A051038 -id:A051038 - cp's OEIS frontend

A080196 13-smooth numbers which are not 11-smooth.

13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208, 234, 260, 273, 286, 312, 325, 338, 351, 364, 390, 416, 429, 455, 468, 507, 520, 546, 572, 585, 624, 637, 650, 676, 702, 715, 728, 780, 819, 832, 845, 858, 910, 936, 975, 1001, 1014, 1040

Offset: 1

Klaus Brockhaus, Feb 10 2003

Numbers of the form 2^r*3^s*5^t*7^u*11^v*13^w with r, s, t, u, v >= 0, w > 0.

			78 = 2*3*13 is a term but 77 = 7*11 is not.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A080197, A051038.

Select[Range[1000], FactorInteger[#][[-1, 1]] == 13 &] (* Amiram Eldar, Nov 10 2020 *)

{m=1040; z=[]; for(r=0,floor(log(m)/log(2)),a=2^r; for(s=0,floor(log(m/a)/log(3)),b=a*3^s; for(t=0, floor(log(m/b)/log(5)),c=b*5^t; for(u=0,floor(log(m/c)/log(7)),d=c*7^u; for(v=0,floor(log(m/d)/log(11)), e=d*11^v; for(w=1,floor(log(m/e)/log(13)),z=concat(z,e*13^w))))))); z=vecsort(z); for(i=1,length(z),print1(z[i],","))}

from sympy import integer_log, prevprime
def A080196(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    def f(x): return n+x-g(x,13)
    return 13*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

From Amiram Eldar, Nov 10 2020: (Start)
a(n) = 13 * A080197(n).
Sum_{n>=1} 1/a(n) = 77/192. (End)

A080195 11-smooth numbers which are not 7-smooth.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 154, 165, 176, 198, 220, 231, 242, 264, 275, 297, 308, 330, 352, 363, 385, 396, 440, 462, 484, 495, 528, 539, 550, 594, 605, 616, 660, 693, 704, 726, 770, 792, 825, 847, 880, 891, 924, 968, 990, 1056, 1078, 1089

Offset: 1

Klaus Brockhaus, Feb 10 2003

Numbers of the form 2^r*3^s*5^t*7^u*11^v with r, s, t, u >= 0, v > 0.

			33 = 3*11 is a term but 35 = 5*7 is not.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A051038, A002473.

N:= 10^6; # to get all terms <= N
A:= NULL;
for v from 1 to floor(log[11](N)) do
   V:= 11^v;
   for u from 0 to floor(log[7](N/V)) do
      U:= 7^u*V;
      for t from 0 to floor(log[5](N/U)) do
        T:= 5^t*U;
        for s from 0 to floor(log[3](N/T)) do
          S:= 3^s*T;
          for r from 0 to floor(log[2](N/S)) do
             A:= A, 2^r*S
          od
         od
       od
    od
od:
{A}; # Robert Israel, May 28 2014

Select[Range[1000], FactorInteger[#][[-1, 1]] == 11 &] (* Amiram Eldar, Nov 10 2020 *)

{m=1100; z=[]; for(r=0,floor(log(m)/log(2)),a=2^r; for(s=0,floor(log(m/a)/log(3)),b=a*3^s; for(t=0, floor(log(m/b)/log(5)),c=b*5^t; for(u=0,floor(log(m/c)/log(7)),d=c*7^u; for(v=1,floor(log(m/d)/log(11)), z=concat(z,d*11^v)))))); z=vecsort(z); for(i=1,length(z),print1(z[i],","))}

from sympy import integer_log, prevprime
def A080195(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    def f(x): return n+x-g(x,11)
    return 11*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

a(n) = 11 * A051038(n). - David A. Corneth, May 27 2017

Sum_{n>=1} 1/a(n) = 7/16. - Amiram Eldar, Nov 10 2020

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

Original entry on oeis.org

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

A155182 Divisors of 12!.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 50, 54, 55, 56, 60, 63, 64, 66, 70, 72, 75, 77, 80, 81, 84, 88, 90, 96, 99, 100, 105, 108, 110, 112, 120, 126, 128, 132, 135, 140, 144, 150, 154, 160

Offset: 1

Reinhard Zumkeller, Jan 21 2009

The sequence is finite with A027423(12) = 792 terms:

a(792) = A000142(12) = 479001600 is the last term.

			Last ten numbers: 47900160, 53222400, 59875200, 68428800, 79833600, 95800320, 119750400, 159667200, 239500800, 479001600. - _Zerinvary Lajos_, Jun 13 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..792 (full sequence)
Index entries for sequences related to divisors of numbers.
Index entries for sequences related to factorial numbers.

Subsequence of A051038.

Cf. A174228. - Reinhard Zumkeller, May 24 2010

Divisors[12!][[;;100]] (* Paolo Xausa, Jul 31 2024 *)

PARI
```
divisors(12!)
```

divisors(factorial(12)) # Zerinvary Lajos, Jun 13 2009

A253572 Rectangular array A read by upward antidiagonals in which row A(n) is the sequence of all numbers divisible by no prime exceeding prime(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 3, 8, 1, 2, 3, 4, 16, 1, 2, 3, 4, 6, 32, 1, 2, 3, 4, 5, 8, 64, 1, 2, 3, 4, 5, 6, 9, 128, 1, 2, 3, 4, 5, 6, 8, 12, 256, 1, 2, 3, 4, 5, 6, 7, 9, 16, 512, 1, 2, 3, 4, 5, 6, 7, 8, 10, 18, 1024, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 2048

Offset: 1

L. Edson Jeffery, Jan 03 2015

Successive rows tend to A000027.

			Array A starts:
{1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ...}
{1, 2, 3, 4,  6,  8,  9,  12,  16,  18,   24,   27,   32,   36, ...}
{1, 2, 3, 4,  5,  6,  8,   9,  10,  12,   15,   16,   18,   20, ...}
{1, 2, 3, 4,  5,  6,  7,   8,   9,  10,   12,   14,   15,   16, ...}
{1, 2, 3, 4,  5,  6,  7,   8,   9,  10,   11,   12,   14,   15, ...}
{1, 2, 3, 4,  5,  6,  7,   8,   9,  10,   11,   12,   13,   14, ...}
{1, 2, 3, 4,  5,  6,  7,   8,   9,  10,   11,   12,   13,   14, ...}

Cf. A000079, A003586, A051037, A002473, A051038 (these are rows 1-5).

Cf. A000027 (natural numbers), A253573.

r = 20; c = 20; cmax = Max[300, Prime[r + 1]]; a[1] = Table[2^j, {j, 0, cmax}]; b[1] = a[1]; For[n = 2, n <= r, n++, a[n_] := a[n] = {}; b[n_] := b[n] = {}; a[n] = Union[Flatten[Table[Prime[n]^j*b[n - 1], {j, 0, cmax}]]]; For[k = 1, k <= cmax, k++, AppendTo[b[n], a[n][[k]]]]]; Table[b[n - k + 1][[k]], {n, 13}, {k, n}] // Flatten (* Array antidiagonals flattened. *)
(* Second program: *)
rows = 13; smoothNumbers[p_, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand @ Log[pp[[j]], max/Times @@ (Take[pp, j-1]^Take[aa, j-1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; t = Table[p = Prime[n]; Take[smoothNumbers[p, If[p == 2, 2^rows, (1/Sqrt[6])* Exp[Sqrt[2*Log[2]*Log[3]*rows]]]], rows-n+1], {n, 1, rows}];  Table[t[[n-k+1, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 09 2016 *)

A(n) = {prime(1)^(i_1)*...*prime(n)^(i_n) : i_1,...,i_n in {0,1,2,...}}.

A(1) subset A(2) subset A(3) subset ... .

First formula corrected by Tom Edgar, Jan 08 2015

A254196 a(n) is the numerator of Product_{i=1..n} (1/(1-1/prime(i))) - 1.

Original entry on oeis.org

1, 2, 11, 27, 61, 809, 13945, 268027, 565447, 2358365, 73551683, 2734683311, 112599773191, 4860900544813, 9968041656757, 40762420985117, 83151858555707, 5085105491885327, 341472595155548909, 24295409051193284539

Offset: 1

Geoffrey Critzer, Jan 26 2015

nonn

The denominators are A038110(n+1).
a(n)/A038110(n+1) = Sum_{k >=2} 1/k where k is a positive integer whose prime factors are among the first n primes. In particular, for n=1,2,3,4,5, a(n)/A038110(n+1) is the sum of the reciprocals of the terms (excepting the first, 1) in A000079, A003586, A051037, A002473, A051038.
Appears to be a duplicate of A161527. - Michel Marcus, Aug 05 2019

			a(1)=1 because 1/2 + 1/4 + 1/8 + 1/16 + ... = 1/1.
a(2)=2 because 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/12 + ... = 2/1.
a(3)=11 because 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/15 + ... = 11/4.
a(4)=27 because Sum_{n>=2} 1/A002473(n) = 27/8.
a(5)=61 because Sum_{n>=2} 1/A051038(n) = 61/16.

Robert Israel, Table of n, a(n) for n = 1..422
Eric Weisstein's World of Mathematics, Smooth Number

Cf. A038110, A000079, A003586, A051037, A002473, A051038, A161527.

seq(numer(mul(1/(1-1/ithprime(i)),i=1..n)-1),n=1..20); # Robert Israel, Jan 28 2015

Numerator[Table[Product[1/(1 - 1/p), {p, Prime[Range[n]]}] - 1, {n,1,20}]]
b[0] := 0; b[n_] := b[n - 1] + (1 - b[n - 1]) / Prime[n]
Numerator@ Table[b[n], {n, 1, 20}] (* Fred Daniel Kline, Jun 27 2017 *)

a(n) = numerator(prod(i=1, n, (1/(1-1/prime(i)))) - 1); \\ Michel Marcus, Jun 29 2017

a(n) = A038111(n+1)/prime(n+1)-A038110(n+1). - Robert Israel, Jan 28 2015, corrected Jul 07 2019.

A080187 Primes p such that 11 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

Original entry on oeis.org

19, 97, 197, 461, 659, 1319, 1451, 2111, 2309, 2969, 3167, 3299, 4157, 5279, 7127, 9239, 10889, 11549, 15971, 16631, 22637, 25409, 26729, 29567, 30491, 34649, 34847, 55439, 55901, 64151, 87119, 92399, 98009, 110879, 118799, 152459, 164999, 176417

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

The sequence appears to consist of 19, 97 and the lesser of twin primes q (A001359) such that q+1 is 11-smooth (A051038) but not 7-smooth (A002473, A080195).

			97 is a term since 98 = 2*7^2, 99 = 3^2*11, 100 = 2^2*5^2 are the numbers between 97 and the next prime 101;
461 is a term since 462 = 2*3*7*11 is the only number between 461 and the next prime 463.

Amiram Eldar, Table of n, a(n) for n = 1..521 (terms below 10^11)

Cf. A052248, A001359, A051038, A002473, A080195.

maxPrime[n1_, n2_] := FactorInteger[#][[-1, 1]] & /@ Range[n1, n2]; Select[Range[180000], PrimeQ[#] && Max[maxPrime[# + 1, NextPrime[#] - 1]] == 11 &] (* Amiram Eldar, Feb 08 2020 *)

{forprime(p=2,180000,q=nextprime(p+1); m=0; j=p+1; while(j

A080188 Primes p such that 13 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

Original entry on oeis.org

23, 311, 349, 857, 1091, 1871, 1949, 2027, 2339, 2729, 3119, 3821, 5849, 6551, 7487, 9437, 10139, 10529, 11699, 15287, 18251, 21059, 21839, 38609, 42899, 49919, 51479, 57329, 61151, 65519, 69497, 70199, 70979, 81899, 97499, 108107, 109199, 114659

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

The sequence appears to consist of 23, 349 and the lesser of twin primes q (A001359) such that q+1 is 13-smooth (A080197) but not 11-smooth (A051038, A080196).

			349 is a term since 350 = 2*5^2*7, 351 = 3^3*13, 352 = 2^5*11 are the numbers between 349 and the next prime 353; 857 is a term since 858 = 2*3*11*13 is the only number between 857 and the next prime 859.

Amiram Eldar, Table of n, a(n) for n = 1..1092 (terms below 10^11)

Cf. A052248, A001359, A080197, A051038, A080196.

maxPrime[n1_, n2_] := FactorInteger[#][[-1, 1]] & /@ Range[n1, n2]; Select[Range[120000], PrimeQ[#] && Max[maxPrime[# + 1, NextPrime[#] - 1]] == 13 &] (* Amiram Eldar, Feb 08 2020 *)

{forprime(p=2,120000,q=nextprime(p+1); m=0; j=p+1; while(j

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

A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 15, 25, 11, 21, 13, 49, 35, 8, 17, 27, 19, 55, 77, 121, 23, 33, 65, 169, 39, 91, 29, 85, 31, 10, 143, 289, 119, 45, 37, 361, 221, 95, 41, 133, 43, 187, 115, 529, 47, 51, 161, 125, 323, 247, 53, 57, 209, 203, 437, 841, 59, 145, 61, 961

Offset: 1

Rémy Sigrist, Jan 27 2018

nonn

This sequence is a permutation of the natural numbers, with inverse A298882.
For any prime p and k > 0:
- if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,
- then a(p * s_p(k)) = p * r_p(k),
- for example: a(11 * A051038(k)) = 11 * A008364(k).

			The first terms, alongside A006530(n), are:
  n     a(n)   gpf(n)
  --    ----   ------
   1      1      1
   2      2      2
   3      3      3
   4      4      2
   5      5      5
   6      9      3
   7      7      7
   8      6      2
   9     15      3
  10     25      5
  11     11     11
  12     21      3
  13     13     13
  14     49      7
  15     35      5
  16      8      2
  17     17     17
  18     27      3
  19     19     19
  20     55      5

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (with prime and semiprime values highlighted)
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A006530(n))
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A176506(k) when a(n) is the k-th semiprime)
Rémy Sigrist, PARI program for A298268
Index entries for sequences that are permutations of the natural numbers

Cf. A006530, A008364, A046022, A051038, A061395, A078899, A083140, A125624, A151800, A176506, A298268, A298882 (inverse).

PARI
```
See Links section.
```

a(1) = 1.
a(A125624(n, k)) = A083140(n, k) for any n > 0 and k > 0.
a(n) = A083140(A061395(n), A078899(n)) for any n > 1.
Empirically:
- a(n) = n iff n belongs to A046022,
- a(2^k) = 2 * k for any k > 0,
- a(2 * p) = p^2 for any prime p,
- a(3 * p) = p * A151800(p) for any odd prime p.

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A080196 13-smooth numbers which are not 11-smooth.

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A080195 11-smooth numbers which are not 7-smooth.

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

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A155182 Divisors of 12!.

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A253572 Rectangular array A read by upward antidiagonals in which row A(n) is the sequence of all numbers divisible by no prime exceeding prime(n).

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A254196 a(n) is the numerator of Product_{i=1..n} (1/(1-1/prime(i))) - 1.

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A080187 Primes p such that 11 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

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