A002473 -id:A002473 - cp's OEIS frontend

A195238 Numbers with at least 2 and not more than 3 distinct prime factors not greater than 7 that are multiples of 7 or of 15.

14, 15, 21, 28, 30, 35, 42, 45, 56, 60, 63, 70, 75, 84, 90, 98, 105, 112, 120, 126, 135, 140, 147, 150, 168, 175, 180, 189, 196, 224, 225, 240, 245, 252, 270, 280, 294, 300, 315, 336, 350, 360, 375, 378, 392, 405, 441, 448, 450, 480, 490, 504, 525, 540, 560

Offset: 1

Harvey P. Dale, Sep 13 2011

A143204 is a subsequence.

Subsequence of A002473.

			a(10) = 60 = 2^2 * 3 * 5.
a(11) = 63 = 3^2 * 7.
a(12) = 70 = 2 * 5 * 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A001221, A006530, A143204, A002473.

a195238 n = a195238_list !! (n-1)
a195238_list = filter (\x -> a001221 x `elem` [2,3] &&
                             a006530 x `elem` [5,7] &&
                             (mod x 7 == 0 || mod x 15 == 0)) [1..]
-- Reinhard Zumkeller, Sep 13 2011

pfsQ[n_]:=Module[{fs=Transpose[FactorInteger[n]][[1]]},Max[fs]<8 && 1Harvey P. Dale, Aug 21 2011 *)

is(n)=my(v=apply(p->valuation(n,p), [2,3,5,7])); n==2^v[1]*3^v[2]*5^v[3]*7^v[4] && (v[4] || v[2]*v[3]) && factorback(v)==0 && !!v[1]+!!v[2]+!!v[3]+!!v[4]>1 \\ Charles R Greathouse IV, Sep 14 2015

2 <= A001221(a(n)) <= 3.
5 <= A006530(a(n)) <= 7.
Sum_{n>=1} 1/a(n) = 11/16. - Amiram Eldar, Oct 25 2024

A198375 Smallest n-digit number whose product of digits is n or 0 if no number exists.

Original entry on oeis.org

1, 12, 113, 1114, 11115, 111116, 1111117, 11111118, 111111119, 1111111125, 0, 111111111126, 0, 11111111111127, 111111111111135, 1111111111111128, 0, 111111111111111129, 0, 11111111111111111145, 111111111111111111137, 0, 0, 111111111111111111111138

Offset: 1

Jaroslav Krizek, Oct 23 2011

			113, 131, and 311 are the 3-digit numbers whose product of digits is 3; 113 is the smallest.

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Cf. A198376 (largest), A002473, A068191.

Table[If[FactorInteger[n][[-1, 1]] > 9, 0, i = (10^n - 1)/9; While[i < 10^n && Times @@ IntegerDigits[i] != n, i++]; If[i == 10^n, 0, i]], {n, 30}] (* T. D. Noe, Oct 24 2011 *)

def A198375(n): return int(str(A198376(n))[::-1])
print([A198375(n) for n in range(1, 25)]) # Michael S. Branicky, Jan 21 2021

a(A068191(n)) = 0 for n >=1.

a(n) <> 0 iff n in { A002473 }. - Michael S. Branicky, Jan 21 2021

A219697 Primes neighboring a 7-smooth number.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 89, 97, 101, 107, 109, 113, 127, 139, 149, 151, 163, 167, 179, 181, 191, 193, 197, 199, 211, 223, 239, 241, 251, 257, 269, 271, 281, 293, 337, 349, 359, 379, 383, 401, 419, 421, 431

Offset: 1

Jonathan Vos Post, Nov 25 2012

This is to the 7-smooth numbers A002473 as A219528 is to the 3-smooth numbers A003586 and as A219669 is to the 5-smooth numbers A051037. The first primes NOT within one of a 7-smooth number are 103, 131, 137, 157, 173, ...

			23 is in the sequence as one of 23-1 = 22 = 2 * 11 and 23+1 = 24 = 2^3 * 3 is 7-smooth and 23 is prime. - _David A. Corneth_, Apr 19 2021

David A. Corneth, Table of n, a(n) for n = 1..10765 (terms <= 10^16)

Cf. A000040, A002473, A051037, A080194, A219528.

mx = 2^10; t7 = Select[Sort[Flatten[Table[2^i * 3^j * 5^k * 7^l, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, Log[5, mx]}, {l, 0, Log[7, mx]}]]], # <= mx &]; Union[Select[t7 + 1, PrimeQ], Select[t7 - 1, PrimeQ]] (* T. D. Noe, Nov 26 2012 *)
Select[Prime[Range[90]],Max[FactorInteger[#-1][[;;,1]]]<11||Max[FactorInteger[#+1][[;;,1]]]<11&] (* Harvey P. Dale, Nov 03 2024 *)

is7smooth(n) = forprime(p = 2, 7, n /= p^valuation(n, p)); n==1
is(n) = isprime(n) && (is7smooth(n - 1) || is7smooth(n + 1)) \\ David A. Corneth, Apr 19 2021

Primes INTERSECTION {2^h 3^i 5^j 7^k +/-1 for h,i,j,k >= 0}.

A237851 a(1)=1; a(n) is the smallest integer not yet in the sequence divisible by all nonzero digits of a(n-1).

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 12, 10, 3, 9, 18, 24, 20, 14, 28, 32, 30, 15, 5, 25, 40, 36, 42, 44, 48, 56, 60, 54, 80, 64, 72, 70, 7, 21, 22, 26, 66, 78, 112, 34, 84, 88, 96, 90, 27, 98, 144, 52, 50, 35, 45, 100, 11, 13, 33, 39, 63, 102, 38, 120, 46, 108, 104, 68, 168

Offset: 1

Eric Angelini, Feb 14 2014

A permutation of the naturals with inverse A237860.

If a(n) is a prime greater than 7 then no digit of a(n-1) is greater than 1, cf. A007088.

Lars Blomberg and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A002796, A180410, A002473, A237860, A007088.

import Data.List (nub, sort, delete)
a237851 n = a237851_list !! (n-1)
a237851_list = 1 : f 1 [2..] where
   f x zs = g zs where
     g (u:us) | all ((== 0) . (mod u)) ds = u : f u (delete u zs)
              | otherwise = g us
              where ds = dropWhile (<= 1) $
                         sort $ nub $ map (read . return) $ show x
-- Reinhard Zumkeller, Feb 14 2014

a[1] = 1;
a[n_] := a[n] = For[k = 1, True, k++, If[FreeQ[Array[a, n-1], k], If[ AllTrue[Select[IntegerDigits[a[n-1]], #>0&] // Union, Divisible[k, #]&], Return[k]]]];
a /@ Range[100] (* Jean-François Alcover, Nov 26 2019 *)

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.

A322447 Numbers k where Sum_{d | k} 1/rad(d) increases to a record.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 72, 96, 144, 192, 288, 384, 576, 864, 1152, 1728, 2304, 3456, 4608, 5184, 6912, 10368, 13824, 20736, 27648, 41472, 55296, 62208, 82944, 124416, 165888, 207360, 248832, 331776, 373248, 414720, 497664, 622080, 746496, 829440, 995328

Offset: 1

David S. Metzler, Dec 08 2018

nonn

Let rad(n) be the radical of n, which equals the product of all prime factors of n (A007947). Let g(n) = 1/rad(n) and let f(n) = Sum_{d | n} g(d). This is a multiplicative function whose value on a prime power is f(p^k) = 1 + k/p. Hence f is a weighted divisor-counting function that weights divisors d higher when they have few and small prime divisors themselves. This sequence lists the values where f(n) increases to a record, analogously to highly composite numbers (A002182) or superabundant numbers (A004394). The numbers in this sequence are much smoother than those in the other two sequences, since the definition of f(n) strongly disfavors a lack of smoothness in n.

			The divisors of 12 are 1,2,3,4,6,12, so f(12) = 1 + (1/2) + (1/3) + (1/2) + (1/6) + (1/6) = 8/3, which exceeds f(n) for n = 1,...,11. Alternately, since f is multiplicative, f(12) = f(4)*f(3) = (1+2/2)*(1+1/3).
f(207360) = f(2^9)*f(3^4)*f(5) = (11/2)*(7/3)*(6/5) = 15.4, which exceeds f(n) for n < 207360. (Note that this is the first value of the sequence that is divisible by 5; earlier values are all 3-smooth.)

Charlie Neder, Table of n, a(n) for n = 1..200

Cf. A007947 (radical), A002182, A004394.

Also smooth numbers: A003586, A051037, A002473.

rad[n_] := Times @@ (First@# & /@ FactorInteger@n); f[n_] := DivisorSum[n, 1/rad[#] &]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, Dec 08 2018 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = {my(m=0, newm); for (n=1, nn, newm = sumdiv(n, d, 1/rad(d)); if (newm > m, m = newm; print1(n, ", ")););} \\ Michel Marcus, Dec 09 2018

A335331 a(n) = prime(k) where k is the n-th 7-smooth number.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 89, 97, 103, 107, 113, 131, 149, 151, 173, 181, 197, 223, 227, 229, 251, 263, 281, 307, 311, 349, 359, 379, 409, 419, 433, 463, 503, 521, 541, 571, 593, 613, 659, 691, 701, 719, 761, 809, 827, 853, 863

Offset: 1

David A. Corneth, Jun 01 2020

nonn

At A110069 we look for numbers of the form n = (d_1 + d_2 + ... + d_k)*prime(d_1*d_2*...*d_k) where d_1 d_2 ... d_k is the decimal expansion of n. As the largest prime that can be among the digits of a base-10 number is 7, the product of digits is 7-smooth. Hence the factor prime(d_1*d_2*...*d_k) is a term from this sequence. As lots of numbers have a product of digits of, say, 210^4, it would help to know prime(210^4) in advance. That's a(5817) of this sequence as 210^4 is the 5817th 7-smooth number. Precomputing such numbers is a computational benefit.

David A. Corneth, Table of n, a(n) for n = 1..8862

Cf. A002473, A006988, A033844, A110069.

A342950 7-smooth numbers not divisible by 10: positive numbers whose prime divisors are all <= 7 but do not contain both 2 and 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 16, 18, 21, 24, 25, 27, 28, 32, 35, 36, 42, 45, 48, 49, 54, 56, 63, 64, 72, 75, 81, 84, 96, 98, 105, 108, 112, 125, 126, 128, 135, 144, 147, 162, 168, 175, 189, 192, 196, 216, 224, 225, 243, 245, 252, 256, 288, 294, 315, 324

Offset: 1

David A. Corneth, Mar 30 2021

nonn

			12 is in the sequence as all of its prime divisors are <= 7 and 12 is not divisible by 10.

David A. Corneth, Table of n, a(n) for n = 1..10195

Union of A108319 and A108347.

Intersection of A002473 and A067251.

Select[Range@500,Max[First/@FactorInteger@#]<=7&&Mod[#,10]!=0&] (* Giorgos Kalogeropoulos, Mar 30 2021 *)

is(n) = if(n%10 == 0, return(0)); forprime(p = 2, 7, n/=p^valuation(n, p)); n==1

A342950_list, n = [], 1
while n < 10**9:
    if n % 10:
        m = n
        for p in (2,3,5,7):
            q, r = divmod(m,p)
            while r == 0:
                m = q
                q, r = divmod(m,p)
        if m == 1:
            A342950_list.append(n)
    n += 1 # Chai Wah Wu, Mar 31 2021

from sympy import integer_log
def A342950(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,3)[0]+1):
                c -= (k:=m//3**j).bit_length()+integer_log(k,5)[0]
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 17 2024

# faster for initial segment of sequence
import heapq
from itertools import islice
def A342950gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5, 7]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                if not (p==2 and v%5==0) and not (p==5 and v&1==0):
                    heapq.heappush(h, v*p)
print(list(islice(A342950gen(), 65))) # Michael S. Branicky, Sep 17 2024

Sum_{n>=1} 1/a(n) = 63/16. - Amiram Eldar, Apr 01 2021

A343597 Numbers divisible by a 7-smooth composite number.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 125, 126, 128, 130

Offset: 1

Peter Munn, Apr 21 2021

Numbers divisible by at least one of 4, 6, 9, 10, 14, 15, 21, 25, 35, 49.
Exactly half of the first 10, first 100 and first 600 positive integers are divisible by a 7-smooth composite number; the largest 7-smooth divisor of the remaining numbers is 1, 2, 3, 5 or 7.
Intervals extending to hundreds of integers with exactly 50% membership of this sequence are far from rare, some notable examples being [3000, 3999], [8000, 8999], [20000, 20999], [21000, 21999] and [23000, 23999]. This reflects the asymptotic density of the corresponding set being close to 0.5, precisely 1847 / 3675 = 0.50258503... (and membership of the set has a periodic pattern). See A343598 for further information.

			33 = 11 * 3 has divisors 1, 3, 11, 33, of which only 33 is composite. 33 is not 7-smooth, as its prime factors include 11, which is greater than 7. So 33 is not in the sequence.
52 = 13 * 2 * 2 is divisible by 4, which is composite and 7-smooth, so 52 is in the sequence.

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

Cf. A002473 (7-smooth numbers), A014673, A020639, A210615 (smallest smoothest composite divisor), A343598.

Subsequence of A080672.

Select[Range[130], Plus @@ IntegerExponent[#, {2, 3, 5, 7}] > 1 &] (* Amiram Eldar, May 04 2021 *)

{a(n)} = {k : k >= 1, 2 <= A014673(k) <= 7}, where A014673(k) = lpf(k/lpf(k)), where lpf(m) = A020639(m), the least prime factor of m.
For n >= 1, a(22164 + n) = 44100 + a(n).
For n < 22164, a(22164 - n) = 44100 - a(n).

cp's OEIS Frontend

A195238 Numbers with at least 2 and not more than 3 distinct prime factors not greater than 7 that are multiples of 7 or of 15.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A198375 Smallest n-digit number whose product of digits is n or 0 if no number exists.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A219697 Primes neighboring a 7-smooth number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A237851 a(1)=1; a(n) is the smallest integer not yet in the sequence divisible by all nonzero digits of a(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A322447 Numbers k where Sum_{d | k} 1/rad(d) increases to a record.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs