A108319 -id:A108319 - cp's OEIS frontend

A108347 Numbers of the form (3^i)(5^j)(7^k), with i, j, k >= 0.

1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63, 75, 81, 105, 125, 135, 147, 175, 189, 225, 243, 245, 315, 343, 375, 405, 441, 525, 567, 625, 675, 729, 735, 875, 945, 1029, 1125, 1215, 1225, 1323, 1575, 1701, 1715, 1875, 2025, 2187, 2205, 2401, 2625, 2835, 3087

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jul 01 2005

nonn

The Heinz numbers of the partitions into parts 2,3, and 4 (including the number 1, the Heinz number of the empty partition). We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [2,3,3,4] the Heinz number is 3*5*5*7 = 525; it is in the sequence. - Emeric Deutsch , May 21 2015

Numbers m | 105^e with integer e >= 0. - Michael De Vlieger, Aug 22 2019

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A003586, A003591-A003595, A051037, A108319, A108513, A215366.

The odd terms of A002473.

[n: n in [1..4000] | PrimeDivisors(n) subset [3,5,7]]; // Bruno Berselli, Sep 24 2012

with(numtheory): S := {}: for j to 3100 do if `subset`(factorset(j), {3, 5, 7}) then S := `union`(S, {j}) else end if end do: S; # Emeric Deutsch, May 21 2015
# alternative
isA108347 := proc(n)
      if n = 1 then
        true;
    else
        return (numtheory[factorset](n) minus {3, 5, 7} = {} );
    end if;
end proc:
A108347 := proc(n)
     option remember;
     if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA108347(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A108347(n),n=1..80); # R. J. Mathar, Jun 06 2024

With[{n = 3087}, Sort@ Flatten@ Table[3^i * 5^j * 7^k, {i, 0, Log[3, n]}, {j, 0, Log[5, n/2^i]}, {k, 0, Log[7, n/(3^i*5^j)]}]] (* Michael De Vlieger, Aug 22 2019 *)

from sympy import integer_log
def A108347(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):
                c -= integer_log(m//5**j,3)[0]+1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = (3*5*7)/((3-1)*(5-1)*(7-1)) = 35/16. - Amiram Eldar, Sep 22 2020

a(n) ~ exp((6*log(3)*log(5)*log(7)*n)^(1/3)) / sqrt(105). - Vaclav Kotesovec, Sep 23 2020

A108513 Numbers of the form (2^i)(5^j)(7^k), with i, j, k >= 0.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 25, 28, 32, 35, 40, 49, 50, 56, 64, 70, 80, 98, 100, 112, 125, 128, 140, 160, 175, 196, 200, 224, 245, 250, 256, 280, 320, 343, 350, 392, 400, 448, 490, 500, 512, 560, 625, 640, 686, 700, 784, 800, 875, 896, 980, 1000, 1024, 1120

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jul 05 2005

Numbers m | 70^e with integer e >= 0. - Michael De Vlieger, Aug 22 2019

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

Cf. A051037, A108319, A108347, A003592, A003591, A003595.

With[{n = 1120}, Sort@ Flatten@ Table[2^i * 5^j * 7^k, {i, 0, Log2@ n}, {j, 0, Log[5, n/2^i]}, {k, 0, Log[7, n/(2^i*5^j)]}]] (* Michael De Vlieger, Aug 22 2019 *)

isok(n) = (n/(2^valuation(n,2)*5^valuation(n,5)*7^valuation(n,7)) == 1); \\ Michel Marcus, Oct 01 2013

Sum_{n>=1} 1/a(n) = (2*5*7)/((2-1)*(5-1)*(7-1)) = 35/12. - Amiram Eldar, Sep 23 2020

a(n) ~ exp((6*log(2)*log(5)*log(7)*n)^(1/3)) / sqrt(70). - Vaclav Kotesovec, Sep 23 2020

A219174 Numbers that have no other prime factors than 2 and/or Mersenne primes.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 27, 28, 31, 32, 36, 42, 48, 49, 54, 56, 62, 63, 64, 72, 81, 84, 93, 96, 98, 108, 112, 124, 126, 127, 128, 144, 147, 162, 168, 186, 189, 192, 196, 216, 217, 224, 243, 248, 252, 254, 256, 279, 288, 294, 324, 336, 343, 372

Offset: 1

Jon Perry, Nov 13 2012

nonn

If k is in the sequence, then so is 2*k.
From Antti Karttunen, Jul 16 2023: (Start)
The original definition was "Numbers whose prime factors are either 2 or Mersenne primes". The new definition admits also {1}.
Multiplicative semigroup. Primitive terms are {1, 2} U A000668.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10001 (original 10000 initial terms from Amiram Eldar)

Cf. A000668, A056652, A108319, A364252 (characteristic function).
Positions of 1's in A336467.
Subsequences: A054784, A335431.

seq[max_] := Module[{e = Floor @ Log2[max + 1], s = {1}, es, ps, n, p, m}, es = Select[MersennePrimeExponent @ Range[20], # <= e &]; ps = Join[{2}, 2^es - 1];  n = Length[ps]; Do[p = ps[[k]]; m = Floor @ Log[p, max]; s = Select[Union @ Flatten@Outer[Times, s, p^Range[0, m]], # <= max &], {k, 1, n}]; s]; seq[10^3] (* Amiram Eldar, Jan 09 2021 *)

isokp(p) = my(q); (p==2) || (isprimepower(p+1, &q) && (q==2));
isok(m) = ((1==m) || vecmin(apply(isokp, factor(m)[, 1]))); \\ Michel Marcus, Jan 09 2021, edited by Antti Karttunen, Jul 16 2023

isok(n) = A364252(n); \\ Antti Karttunen, Jul 16 2023

Sum_{n>=1} 1/a(n) = 2 * Product_{p in A000668} p/(p-1) = 3.6458502419452069302... - Amiram Eldar, Jan 09 2021

a(1) = 1 prepended, and definition changed accordingly by Antti Karttunen, Jul 16 2023

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

A382659 Numbers k such that k < A053669(k)^2 * A380539(k), i.e., k < A382248(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 48, 50, 54, 60, 66, 70, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 210, 240, 252, 270, 300, 330, 360, 390

Offset: 1

Michael De Vlieger, Apr 14 2025

Numbers k whose reduced residue system (RRS) does not intersect A126706 (i.e., the sequence of numbers that are neither squarefree nor prime powers). Alternatively, numbers k whose RRS is a subset of A303554 (i.e., the union of powers of primes and squarefree numbers).
Let p = A053669(k), and let q = A380539(k). Thus, p and q are the smallest and second smallest primes, respectively, that do not divide k. Let m = p^2 * q = A382248(k). Then this sequence is that of k such that k < m.
There are 72 terms in this sequence.
Sequences A048597 and A051250 are proper subsets of this sequence.

			Let omega = A001221.
For omega = 0, we have the subset {1}. 1 is in the sequence since 1 < m, m = 2^2 * 3 = 12.
For omega = 1, we have the subset {2, 3, 4, 5, 7, 8, 9, 11, 16, 32}.
  11 is in the sequence since 11 < m, m = 2^2 * 3 = 12, but 13 is not, since 13 > 12.
  9 is in the sequence since 9 < m, m = 2^2 * 5 = 20.
  25 is not a term since 25 > 12, and 27 is not a term since 27 > 20.
For omega = 2, we have the subset {6, 10, 12, 14, 15, 18, 20, 22, 24, 26, 28, 34, 36, 38, 40, 44, 48, 50, 54, 72, 96, 108, 144, 162}.
  38 = 2*19 is a term since 38 < 45, 45 = 3^2 * 5, but 46 = 2*23 is not, since 46 > 45.
  15 = 3*5 is a term since 15 < 20, but 21 is not, since 21 > 20 and 35 is not, since 35 > 12.
  Intersection with A033845 = {k : rad(k) = 6} is {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162}, since m = 5^2 * 7 = 175.
  Intersection with A033846 = {k : rad(k) = 10} is {10, 20, 40, 50}, since m = 3^2 * 7 = 63.
  Intersection with A033847 = {k : rad(k) = 14} is {14, 28}, since m = 3^2 * 5 = 45, etc.
For omega = 3, we have the subset {30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 156, 168, 174, 180, 240, 252, 270, 300, 360, 450, 480}, of which {30, 42, 66, 70, 78, 102, 114, 138, 174} are squarefree.
  Intersection with A143207 = {k : rad(k) = 30} is {30, 60, 90, .., 480} because m = 7^2 * 11 = 539.
  Intersection with 42*A108319 = {k : rad(k) = 42} is {42, 84, 126, 168}, since m = 5^2 * 11 = 275, etc.
For omega = 4, we have the subset {210, 330, 390, 420, 510, 630, 840, 1050, 1260, 1470}, of which {210, 330, 390, 510} are squarefree.
  Intersection with A147571 = {k : rad(k) = 210} is {210, 420, 630, 840, 1050, 1260, 1470} since m = 11^2 * 13 = 1573, etc.
For omega = 5, we have 2310 = 2*3*5*7*11, a term since 2310 < 13*17 = 2873; 2730 = 2*3*5*7*13 is not a term.
There are no terms larger than 2310, since the intersection with A147572 = {2310}, 2730 is not a term, and k = Product_{i=1..j} prime(i), k > prime(j+1)^2 * prime(j+2) for j > 5. Therefore the sequence is finite like A051250.

Michael De Vlieger, Table of n, a(n) for n = 1..72

Cf. A048597 (k such that k < p^2), A051250 (k such that k < p*q), A053669, A126706, A303554, A380539, A382248, A382960.

Select[Range[30030], Function[n, c = 0; q = 2; n < Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^(2 - c)]; q = NextPrime[q]; c++]][[-1, 1]] ] ]

A362010 Numbers k such that 1 < gcd(k, 42) < k and A007947(k) does not divide 42.

Original entry on oeis.org

10, 15, 20, 22, 26, 30, 33, 34, 35, 38, 39, 40, 44, 45, 46, 50, 51, 52, 57, 58, 60, 62, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 82, 86, 87, 88, 90, 91, 92, 93, 94, 99, 100, 102, 104, 105, 106, 110, 111, 114, 116, 117, 118, 119, 120, 122, 123, 124, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142

Offset: 1

Michael De Vlieger, Apr 04 2023

nonn

The asymptotic density of this sequence is 5/7. - Amiram Eldar, Dec 02 2023

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Diagram showing numbers k in this sequence instead as k mod 42, labeled and in large black circles, else gray dots if coprime to 42, purple if k = 1, red if k | 42, and gold if rad(k) | 42.

Cf. A007947, A108319, A206547.

With[{n = 42}, Select[Range[200], And[! CoprimeQ[#, n], ! Divisible[n, Times @@ FactorInteger[#][[All, 1]]]] & ] ]

This sequence is { N \ { A108319 U A206547 } }.

A241681 Numbers n such that the decimal digits of n are also the prime divisors of n.

Original entry on oeis.org

2, 3, 5, 7, 735, 2333772

Offset: 1

Michel Lagneau, Apr 27 2014

The sequence is given for a(n) < 10^11.
No more terms <= 10^150. Terms are of the form 2^e2 * 3^e3 * 7^e7 or of the form 3^e3 * 5^e5 * 7^e7, for which no other number <= 10^150 than those listed is a term. - David A. Corneth, Sep 28 2019
No more terms <= 10^1000. - Michael S. Branicky, May 30 2025

			735 = 3*5*7^2 is in the sequence because the digits 7, 3 and 5 are also the prime divisors of 735.

Cf. A108319, A108347.

Subsequence of A046034.

with(numtheory):nn:=1000000:for n from 1 to 10^11 do:lst:={}:x:=factorset(n):y:=convert(n,base,10):n1:=nops(x):n2:=nops(y): for j from 1 to n2 do:lst:=lst union {y[j]}:od:if x=lst then print(n):else fi:od:

cp's OEIS Frontend

A108347 Numbers of the form (3^i)*(5^j)*(7^k), with i, j, k >= 0.

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A108513 Numbers of the form (2^i)*(5^j)*(7^k), with i, j, k >= 0.

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A219174 Numbers that have no other prime factors than 2 and/or Mersenne primes.

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

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A382659 Numbers k such that k < A053669(k)^2 * A380539(k), i.e., k < A382248(k).

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A362010 Numbers k such that 1 < gcd(k, 42) < k and A007947(k) does not divide 42.

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A241681 Numbers n such that the decimal digits of n are also the prime divisors of n.

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A108347 Numbers of the form (3^i)(5^j)(7^k), with i, j, k >= 0.

A108513 Numbers of the form (2^i)(5^j)(7^k), with i, j, k >= 0.