A002473 -id:A002473 - cp's OEIS frontend

A343598 Positive integers k such that exactly half the integers in [1..k] are divisible by a 7-smooth composite number.

10, 12, 14, 62, 74, 86, 88, 90, 92, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 126, 128, 130, 132, 136, 138, 140, 154, 156, 172, 174, 178, 180, 182, 184, 186, 188, 194, 202, 204, 208, 210, 212, 246, 248, 250, 252, 256, 258, 260, 262, 264, 266, 268, 270

Offset: 1

Peter Munn, Apr 21 2021

In every interval of 44100 integers, exactly 22164 are divisible by a 7-smooth composite number. 44100 = (2*3*5*7)^2 = A002110(4)^2 and 22164 = A281891(4,2). See A281891 for more details.

The sequence is finite with largest term a(136) = 1406.

			The numbers divisible by a 7-smooth composite number are given in A343597. List in a row the numbers that are present, with the absent numbers (aligned) in a row below. Where the count of absent numbers matches that of those present, draw a vertical line, such that all the numbers to the left are less than all the numbers to the right. See the figure below, where the rows are segmented for practical reasons:
--------------
Present :   4   6   8   9  10 | 12 | 14 | 15  16  18  20  21
Missing :   1   2   3   5   7 | 11 | 13 | 17  19  22  23  26
----------
Present :  24  25  27  28  30  32  35  36  40  42  44  45  48
Missing :  29  31  33  34  37  38  39  41  43  46  47  51  53
----------
Present :  49  50  52  54  56  60 | 63  64  66  68  70  72 |
Missing :  55  57  58  59  61  62 | 65  67  69  71  73  74 |
----------
  ...
--------------
Listing the largest number to the left of each vertical line gives this sequence: 10, 12, 14, 62, 74, ... .

Jinyuan Wang, Table of n, a(n) for n = 1..136
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A002110, A002473, A281891, A343597.

upto(n) = { my(t = 0, res = List()); for(i = 1, n, if(isdivby(i), t++; ); if(2*t == i, listput(res, i))); res }
isdivby(n) = { my(v = [4, 6, 9, 10, 14, 15, 21, 25, 35, 49]); for(i = 1, #v, if(n%v[i] == 0, return(1))); 0 } \\ David A. Corneth, Apr 24 2021

{a(n)} = {k : k = 2*m, A343597(m) <= k < A343597(m + 1)}.

A067183 Product of the prime factors of n equals the product of the digits of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 135, 175, 735, 1176, 1715, 131712

Offset: 1

Joseph L. Pe, Feb 18 2002

Terms are zeroless 7-smooth numbers (cf. A238985). - David A. Corneth, Sep 14 2022

			The prime factors of 1176 are 2,3,7 which have product = 42, the product of the digits of 1176, so 1176 is a term of the sequence.

Cf. A002473, A006753, A052382, A075048, A238985, A357132.

Do[ If[ Apply[ Times, Transpose[ FactorInteger[n]] [[1]] ] == Apply[ Times, IntegerDigits[n]], Print[n]], {n, 2, 2*10^7} ]
Select[Range[2,1000000],Times@@Transpose[FactorInteger[#]][[1]] == Times@@ IntegerDigits[#]&] (* Harvey P. Dale, Mar 19 2012 *)

is(n) = {if(n == 1, return(1)); my(f = factor(n, 7), d = digits(n)); if(f[#f~, 1] > 7, return(0)); vecprod(f[,1]) == vecprod(d)} \\ David A. Corneth, Sep 14 2022

Edited and extended by Robert G. Wilson v, Feb 19 2002

a(1)=1 inserted by Alois P. Heinz, Sep 14 2022

A075048 10-smooth numbers that show their prime factors.

Original entry on oeis.org

1, 2, 3, 5, 7, 135, 175, 735, 1715, 13122, 131712, 2333772

Offset: 1

Amarnath Murthy, Sep 03 2002

A number n is in the sequence if it has only noncomposite digits (1,2,3,5,7) and a prime p divides n if and only if p is a digit of n.
No other terms below 10^13.
No more terms < 10^100. - David Wasserman, Jan 04 2005
No more terms < 10^238. - Michael S. Branicky, Jul 03 2022

			131712 is a member because 131712 = 2*2*2*2*2*2*2*3*7*7*7; the prime factors are digits and the digits are factors.

Cf. A002473, A052382, A067183, A357132.

from sympy import primefactors
def ok(n):
    digset = set(map(int, str(n)))
    if not digset <= {1, 2, 3, 5, 7}: return False
    return set(primefactors(n)) == digset - {1}
print([k for k in range(10**7) if ok(k)]) # Michael S. Branicky, Jul 03 2022

Edited by Don Reble, Jun 07 2003

Offset 1 from Alois P. Heinz, Sep 15 2022

A080186 Primes p such that 7 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

Original entry on oeis.org

13, 41, 419, 881, 1049, 2267, 2687, 3359, 3527, 5879, 6299, 7349, 7559, 8231, 8819, 10499, 18521, 26249, 26879, 28349, 29399, 30869, 33599, 35279, 49391, 81647, 100799, 102059, 131249, 131711, 134399, 158759, 170099, 183707, 197567, 241919

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

The sequence appears to consist of 13 and the lesser of twin primes q (A001359) such that q+1 is 7-smooth (A002473) but not 5-smooth (A051037, A080194).

			13 is a term since 14 = 2*7, 15 = 3*5, 16 = 2^4 are the numbers between 13 and the next prime 17; 419 is a term since 420 = 2^2*3*5*7 is the only number between 419 and the next prime 421.

Amiram Eldar, Table of n, a(n) for n = 1..200 (terms 1..100 from Harvey P. Dale)

Cf. A052248, A001359, A002473, A051037, A080194.

lpf7Q[n_]:=Max[Flatten[Transpose[FactorInteger[#]][[1]]&/@Range[ n+1, NextPrime[ n]-1]]]==7; Select[Prime[Range[22000]],lpf7Q] (* Harvey P. Dale, Sep 25 2015 *)

{forprime(p=2,250000,q=nextprime(p+1); m=0; j=p+1; while(j

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

A086286 Smallest prime factor of 7-smooth numbers.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 5, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 7, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 5, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2

Offset: 1

Reinhard Zumkeller, Jul 15 2003

nonn

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

Cf. A002473, A020639, A086287.

a(n) = A020639(A002473(n));

a(n) <= A086287(n) <= 7.

A086287 Greatest prime factor of 7-smooth numbers.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 3, 7, 5, 2, 3, 5, 7, 3, 5, 3, 7, 5, 2, 7, 3, 5, 7, 5, 3, 7, 5, 3, 7, 5, 7, 2, 7, 3, 5, 5, 3, 7, 5, 3, 7, 5, 7, 3, 7, 5, 5, 7, 2, 5, 7, 3, 7, 5, 5, 3, 7, 7, 5, 7, 3, 7, 5, 7, 3, 7, 5, 5, 3, 7, 5, 7, 2, 5, 7, 3, 7, 5, 7, 5, 3, 7, 7, 7, 5, 5, 7, 3, 7, 5, 5, 7, 3, 7, 7, 5

Offset: 1

Reinhard Zumkeller, Jul 15 2003

nonn

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

Cf. A002473, A006530, A086286.

Reap[Do[p = FactorInteger[n][[-1, 1]]; If[p < 11, Sow[p]], {n, 1, 500}] ][[2, 1]] (* Jean-François Alcover, Dec 17 2017 *)

a(n) = A006530(A002473(n)).

A086286(n) <= a(n) <= 7.

A086292 Number of divisors of 7-smooth numbers.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 6, 4, 4, 5, 6, 6, 4, 8, 3, 4, 6, 8, 6, 4, 9, 8, 8, 6, 10, 3, 6, 8, 8, 12, 6, 7, 8, 12, 6, 10, 5, 12, 12, 12, 6, 9, 8, 12, 10, 16, 4, 12, 8, 8, 12, 15, 6, 12, 12, 10, 16, 6, 18, 8, 14, 9, 12, 16, 16, 12, 9, 20, 6, 6, 8, 18, 9, 16, 16, 18, 12, 18, 12, 14

Offset: 1

Reinhard Zumkeller, Jul 15 2003

nonn

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

Cf. A000005, A002473, A086293.

DivisorSigma[0, Select[Range[500], Max[Transpose[FactorInteger[#]][[1]]] <= 7 &]] (* Amiram Eldar, Jan 06 2020 after G. C. Greubel at A086288 *)

a(n) = A000005(A002473(n)).

A086294 Sum of distinct prime factors of 7-smooth numbers.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 5, 9, 8, 2, 5, 7, 10, 5, 5, 3, 9, 10, 2, 12, 5, 7, 12, 8, 5, 7, 7, 5, 9, 10, 10, 2, 14, 5, 8, 7, 3, 12, 10, 5, 9, 7, 15, 5, 9, 10, 5, 12, 2, 8, 14, 5, 10, 10, 7, 5, 12, 12, 10, 10, 5, 9, 7, 17, 5, 9, 8, 10, 3, 12, 7, 12, 2, 10, 14, 5, 12, 10, 15, 7, 5, 12, 7

Offset: 1

Reinhard Zumkeller, Jul 15 2003

nonn

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

Cf. A002473, A008472, A086288, A086293, A086295.

sumPrimes[1] = 0; sumPrimes[n_] := Plus @@ First[Transpose[FactorInteger[n]]]; sumPrimes/@Select[Range[500], Max[Transpose[FactorInteger[#]][[1]]] <= 7 &] (* Amiram Eldar, Jan 06 2020 *)
dpf7[n_]:=Module[{fi=FactorInteger[n][[All,1]]},If[Max[fi]<11,Total[ fi],Nothing]]; Join[{0},Array[dpf7,400,2]] (* Harvey P. Dale, Feb 26 2022 *)

a(n) = A008472(A002473(n)).

A086295 Sum of all prime factors of 7-smooth numbers.

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 7, 9, 8, 8, 8, 9, 10, 9, 10, 9, 11, 10, 10, 12, 10, 11, 12, 11, 11, 14, 12, 11, 13, 12, 13, 12, 14, 12, 13, 13, 12, 14, 13, 13, 16, 14, 15, 13, 15, 14, 15, 15, 14, 14, 16, 14, 17, 15, 15, 14, 16, 17, 15, 16, 15, 18, 16, 17, 15, 17, 16, 16

Offset: 1

Reinhard Zumkeller, Jul 15 2003

nonn

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

Cf. A002473, A001414, A086289, A086293, A086294.

sumPrimes[1] = 0; sumPrimes[n_] := Plus @@ Times @@@ FactorInteger[n]; sumPrimes/@Select[Range[500], Max[Transpose[FactorInteger[#]][[1]]] <= 7 &] (* Amiram Eldar, Jan 06 2020 *)

a(n) = A001414(A002473(n)).

cp's OEIS Frontend

A343598 Positive integers k such that exactly half the integers in [1..k] are divisible by a 7-smooth composite number.

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A067183 Product of the prime factors of n equals the product of the digits of n.

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A075048 10-smooth numbers that show their prime factors.

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A080186 Primes p such that 7 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

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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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A086286 Smallest prime factor of 7-smooth numbers.

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A086287 Greatest prime factor of 7-smooth numbers.

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A086292 Number of divisors of 7-smooth numbers.

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