A080197 -id:A080197 - cp's OEIS frontend

A128605 Smallest number m having exactly n divisors d with sqrt(m/2) <= d < sqrt(2*m).

3, 1, 6, 72, 120, 1800, 840, 3600, 2520, 28800, 10080, 88200, 27720, 259200, 50400, 176400, 83160, 352800, 138600, 3484800, 277200, 1411200, 360360, 2822400, 831600, 3175200, 720720, 6350400, 1663200, 31363200, 1441440, 28576800, 2162160, 12700800, 3326400, 21344400, 4324320

Offset: 0

Reinhard Zumkeller, Mar 14 2007

nonn

A067742(a(n)) = n and A067742(m) <> n for m < a(n).
From Hartmut F. W. Hoft, Feb 06 2017: (Start)
a(66)=86486400 has the largest index n with a(n) <= 100000000, but there are 12 values from a(38) to a(67) that are larger than 100000000.
Conjecture: a(n) = k where p(k) and p(k-1) are the first pair of Dyck paths for the symmetric representation of sigma(k) and sigma(k-1), as described in A237593, having a gap of exactly n units on the diagonal, i.e., it is the sequence of record gaps in sequence A240542; tested through 2000000 with a variant of function A279286. (End)
The first 37 terms are 13-smooth (see A080197). - David A. Corneth, Apr 07 2018

			A067742(a(5)) = A067742(1800) = #{30,36,40,45,50} = 5;
A067742(a(6)) = A067742(840) = #{21,24,28,30,35,40} = 6;
A067742(a(7)) = A067742(3600) = #{45,48,50,60,72,75,80} = 7.
a(0)=3 since 3 has no middle divisors. - _Hartmut F. W. Hoft_, Feb 06 2017

David A. Corneth, Upper bounds on a(0)..a(376) and some more values
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015, see page 29 Remarks 6.8(b). [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016, see Remark 1.3.

Cf. A067742.

Related to Dyck paths: A237593, A240542, A279286.

(* computation based on the function of Michael Somos in A067742 *)
a128605[pL_,b_] := Module[{posL=Map[0&, Range[pL]], k=1, mCur, count}, While[k<=b, mCur=DivisorSum[k, 1&, k/2 <= #^2 < 2k&]; If[posL[[mCur]]==0, posL[[mCur]]=k]; k++]; Prepend[posL, 3]]
a128605[70,100000000] (* computes those a(0) .. a(66) <= 100000000 *)
(* Hartmut F. W. Hoft, Feb 06 2017 *)

ct(m)=my(lower=if(m%2==0&&issquare(m/2), sqrtint(m/2), sqrtint(m\2)+1), upper=sqrtint(2*m)); sumdiv(m, d, lower<=d && d<=upper)
v=vector(10^3); need=1; for(m=1, 1e9, t=ct(m); if(t>=need && v[t]==0, v[t]=m; print("a("t") = "n); while(v[need], need++))) \\ Charles R Greathouse IV, Feb 06 2017

a(33)-a(37) from Hartmut F. W. Hoft, Feb 06 2017

A147573 Numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13}.

Original entry on oeis.org

30030, 60060, 90090, 120120, 150150, 180180, 210210, 240240, 270270, 300300, 330330, 360360, 390390, 420420, 450450, 480480, 540540, 600600, 630630, 660660, 720720, 750750, 780780, 810810, 840840, 900900, 960960, 990990, 1051050, 1081080, 1171170, 1201200, 1261260

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Successive numbers k such that EulerPhi(x)/x = m:
( Family of sequences for successive n primes )
m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079
m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845
m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207
m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571
m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572
m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573
m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574
m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575
Although 39270 has exactly 6 distinct prime divisors (39270=2*3*5*7*11*17), it is not in this sequence because the 6 distinct prime divisors may only comprise 2, 3, 5, 7, 11, and 13. - Harvey P. Dale, Oct 11 2014

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

Subsequence of A067885 and of A080197.

Cf. A060735, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x]/x == 192/1001, AppendTo[a, x]], {x, 1, 100000}]; a

is(n)=if(n%30030, return(0)); my(g=30030); while(g>1, n/=g; g=gcd(n,30030)); n==1 \\ Charles R Greathouse IV, Sep 14 2015

a(n) = 30030 * A080197(n). - Charles R Greathouse IV, Sep 14 2015

Sum_{n>=1} 1/a(n) = 1/5760. - Amiram Eldar, Nov 12 2020

More terms from Amiram Eldar, Mar 10 2020

A326320 Numbers that cannot be written as a difference of 13-smooth numbers.

Original entry on oeis.org

35803, 36349, 41299, 43591, 45109, 45583, 53821

Offset: 1

Esteban Crespi de Valldaura, Jun 26 2019

Terms were found by generating in ascending order the 13-smooth numbers up to some limit and checking the differences. The first 7 candidates k have been proved true by showing a set of 31 congruences to be impossible (see link below). For a discussion iof this method of proof see A308247.

			35803 = A308247(6) cannot be written as the difference of 13-smooth numbers. All smaller numbers can; for example, 9007 = 3^2*7*11*13 - 2, 10091 = 2^2*3*5*13^2 - 7^2.

Esteban Crespi de Valldaura, Congruences needed in proof of a(1)..a(7)

Cf. A080197 (13-smooth numbers).
Cf. numbers not the difference of p-smooth numbers for other values of p: A101082 (p=2), A290365 (p=3), A308456 (p=5), A326318 (p=7), A326319 (p=11).
Cf. A308247.

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

Original entry on oeis.org

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)

A106629 Number of positive integers <= 10^n that are divisible by no prime exceeding 13.

Original entry on oeis.org

1, 10, 62, 242, 733, 1848, 4106, 8289, 15519, 27365, 45914, 73908, 114831, 173077, 254065, 364385, 511985, 706293, 958460, 1281500, 1690506, 2202871, 2838489, 3620013, 4573071, 5726533, 7112760, 8767880, 10732089, 13049906, 15770500, 18948010, 22641849, 26917042, 31844560

Offset: 0

Robert G. Wilson v, May 27 2005

nonn

Row 6 of A253635.

Cf. A011557, A080197.

n = 10; t = 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 &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 10}];

a(21)-a(34) from Daniel Suteu, Nov 14 2019

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

A076848 Smallest prime factor of googol + n that exceeds 13, or 1 if googol + n is 13-smooth.

Original entry on oeis.org

73, 4832936419, 157, 20794121, 127, 859493, 557, 113, 3221, 19, 7549, 43, 17, 61, 211, 241, 18617, 907, 419, 47, 443, 911, 47955653711170550856726386495271851, 1109, 53, 31, 37, 2543, 19, 17, 617, 33521, 23, 7906914473, 38851, 421, 39640576062095087

Offset: 1

Jason Earls, Nov 23 2002

nonn

A googol is 10^100. Factors found using PARI and Dario Alpern's ECM factorization applet.

Sean A. Irvine, Table of n, a(n) for n = 1..1000 (terms 1..400 from Charles R Greathouse IV, n=260 corrected by Sean A. Irvine)
Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A049014 (n such that googol + n is prime), A066298, A080197 (relates to positions of 1's).
Equivalent sequences: A072288 (googolplex + n), A078813 (googol - n).
See the formula section for the relationships with A007947, A020639, A034386.

a(n) = A020639(A007947(10^100 + n)/gcd(10^100 + n, A034386(13))), where A020639(m) = lpf(m), smallest prime factor of m. - Peter Munn, Feb 20 2025

a(23) found by Sean A. Irvine on Dec 08 2002 by employing SNFS, using the polynomials x^5+23 and x-10^20.
Edited by Robert G. Wilson v, Dec 09 2002
Edited by Peter Munn, Feb 20 2025

A078813 Smallest prime factor of googol - n that exceeds 13, or 1 if googol - n is 13-smooth.

Original entry on oeis.org

1, 41, 220217, 596275259857, 17, 31, 7583, 167988019, 1898431, 19, 37, 8747, 433, 23, 4647535350279428239, 1637, 29, 1997, 569, 383, 71, 17, 179, 683592593118601, 601, 1259, 109, 47, 19, 83, 367, 43, 151, 8633431, 103, 20859069935591, 23

Offset: 0

Robert G. Wilson v, Dec 06 2002

nonn

			From _Zhuorui He_, Jul 15 2025: (Start)
Googol = 10^100 = 2^100 * 5^100 is 13-smooth so a(0)=1.
10^100 - 1 = 3^2 * 11 * 41 * 101 * 251 * 271 * ... so a(1)=41. (End)

Sean A. Irvine, Table of n, a(n) for n = 0..1000
Dario Alejandro Alpern, Factors of 1000 numbers starting from googolplex
Robert Harley, Factors of Googolplex+1
Eric Weisstein's World of Mathematics, Googol

Cf. A108251 (n such that googol - n is prime), A080197 (relates to positions of 1's).
Equivalent sequences: A076848 (googol + n), A078814 (googolplex - n).
See the formula section for the relationships with A007947, A020639, A034386.

/* using M. F. Hasler's definition for A020639 */
A078813(n)={n=10^100-n; my(p=[2,3,5,7,11,13]); for(i=1, 6, n=n/(p[i]^valuation(n,p[i]))); A020639(n)} /* Zhuorui He , Jul 17 2025 */

For n >= 1, a(n) = A020639(A007947(10^100 - n)/gcd(10^100 - n, A034386(13))), where A020639(m) = lpf(m), smallest prime factor of m. - Peter Munn, Feb 20 2025

a(-n) = A076848(n). - Zhuorui He, Jul 15 2025

Name edited by Peter Munn, Feb 20 2025

a(0) prepended by Zhuorui He, Jul 15 2025

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

cp's OEIS Frontend

A128605 Smallest number m having exactly n divisors d with sqrt(m/2) <= d < sqrt(2*m).

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A147573 Numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13}.

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A326320 Numbers that cannot be written as a difference of 13-smooth numbers.

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

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A106629 Number of positive integers <= 10^n that are divisible by no prime exceeding 13.

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

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A076848 Smallest prime factor of googol + n that exceeds 13, or 1 if googol + n is 13-smooth.

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A078813 Smallest prime factor of googol - n that exceeds 13, or 1 if googol - n is 13-smooth.

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