A002473 -id:A002473 - cp's OEIS frontend

A080197 13-smooth numbers: numbers whose prime divisors are all <= 13.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 56, 60, 63, 64, 65, 66, 70, 72, 75, 77, 78, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 104, 105, 108, 110, 112, 117, 120

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, w >= 0.

			33 = 3*11 and 39 = 3*13 are terms but 34 = 2*17 is not.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000079, A080196. For p-smooth numbers with other values of p, see A003586, A051037, A002473, A051038, A080681, A080682, A080683.

[n: n in [1..150] | PrimeDivisors(n) subset PrimesUpTo(13)]; // Bruno Berselli, Sep 24 2012

mx = 120; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l*11^m*13^n, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[5, mx/(2^i*3^j)]}, {l, 0, Log[7, mx/(2^i*3^j*5^k)]}, {m, 0, Log[11, mx/(2^i*3^j*5^k*7^l)]}, {n, 0, Log[13, mx/(2^i*3^j*5^k*7^l*11^m)]}] (* Robert G. Wilson v, Aug 17 2012 *)

test(n)=m=n; forprime(p=2,13, while(m%p==0,m=m/p)); return(m==1)
for(n=1,200,if(test(n),print1(n",")))

is_A080197(n,p=13)=n<=p||vecmax(factor(n,p+1)[,1])<=p \\ M. F. Hasler, Jan 16 2015

list(lim,p=13)=if(p==2, return(powers(2, logint(lim\1,2)))); my(v=[],q=precprime(p-1),t=1); for(e=0,logint(lim\=1,p), v=concat(v, list(lim\t,q)*t); t*=p); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

import heapq
from itertools import islice
from sympy import primerange
def agen(p=13): # generate all p-smooth terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(agen(), 69))) # Michael S. Branicky, Nov 20 2022

from sympy import integer_log, prevprime
def A080197(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 bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = Product_{primes p <= 13} p/(p-1) = (2*3*5*7*11*13)/(1*2*4*6*10*12) = 1001/192. - Amiram Eldar, Sep 22 2020

A080682 19-smooth numbers: numbers whose prime divisors are all <= 19.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 88, 90, 91, 95, 96, 98, 99, 100

Offset: 1

Cino Hilliard, Mar 02 2003

William A. Tedeschi, Table of n, a(n) for n = 1..10000

For p-smooth numbers with other values of p, see A003586, A051037, A002473, A051038, A080197, A080681, A080683.

[n: n in [1..100] | PrimeDivisors(n) subset PrimesUpTo(19)]; // Bruno Berselli, Sep 24 2012

mx = 120; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l*11^m*13^n*17^o*19^p, {i, 0, Log[2,mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[5, mx/(2^i*3^j)]},{l, 0, Log[7, mx/(2^i*3^j*5^k)]}, {m, 0, Log[11, mx/(2^i*3^j*5^k*7^l)]}, {n, 0, Log[13, mx/(2^i*3^j*5^k*7^l*11^m)]}, {o, 0, Log[17, mx/(2^i*3^j*5^k*7^l*11^m*13^n)]}, {p, 0, Log[19, mx/(2^i*3^j*5^k*7^l*11^m*13^n*17^o)]}] (* Robert G. Wilson v, Jan 19 2016 *)
Select[Range[100],Max[FactorInteger[#][[All,1]]]<20&] (* Harvey P. Dale, Sep 20 2018 *)

test(n)= {m=n; forprime(p=2,19, while(m%p==0,m=m/p)); return(m==1)}
for(n=1,200,if(test(n),print1(n",")))

list(lim,p=19)=if(p==2, return(powers(2, logint(lim\1,2)))); my(v=[],q=precprime(p-1),t=1); for(e=0,logint(lim\=1,p), v=concat(v, list(lim\t,q)*t); t*=p); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

import heapq
from itertools import islice
from sympy import primerange
def agen(p=19): # generate all p-smooth terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(agen(), 72))) # Michael S. Branicky, Nov 20 2022

from sympy import integer_log
def A080682(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,19)
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = Product_{primes p <= 19} p/(p-1) = (2*3*5*7*11*13*17*19)/(1*2*4*6*10*12*16*18) = 323323/55296. - Amiram Eldar, Sep 22 2020

A080683 23-smooth numbers: numbers whose prime divisors are all <= 23.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 88, 90, 91, 92, 95

Offset: 1

Cino Hilliard, Mar 02 2003

Coincides for the first 111 terms with A174228 (divisors of 24!). - Bruno Berselli, Sep 24 2012

William A. Tedeschi, Table of n, a(n) for n = 1..10000

For p-smooth numbers with other values of p, see A003586, A051037, A002473, A051038, A080197, A080681, A080682.

[n: n in [1..100] | PrimeDivisors(n) subset PrimesUpTo(23)]; // Bruno Berselli, Sep 24 2012

select(t -> max(numtheory:-factorset(t)) <= 23, [$1..1000]); # Robert Israel, Jan 22 2016

mx = 100; Sort@ Flatten@ Table[ 2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h*23^i, {a, 0, Log[2, mx]}, {b, 0, Log[3, mx/2^a]}, {c, 0, Log[5, mx/(2^a*3^b)]}, {d, 0, Log[7, mx/(2^a*3^b*5^c)]}, {e, 0, Log[11, mx/(2^a*3^b*5^c*7^d)]}, {f, 0, Log[13, mx/(2^a*3^b*5^c*7^d*11^e)]}, {g, 0, Log[17, mx/(2^a*3^b*5^c*7^d*11^e*13^f)]}, {h, 0, Log[19, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g)]}, {i, 0, Log[23, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h)]}] (* Robert G. Wilson v, Jan 19 2016 *)

test(n)=m=n; forprime(p=2,23, while(m%p==0,m=m/p)); return(m==1)
for(n=1,100,if(test(n),print1(n",")))

list(lim,p=23)=if(p==2, return(powers(2, logint(lim\1,2)))); my(v=[],q=precprime(p-1),t=1); for(e=0,logint(lim\=1,p), v=concat(v, list(lim\t,q)*t); t*=p); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

import heapq
from itertools import islice
from sympy import primerange
def agen(p=23): # generate all p-smooth terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(agen(), 72))) # Michael S. Branicky, Nov 20 2022

from sympy import integer_log, prevprime
def A080683(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,23)
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = Product_{primes p <= 23} p/(p-1) = (2*3*5*7*11*13*17*19*23)/(1*2*4*6*10*12*16*18*22) = 676039/110592. - Amiram Eldar, Sep 22 2020

A147571 Numbers with exactly 4 distinct prime divisors {2,3,5,7}.

Original entry on oeis.org

210, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2520, 2940, 3150, 3360, 3780, 4200, 4410, 5040, 5250, 5670, 5880, 6300, 6720, 7350, 7560, 8400, 8820, 9450, 10080, 10290, 10500, 11340, 11760, 12600, 13230, 13440, 14700, 15120, 15750, 16800

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

David A. Corneth, Table of n, a(n) for n = 1..10688

Cf. A002473, A086780, A143207, A147572, A147573, A147574, A147575, A147576, A147577, A147578, A147579, A147580.

[n: n in [1..2*10^4] | PrimeDivisors(n) eq [2,3,5,7]]; // Vincenzo Librandi, Sep 15 2015

a = {}; Do[If[EulerPhi[x]/x == 8/35, AppendTo[a, x]], {x, 1, 100000}]; a
Select[Range[20000],PrimeNu[#]==4&&Max[FactorInteger[#][[;;,1]]]<11&] (* Harvey P. Dale, Nov 05 2024 *)

is(n)=n%210==0 && n==2^valuation(n,2) * 3^valuation(n,3) * 5^valuation(n,5) * 7^valuation(n,7) \\ Charles R Greathouse IV, Jun 19 2016

a(n) = 210 * A002473(n). - David A. Corneth, May 14 2019

Sum_{n>=1} 1/a(n) = 1/48. - Amiram Eldar, Nov 12 2020

A080681 17-smooth numbers: numbers whose prime divisors are all <= 17.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 60, 63, 64, 65, 66, 68, 70, 72, 75, 77, 78, 80, 81, 84, 85, 88, 90, 91, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Cino Hilliard, Mar 02 2003

William A. Tedeschi, Table of n, a(n) for n = 1..10000

For p-smooth numbers with other values of p, see A003586, A051037, A002473, A051038, A080197, A080682, A080683.

[n: n in [1..150] | PrimeDivisors(n) subset PrimesUpTo(17)]; // Bruno Berselli, Sep 24 2012

mx = 120; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l*11^m*13^n*17^o, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[5, mx/(2^i*3^j)]}, {l, 0, Log[7, mx/(2^i*3^j*5^k)]}, {m, 0, Log[11, mx/(2^i*3^j*5^k*7^l)]}, {n, 0, Log[13, mx/(2^i*3^j*5^k*7^l*11^m)]}, {o, 0, Log[17, mx/(2^i*3^j*5^k*7^l*11^m*13^n)]}] (* Robert G. Wilson v, Aug 17 2012 *)

test(n)= {m=n; forprime(p=2,17, while(m%p==0,m=m/p)); return(m==1)}
for(n=1,200,if(test(n),print1(n",")))

list(lim,p=17)=if(p==2, return(powers(2, logint(lim\1,2)))); my(v=[],q=precprime(p-1),t=1); for(e=0,logint(lim\=1,p), v=concat(v, list(lim\t,q)*t); t*=p); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

import heapq
from itertools import islice
from sympy import primerange
def agen(p=17): # generate all p-smooth terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(agen(), 70))) # Michael S. Branicky, Nov 20 2022

from sympy import integer_log, prevprime
def A080681(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,17)
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = Product_{primes p <= 17} p/(p-1) = (2*3*5*7*11*13*17)/(1*2*4*6*10*12*16) = 17017/3072. - Amiram Eldar, Sep 22 2020

A080194 7-smooth numbers which are not 5-smooth.

Original entry on oeis.org

7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 84, 98, 105, 112, 126, 140, 147, 168, 175, 189, 196, 210, 224, 245, 252, 280, 294, 315, 336, 343, 350, 378, 392, 420, 441, 448, 490, 504, 525, 560, 567, 588, 630, 672, 686, 700, 735, 756, 784, 840, 875, 882, 896, 945, 980

Offset: 1

Klaus Brockhaus, Feb 10 2003

Numbers of the form 7*2^r*3^s*5^t*7^u with r, s, t, u >= 0.
Multiples of 7 which are members of A002473. Or multiples of 7 with the largest prime divisor < 10.
Numbers whose greatest prime factor (A006530) is 7. - M. F. Hasler, Nov 21 2018

			28 = 2^2*7 is a term but 30 = 2*3*5 is not.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A002473, A051037.

Cf. A085125, A085126, A085127, A085128, A085129, A085131, A085132.

Select[Range[999], FactorInteger[#][[-1, 1]] == 7 &] (* Giovanni Resta, Nov 22 2018 *)

A080194_list(M)={my(L=List(),a,b,c); for(r=1,logint(M\1,7), a=7^r; for(s=0, logint(M\a,3), b=a*3^s; for(t=0,logint(M\b,5), c=b*5^t; for(u=0,logint(M\c,2), listput(L,c<A051037. - Edited by M. F. Hasler, Nov 22 2018

select( is_A080194(n)={n>1 && vecmax(factor(n,7)[,1])==7}, [0..10^3]) \\ Defines is_A080194(), used elsewhere. The select() command is a check and illustration. For longer lists, use list() above. - M. F. Hasler, Nov 21 2018

from sympy import integer_log
def A080194(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 f(x):
        c = n+x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                for k in range(integer_log(r:=m//5**j,3)[0]+1):
                    c -= (r//3**k).bit_length()
        return c
    return bisection(f,n,n)*7 # Chai Wah Wu, Sep 17 2024

# faster for initial segment of sequence
import heapq
from itertools import islice
from sympy import primerange
def A080194gen(p=7): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield 7*v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A080194gen(), 65))) # Michael S. Branicky, Sep 17 2024

a(n) = 7 * A002473(n). - David A. Corneth, Nov 22 2018

Sum_{n>=1} 1/a(n) = 5/8. - Amiram Eldar, Nov 10 2020

A068189 Smallest positive number whose product of digits equals n, or a(n)=0 if no such number exists, i.e. when n has a prime divisor greater than 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 0, 26, 0, 27, 35, 28, 0, 29, 0, 45, 37, 0, 0, 38, 55, 0, 39, 47, 0, 56, 0, 48, 0, 0, 57, 49, 0, 0, 0, 58, 0, 67, 0, 0, 59, 0, 0, 68, 77, 255, 0, 0, 0, 69, 0, 78, 0, 0, 0, 256, 0, 0, 79, 88, 0, 0, 0, 0, 0, 257, 0, 89, 0, 0, 355, 0, 0, 0, 0, 258, 99, 0, 0, 267, 0

Offset: 1

Labos Elemer, Feb 19 2002

a(n) > 0 if and only if n is in A002473.

			n=2,10,50,250 gives a(n)=2,25,255,2555; n=11,39,78, etc..a(n)=0.
10000 = 2 * 5 * 5 * 5 * 5 * 8. No product of two of these factors is less than 10 so a(10000) = 255558 (the concatenation of these factors in nondecreasing order). - _David A. Corneth_, Jul 31 2017

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001222, A002473, A007954, A067734, A068183-A068187, A068189-A068191.

Cf. A085123, A280249.

f[x_] := Apply[Times, IntegerDigits[x]] a = Table[0, {256} ]; Do[ b = f[n]; If[b < 257 && a[[b]] == 0, a[[b]] =n], {n, 1, 10000} ]; a

a(n) = {if(n==1, return(1)); my(res = []); forstep(i=9,2,-1, v = valuation(n, i); if(v > 0, res = concat(vector(v, j, i), res); n/=i^v)); if(n==1,fromdigits(res), 0)} \\ David A. Corneth, Jul 31 2017

def convert(n):
    if n == 1:
        return 1
    result = 0
    cur = 1
    while n > 1:
        found = False
        for i in range(9, 1, -1):
            if n % i == 0:
                result += cur * i
                cur *= 10
                n //= i
                found = True
                break
        if not found:
            return 0
    return result
N = 256
for n in range(1, N):
    print(n, convert(n))
# Dmitry Kamenetsky, Oct 20 2008

A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

Original entry on oeis.org

1, 2, 1, 2, 3, 6, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210, 1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 110, 154, 165, 210, 231, 330, 385, 462, 770, 1155, 2310, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 26, 30, 33, 35, 39, 42

Offset: 1

Jean-François Alcover, Nov 26 2015

If we define a triangle whose n-th row consists of all squarefree numbers whose prime factors are all less than prime(k), we get this same triangle except starting with a row {1}, with offset 1. - Gus Wiseman, Aug 24 2021

			Triangle begins:
1, 2;                        squarefree and 2-smooth
1, 2, 3, 6;                  squarefree and 3-smooth
1, 2, 3, 5, 6, 10, 15, 30;
1, 2, 3, 5, 6,  7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210;
...

Jean-François Alcover, Table of n, a(n) for n = 1..2046 (first 10 rows)
A. Hildebrand and G. Tenenbaum, Integers without large prime factors, Journal de théorie des nombres de Bordeaux (1993) Volume:5, Issue:2, p. 411-484.
Eric Weisstein's MathWorld, Smooth number.
Wikipedia, Smooth number

Cf. A000079 (2-smooth), A003586 (3-smooth), A051037 (5-smooth), A002473 (7-smooth), A018336 (7-smooth & squarefree), A051038 (11-smooth), A087005 (11-smooth & squarefree), A080197 (13-smooth), A087006 (13-smooth & squarefree), A087007 (17-smooth & squarefree), A087008 (19-smooth & squarefree).
Row lengths are A000079.
Rightmost terms (or column k = 2^n) are A002110.
Rows are partial unions of rows of A019565.
Row n is A027750(A002110(n)), i.e., divisors of primorials.
Row sums are A054640.
Column k = 2^n-1 is A070826.
Multiplying row n by prime(n+1) gives A339195, row sums A339360.
A005117 lists squarefree numbers.
A056239 adds up prime indices, row sums of A112798.
A072047 counts prime factors of squarefree numbers.
A246867 groups squarefree numbers by Heinz weight, row sums A147655.
A329631 lists prime indices of squarefree numbers, sums A319246.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
Cf. A000040, A001221, A006881, A071403, A209862, A319247.

b:= proc(n) option remember; `if`(n=0, [1],
      sort(map(x-> [x, x*ithprime(n)][], b(n-1))))
    end:
T:= n-> b(n)[]:
seq(T(n), n=1..7);  # Alois P. Heinz, Nov 28 2015

primorial[n_] := Times @@ Prime[Range[n]]; row[n_] := Select[ Divisors[ primorial[n]], SquareFreeQ]; Table[row[n], {n, 1, 10}] // Flatten

T(n-1,k) = A339195(n,k)/prime(n). - Gus Wiseman, Aug 24 2021

A067734 Number of ways writing n as a product of decimal digits of some other number which has no digits equal to 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 2, 0, 7, 0, 2, 2, 7, 0, 7, 0, 5, 2, 0, 0, 17, 1, 0, 3, 5, 0, 8, 0, 13, 0, 0, 2, 21, 0, 0, 0, 12, 0, 8, 0, 0, 5, 0, 0, 38, 1, 3, 0, 0, 0, 15, 0, 12, 0, 0, 0, 24, 0, 0, 5, 24, 0, 0, 0, 0, 0, 6, 0, 58, 0, 0, 3, 0, 0, 0, 0, 26, 5, 0, 0, 24, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 82, 0

Offset: 1

Labos Elemer, Jan 28 2002

For n=36, this was given as an exercise for children of age 14 years.

			There are 21 other numbers with no digit 1 whose digit product equals 36: 49, 66, 94, 229, 236, 263, 292, 326, 334, 343, 362, 433, 623, 632, 922, 2233, 2323, 2332, 3223, 3232, 3322. If 1-digits were permitted then an infinite number of solutions would exist, e.g., 111114111113111113. If n has a prime divisor larger than 7, i.e., a prime divisor that is two or more digits in length, such as 11, then no solutions exist at all. The largest solution is a (decimal) number created by concatenating not-necessarily-distinct prime factors, such as 36 = 3*2*2*2. [edited by _Jon E. Schoenfield_, Jun 14 2017]

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000073, A001222, A002473, A068183-A068187, A068189-A068191.

id1[x_] := IntegerDigits[x]; id2[x_] := DeleteCases[id1[x], 1] f[x_] := Apply[Times, IntegerDigits[x]]; k=0; Do[s=f[n]; If[Equal[s, 36]&&!Greater[Length[id1[n]], Length[id2[n]]], k=k+1; Print[{k, n}]], {n, 1, 3400}]

{ A067734(n) = local(v,r,i2,i3); v=vector(4,i,valuation(n,prime(i))); if(n==1||n!=prod(i=1,4,prime(i)^v[i]), return(0)); r=0; for(i6=0,min(v[1],v[2]), for(i8=0,(v[1]-i6)\3, for(i4=0,(v[1]-i6-3*i8)\2, i2=v[1]-i6-3*i8-2*i4; for(i9=0,(v[2]-i6)\2, i3=v[2]-i6-2*i9; r += (i2+i3+i4+v[3]+i6+v[4]+i8+i9)! / i2! / i3! / i4! / v[3]! / i6! / v[4]! / i8! / i9! )))); r } \\ Max Alekseyev, Sep 19 2009

a(A002473(n)) > 0 for n > 1. - David A. Corneth, Jun 14 2017

A084034 Numbers which are a product of repeated-digit numbers A010785.

Original entry on oeis.org

0, 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, 49, 50, 54, 55, 56, 60, 63, 64, 66, 70, 72, 75, 77, 80, 81, 84, 88, 90, 96, 98, 99, 100, 105, 108, 110, 111, 112, 120, 121, 125, 126, 128

Offset: 1

Amarnath Murthy, May 26 2003

Numbers which can be written as a*b*c*... where a, b, c are numbers whose decimal expansions are repetitions of a single digit.
Superset of A051038. The first numbers in this sequence but not in A051038 are 111, 222, 333, 444, 555. - R. J. Mathar, Sep 17 2008
From David A. Corneth, Aug 03 2017: (Start)
Closed under multiplication.
Multiples of 1-digit primes and numbers of the form (10^n - 1) / 9. (End)

			99 is a member since 99 = 3*33.
9768 is a member since 9768= 2*22*222.
111*2*33*44 = 322344 is a member.

David A. Corneth, Table of n, a(n) for n = 1..12917 (Terms <= 10^8)

Cf. A002275, A010785, A051038, A084348.

A002473 gives products of single-digit numbers.

isA010786 := proc(n) if nops(convert(convert(n,base,10),set)) = 1 then true; else false ; fi; end: isA084034 := proc(n,a010785) local d ; if n = 1 then RETURN(true) ; fi; for d in ( numtheory[divisors](n) minus{1} ) do if d in a010785 then if isA084034(n/d,a010785) then RETURN(true) ; fi; fi; od: RETURN(false) ; end: nmax := 1000: a010785 := [] : for k from 1 to nmax do if isA010786(k) then a010785 := [op(a010785),k] ; fi; od: for n from 1 to nmax do if isA084034(n,a010785) then printf("%d,",n) ; fi; end: # R. J. Mathar, Sep 17 2008

Corrected and extended by David Wasserman, Dec 09 2004
Corrected data, offset changed to 1 by David A. Corneth, Aug 03 2017
Edited by N. J. A. Sloane, Jul 02 2017 and Oct 10 2018

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