A147572 -id:A147572 - cp's OEIS frontend

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

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

A147575 Numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19}.

Original entry on oeis.org

9699690, 19399380, 29099070, 38798760, 48498450, 58198140, 67897830, 77597520, 87297210, 96996900, 106696590, 116396280, 126095970, 135795660, 145495350, 155195040, 164894730, 174594420, 184294110, 193993800, 203693490, 213393180, 232792560, 242492250, 252191940

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

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

Cf. A060735, A080682, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[9699690 x] == 1658880 x, AppendTo[a, 9699690 x]], {x, 1, 100}]; a

a(n) = 9699690 * A080682(n). - Amiram Eldar, Mar 10 2020

Sum_{n>=1} 1/a(n) = 1/1658880. - Amiram Eldar, Nov 12 2020

More terms from Amiram Eldar, Mar 10 2020

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

A147574 Numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17}.

Original entry on oeis.org

510510, 1021020, 1531530, 2042040, 2552550, 3063060, 3573570, 4084080, 4594590, 5105100, 5615610, 6126120, 6636630, 7147140, 7657650, 8168160, 8678670, 9189180, 10210200, 10720710, 11231220, 12252240, 12762750, 13273260, 13783770

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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A060735, A080681, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x 510510] == 92160 x, AppendTo[a, 510510 x]], {x, 1, 100}]; a
sdpdQ[n_]:=Module[{f=FactorInteger[n][[All,1]]},Length[f]==7&&Max[f]==17]; Select[Range[510510,138*10^5,510510],sdpdQ] (* Harvey P. Dale, Aug 03 2019 *)

a(n) = 510510 * A080681(n). - Amiram Eldar, Mar 10 2020

Sum_{n>=1} 1/a(n) = 1/92160. - Amiram Eldar, Nov 12 2020

A372402 Position of 2310^n among 11-smooth numbers A051038.

Original entry on oeis.org

1, 283, 3847, 20996, 74228, 203084, 469053, 960396, 1797086, 3135610, 5173909, 8156188, 12377846, 18190320, 26005929, 36302854, 49629820, 66611231, 87951744, 114441450, 146960432, 186483973, 234087084, 290949702, 358361266, 437725888, 530566933, 638532124, 763398291, 907076258

Offset: 0

Michael De Vlieger, Jun 03 2024

nonn

Also position of 2310^(n+1) in A147572.

Cf. A022330, A051038, A147572, A372400, A372401.

Table[
  Sum[Floor@ Log[11, 2310^n/(2^i*3^j*5^k*7^m)] + 1,
    {i, 0, Log[2, 2310^n]},
    {j, 0, Log[3, 2310^n/2^i]},
    {k, 0, Log[5, 2310^n/(2^i*3^j)]},
    {m, 0, Log[7, 2310^n/(2^i*3^j*5^k)]}],
  {n, 0, 8}]

# uses imports/function in A372401
print(list(islice(A372401gen(p=11), 7))) # Michael S. Branicky, Jun 05 2024

from sympy import integer_log, prevprime
def A372402(n):
    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))
    return g(2310**n,11) # Chai Wah Wu, Sep 16 2024

a(14)-a(18) from Michael S. Branicky, Jun 05 2024

More terms from David A. Corneth, Jun 05 2024

A380446 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k), where omega = A001221.

Original entry on oeis.org

36, 144, 216, 324, 576, 900, 1296, 1728, 2304, 2916, 3600, 5184, 5832, 7776, 8100, 9216, 11664, 13824, 14400, 20736, 22500, 26244, 27000, 32400, 36864, 44100, 46656, 57600, 72900, 82944, 90000, 104976, 110592, 129600, 147456, 157464, 176400, 186624, 202500, 216000

Offset: 1

Michael De Vlieger, Jul 25 2025

nonn

Perfect powers k^m, m > 1, for k in A055932.
Union of {k^m : rad(k) | P(i), m >= 2}, rad = A007947, P = A002110. Therefore perfect powers in A033845, A143207, A147571, A147572, etc. are proper subsets.
Terms are even. For a(n) such that omega(a(n)) > 2, a(n) mod 10 = 0, where omega = A001221.

			Table of n, a(n) for select n, showing exponents m of prime power factors p^m | a(n) for primes p listed in the heading. Terms that also appear in A368682 are marked by "#":
                         Exponents
 n      a(n)             2.3.5.7.11
-----------------------------------
 1       36 =    6^2  #  2.2
 2      144 =   12^2  #  4.2
 3      216 =    6^3  #  3.3
 4      324 =   18^2     2.4
 5      576 =   24^2  #  6.2
 6      900 =   30^2  #  2.2.2
 7     1296 =    6^4  #  4.4
 8     1728 =   12^3  #  6.3
 9     2304 =   48^2  #  8.2
10     2916 =   54^2     2.6
11     3600 =   60^2  #  4.2.2
12     5184 =   72^2  #  6.4
26    44100 =  210^2  #  2.2.2.2
90  5336100 = 2310^2  #  2.2.2.2.2

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Fast Mathematica algorithm for A055932.

Cf. A001597, A002110, A006530, A007947, A033845, A053669, A055932, A126706, A131605, A143207, A147571, A147572, A246547, A304250, A365308 (subset), A368682 (subset), A369374 (superset).

(* Load linked Mathematica algorithm, then: *)
Select[Union@ Flatten[a055932[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[All, -1]] > 1] & @@ {#, FactorInteger[#]} &]

Intersection of A131605 and A055932 = A304250 \ A246547.

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]] ] ]

A373559 Squares k such that rad(k) is a primorial number.

Original entry on oeis.org

1, 4, 16, 36, 64, 144, 256, 324, 576, 900, 1024, 1296, 2304, 2916, 3600, 4096, 5184, 8100, 9216, 11664, 14400, 16384, 20736, 22500, 26244, 32400, 36864, 44100, 46656, 57600, 65536, 72900, 82944, 90000, 104976, 129600, 147456, 176400, 186624, 202500, 230400, 236196, 262144

Offset: 1

David James Sycamore, Jun 09 2024

Squares k such that the squarefree kernel of k is primorial.
Intersection of A000290 and A055932.
1 is the only primorial term.
From Michael De Vlieger, Jun 09 2024: (Start)
Contains k^2 for k in each of A000079, A033845, A143207, A147571, A147572, etc.
Contains k^2 such that k is a product of primorials, i.e., A025487(i)^2, i >= 1, since A025487 is a proper subset of A055932.
(End)

			1 is a square, rad(1) = 1 = A002110(0).
4 is a square and rad(4) = 2 = A002110(1).
36 is a square and rad(36) = 6 = A002110(2).

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

Cf. A000290, A002110, A007947, A055932,

{1}~Join~Select[Range[2, 512, 2], Or[# == {2}, Union@ Differences@ PrimePi[#] == {1}] &@ FactorInteger[#][[All, 1]] &]^2 (* Michael De Vlieger, Jun 09 2024 *)

a(n) = A055932(n)^2.

More terms from David A. Corneth, Jun 09 2024

cp's OEIS Frontend

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

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A147575 Numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19}.

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

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A147574 Numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17}.

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A372402 Position of 2310^n among 11-smooth numbers A051038.

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A380446 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k), where omega = A001221.

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

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A373559 Squares k such that rad(k) is a primorial number.

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