 27886608750, 970453984500, 37905932634570

Offset: 1

Michael De Vlieger, Jun 22 2023

Let R = r(P(n)) = A010846(A002110(n)) = A363061(n).
Let S(n) be the sorted tensor product of prime power ranges {p(i)^e : i<=n, e>=0}, e.g., S(1) = A000079, S(2) = A003586, S(3) = A051037, etc.
Let T(n) = A002110(n)*S(n). Note that S(1) = T(1) since omega(A002110(1)) = 1.
Let S(n,i) be the i-th term in S(n).
Then a(n) is the smallest S(n,i), i >= R, such that S(n,i) is also in T. Equivalently, a(n) is the smallest S(n,i), i >= R, such that rad(S(n,i)) = A002110(n), where rad(n) = A007947(n).

			a(1) = 16 since r(2^4) = 5 and r(6) = 5; numbers in row 16 of A162306 are its divisors {1, 2, 4, 8, 16}, while row 6 of A162306 is {1, 2, 3, 4, 6}.
a(2) = 72 = A003586(18) since r(72) = r(30) = 18. 72 is the 8th term in A003586 that is not in A000961.
a(3) = 540 since r(540) = 69 which exceeds r(210) = 68.
a(4) = 6300 since r(6300) = 290 which exceeds r(2310) = 283, etc.
Table showing the relationship of a(n) to r(P(n)) = A363061(n), with p(n) = prime(n), P(n+1) = A002110(n+1), r(a(n)) = A010846(a(n)), and j the index such that S(r(a(n))) = T(j) = a(n). a(n) = m*P(n).
   n p(n)        P(n+1)          a(n)  r(P(n))  r(a(n))   j    m
  --------------------------------------------------------------
   1   2             6            16        5        5    4    8
   2   3            30            72       18       18    8   12
   3   5           210           540       68       69   13   18
   4   7          2310          6300      283      290   22   30
   5  11         30030         92400     1161     1165   29   40
   6  13        510510       1681680     4843     4848   42   56
   7  17       9699690      36756720    19985    19994   53   72
   8  19     223092870     921470550    83074    83435   68   95
   9  23    6469693230   27886608750   349670   351047   89  125
  10  29  200560490130  970453984500  1456458  1457926  107  150

Cf. A000079, A000961, A002110, A002473, A003586, A007947, A010846, A051037, A051038, A080197, A080681, A080682, A080683, A162306, A363061.

nn = 6; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; f[x_] := FactorInteger[x][[-1, 1]]; S = Array[Product[Prime[i], {i, #}] &, nn + 1]; Table[Set[{p, q}, Prime[n + {0, 1}]]; r = Count[Range[S[[n + 1]]], _?(f[#] <= q &)]; c = k = 1; While[Or[c < r, rad[k] != S[[n]]], If[f[k] <= p, c++]; k++]; k, {n, nn}]

a(n) >= A363061(n).

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A080786 Triangle T(n,k) = number of k-smooth numbers <= n, read by rows.

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A108275 Number of positive integers <= 10^n that are divisible by no prime exceeding 17.

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A184677 Number of numbers <= p^2 with largest prime factor <= p, where p is the n-th prime; a(0) = 1.

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A080785 Least p-smooth number not less n, where p is the smallest prime factor of n.

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A373944 Lexicographically earliest sequence of distinct positive integers such that for A(k) <= n < A(k+1); rad(Product_{i = 1..n} a(i)) = A002110(k), where A = A002110, rad = A007947, k >= 0, n >= 1.

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A363794 a(n) = smallest prime(n)-smooth number k such that r(k) >= r(P(n+1)), where r(n) = A010846(n) and P(n) = A002110(n).

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