A033849 -id:A033849 - cp's OEIS frontend

A033848 Numbers whose prime factors are 2 and 11.

22, 44, 88, 176, 242, 352, 484, 704, 968, 1408, 1936, 2662, 2816, 3872, 5324, 5632, 7744, 10648, 11264, 15488, 21296, 22528, 29282, 30976, 42592, 45056, 58564, 61952, 85184, 90112, 117128, 123904, 170368, 180224, 234256, 247808, 322102

Offset: 1

Jeff Burch

nonn

Numbers k such that phi(k)/k = 5/11. - Michel Marcus, Sep 22 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A033845, A033846, A033847, A033849, A033850, A033851, A143201.

import Data.Set (singleton, deleteFindMin, insert)
a033848 n = a033848_list !! (n-1)
a033848_list = f (singleton (2*11)) where
   f s = m : f (insert (2*m) $ insert (11*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

N:= 10^6: # to get all terms <= N
S:= {seq(seq(2^i*11^j, i=1..ilog2(floor(N/11^j))),j=1..floor(log[11](N/2)))}:
sort(convert(S,list)); # Robert Israel, Oct 26 2017

Select[Range[10^6], FactorInteger[#][[All, 1]] == {2, 11} &] (* Michael De Vlieger, Oct 26 2017 *)
Sort[Flatten[Table[Table[2^j 11^k, {j, 1, 8}], {k, 1, 8}]]] (* Vincenzo Librandi, Oct 27 2017 *)

A143201(a(n)) = 10. - Reinhard Zumkeller, Sep 13 2011

Sum_{n>=1} 1/a(n) = 1/10. - Amiram Eldar, Dec 22 2020

A033851 Numbers whose prime factors are 5 and 7.

Original entry on oeis.org

35, 175, 245, 875, 1225, 1715, 4375, 6125, 8575, 12005, 21875, 30625, 42875, 60025, 84035, 109375, 153125, 214375, 300125, 420175, 546875, 588245, 765625, 1071875, 1500625, 2100875, 2734375, 2941225, 3828125, 4117715, 5359375, 7503125

Offset: 1

Jeff Burch

nonn

Numbers k such that phi(k)/k == 24/35. - Artur Jasinski, Nov 09 2008

Subsequence of A143202. - Reinhard Zumkeller, Sep 13 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003595, A147571-A147575, A147576-A147580. [Artur Jasinski, Nov 09 2008]
Cf. A033845, A033846, A033847, A033848, A033849, A033850, A143201, A143202.
Subsequence of A256617.

import Data.Set (singleton, deleteFindMin, insert)
a033851 n = a033851_list !! (n-1)
a033851_list = f (singleton (5*7)) where
   f s = m : f (insert (5*m) $ insert (7*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

a = {}; Do[If[EulerPhi[x]/x == 24/35, AppendTo[a, x]], {x, 1, 10000}]; a (* Artur Jasinski, Nov 09 2008 *)
Take[With[{nn=10},Sort[Flatten[Table[5^i 7^j,{i,nn},{j,nn}]]]],40] (* Harvey P. Dale, Feb 09 2013 *)

a(n) = 35*A003595(n). - Artur Jasinski, Nov 09 2008
A143201(a(n)) = 3. - Reinhard Zumkeller, Sep 13 2011
Sum_{n>=1} 1/a(n) = 1/24. - Amiram Eldar, Dec 22 2020

Offset fixed by Reinhard Zumkeller, Sep 13 2011

A147576 Numbers with exactly 3 distinct odd prime divisors {3,5,7}.

Original entry on oeis.org

105, 315, 525, 735, 945, 1575, 2205, 2625, 2835, 3675, 4725, 5145, 6615, 7875, 8505, 11025, 13125, 14175, 15435, 18375, 19845, 23625, 25515, 25725, 33075, 36015, 39375, 42525, 46305, 55125, 59535, 65625, 70875, 76545, 77175, 91875, 99225

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Harvey P. Dale)

Cf. A060735, A108347, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x]/x == 16/35, AppendTo[a, x]], {x, 1, 100000}]; a
Select[Range[100000],EulerPhi[#]/#==16/35&] (* Harvey P. Dale, Dec 01 2013 *)

a(n) = 105 * A108347(n). - Amiram Eldar, Mar 10 2020

Sum_{n>=1} 1/a(n) = 1/48. - Amiram Eldar, Dec 22 2020

A147580 Numbers with exactly 7 distinct odd prime divisors {3,5,7,11,13,17,19}.

Original entry on oeis.org

4849845, 14549535, 24249225, 33948915, 43648605, 53348295, 63047985, 72747675, 82447365, 92147055, 101846745, 121246125, 130945815, 160044885, 169744575, 189143955, 218243025, 237642405, 247342095, 266741475, 276441165, 305540235, 315239925, 363738375, 373438065

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

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

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

a = {}; Do[If[EulerPhi[4849845 x] == 1658880 x, AppendTo[a, 4849845 x]], {x, 1, 100}]; a

Sum_{n>=1} 1/a(n) = 1/1658880. - Amiram Eldar, Dec 22 2020

More terms from Amiram Eldar, Mar 11 2020

A284311 Array T(n,k) read by antidiagonals (downward): T(1,k) = A005117(k+1) (squarefree numbers > 1); for n > 1, columns are nonsquarefree numbers (in ascending order) with exactly the same prime factors as T(1,k).

Original entry on oeis.org

2, 3, 4, 5, 9, 8, 6, 25, 27, 16, 7, 12, 125, 81, 32, 10, 49, 18, 625, 243, 64, 11, 20, 343, 24, 3125, 729, 128, 13, 121, 40, 2401, 36, 15625, 2187, 256, 14, 169, 1331, 50, 16807, 48, 78125, 6561, 512, 15, 28, 2197, 14641, 80, 117649, 54, 390625, 19683, 1024

Offset: 1

Bob Selcoe, Mar 24 2017

A permutation of the natural numbers > 1.

T(1,k)= A005117(m) with m > 1; terms in column k = T(1,k) * A162306(T(1,k)) only not bounded by T(1,k). Let T(1,k) = b. Terms in column k are multiples of b and numbers c such that c | b^e with e >= 0. Alternatively, terms in column k are multiples bc with c those numbers whose prime divisors p also divide b. - Michael De Vlieger, Mar 25 2017

			Array starts:
    2    3     5  6      7  10       11        13  14  15
    4    9    25 12     49  20      121       169  28  45
    8   27   125 18    343  40     1331      2197  56  75
   16   81   625 24   2401  50    14641    371293  98 135
   32  243  3125 36  16807  80   161051   4826809 112 225
   64  729 15625 48 117649 100  1771561  62748517 196 375
  128 2187 78125 54 823543 160 19487171 815730721 224 405
Column 6 is: T(1,6) = 2*5; T(2,6) = 2^2*5; T(3,6) = 2^3*5; T(4,6) = 2*5^2; T(5,6) = 2^4*5, etc.

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A005117 (squarefree numbers), A033845 (column 4), columns 1,2,3,5 are powers of primes, A033846 (column 6), A033847 (column 9), A033849 (column 10).
The columns that are powers of primes have indices A071403(n) - 1. - Michel Marcus, Mar 24 2017
See also A007947; the k-th column of the array corresponds to the numbers with radical A005117(k+1). - Rémy Sigrist, Mar 24 2017
Cf. A284457 (this sequence read by antidiagonals upwards), A285321 (a similar array, but columns come in different order).
Cf. A065642.
Cf. A008479 (index of the row where n is located), A285329 (of the column).

f[n_, k_: 1] := Block[{c = 0, sgn = Sign[k], sf}, sf = n + sgn; While[c < Abs[k], While[! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[sgn < 0, sf--, sf++]; c++]; sf + If[sgn < 0, 1, -1]] (* after Robert G. Wilson v at A005117 *); T[n_, k_] := T[n, k] = Which[And[n == 1, k == 1], 2, k == 1, f@ T[n - 1, k], PrimeQ@ T[n, 1], T[n, 1]^k, True, Module[{j = T[n, k - 1]/T[n, 1] + 1}, While[PowerMod[T[n, 1], j, j] != 0, j++]; j T[n, 1]]]; Table[T[n - k + 1, k], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Mar 25 2017 *)

(define (A284311 n) (A284311bi  (A002260 n) (A004736 n)))
(define (A284311bi row col) (if (= 1 row) (A005117 (+ 1 col)) (A065642 (A284311bi (- row 1) col))))
;; Antti Karttunen, Apr 17 2017

From Antti Karttunen, Apr 17 2017: (Start)
A(1,k) = A005117(1+k), A(n,k) = A065642(A(n-1,k)).
A(A008479(n), A285329(n)) = n for all n >= 2.
(End)

A147577 Numbers with exactly 4 distinct odd prime divisors {3,5,7,11}.

Original entry on oeis.org

1155, 3465, 5775, 8085, 10395, 12705, 17325, 24255, 28875, 31185, 38115, 40425, 51975, 56595, 63525, 72765, 86625, 88935, 93555, 114345, 121275, 139755, 144375, 155925, 169785, 190575, 202125, 218295, 259875, 266805, 280665, 282975, 317625

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

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

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

a = {}; Do[If[EulerPhi[x]/x == 32/77, AppendTo[a, x]], {x, 1, 1000000}]; a
Select[Range[350000],EulerPhi[#]/#==32/77&] (* Harvey P. Dale, Mar 25 2016 *)

from sympy import integer_log
def A147577(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 i11 in range(integer_log(x,11)[0]+1):
            for i7 in range(integer_log(x11:=x//11**i11,7)[0]+1):
                for i5 in range(integer_log(x7:=x11//7**i7,5)[0]+1):
                    c -= integer_log(x7//5**i5,3)[0]+1
        return c
    return 1155*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Sum_{n>=1} 1/a(n) = 1/480. - Amiram Eldar, Dec 22 2020

A147578 Numbers with exactly 5 distinct odd prime divisors {3,5,7,11,13}.

Original entry on oeis.org

15015, 45045, 75075, 105105, 135135, 165165, 195195, 225225, 315315, 375375, 405405, 495495, 525525, 585585, 675675, 735735, 825825, 945945, 975975, 1126125, 1156155, 1216215, 1366365, 1486485, 1576575, 1756755, 1816815, 1876875, 2027025, 2147145, 2207205, 2477475

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244;
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849;
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576;
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577;
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578;
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579;
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580;
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581.

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

Cf. A000010 (EulerPhi), A060735, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x]/x == 384/1001, AppendTo[a, x]], {x, 1, 1000000}]; a

Sum_{n>=1} 1/a(n) = 1/5760. - Amiram Eldar, Dec 22 2020

More terms from Amiram Eldar, Mar 11 2020

A147579 Numbers with exactly 6 distinct odd prime divisors {3,5,7,11,13,17}.

Original entry on oeis.org

255255, 765765, 1276275, 1786785, 2297295, 2807805, 3318315, 3828825, 4339335, 5360355, 6381375, 6891885, 8423415, 8933925, 9954945, 11486475, 12507495, 13018005, 14039025, 16081065, 16591575, 19144125, 19654635, 20675655, 21696675, 23228205, 25270245, 26801775

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

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

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

a = {}; Do[If[EulerPhi[255255 x] == 92160 x, AppendTo[a, 255255 x]], {x, 1, 100}]; a

Sum_{n>=1} 1/a(n) = 1/92160. - Amiram Eldar, Dec 22 2020

More terms from Amiram Eldar, Mar 11 2020

A147581 Numbers with exactly 8 distinct odd prime divisors {3,5,7,11,13,17,19,23}.

Original entry on oeis.org

111546435, 334639305, 557732175, 780825045, 1003917915, 1227010785, 1450103655, 1673196525, 1896289395, 2119382265, 2342475135, 2565568005, 2788660875, 3011753745, 3681032355, 3904125225, 4350310965, 5019589575, 5465775315, 5688868185, 6135053925, 6358146795

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

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

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

a = {}; Do[If[EulerPhi[111546435 x] == 36495360 x, AppendTo[a, 111546435 x]], {x, 1, 100}]; a

from sympy import integer_log
def A147581(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 i23 in range(integer_log(x,23)[0]+1):
            for i19 in range(integer_log(x23:=x//23**i23,19)[0]+1):
                for i17 in range(integer_log(x19:=x23//19**i19,17)[0]+1):
                    for i13 in range(integer_log(x17:=x19//17**i17,13)[0]+1):
                        for i11 in range(integer_log(x13:=x17//13**i13,11)[0]+1):
                            for i7 in range(integer_log(x11:=x13//11**i11,7)[0]+1):
                                for i5 in range(integer_log(x7:=x11//7**i7,5)[0]+1):
                                    c -= integer_log(x7//5**i5,3)[0]+1
        return c
    return 111546435*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Sum_{n>=1} 1/a(n) = 1/36495360. - Amiram Eldar, Dec 22 2020

More terms from Amiram Eldar, Mar 11 2020

A288162 Numbers whose prime factors are 2 and 13.

Original entry on oeis.org

26, 52, 104, 208, 338, 416, 676, 832, 1352, 1664, 2704, 3328, 4394, 5408, 6656, 8788, 10816, 13312, 17576, 21632, 26624, 35152, 43264, 53248, 57122, 70304, 86528, 106496, 114244, 140608, 173056, 212992, 228488, 281216, 346112, 425984, 456976, 562432, 692224, 742586, 851968, 913952

Offset: 1

Bernard Schott, Jun 06 2017

nonn

Numbers k such that phi(k)/k = 6/13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A033845, A033846, A033847, A033848, A033849, A033850, A033851, A086780, A107326, A143207.

[n:n in [1..100000] | Set(PrimeDivisors(n)) eq {2,13}];  // Marius A. Burtea, May 10 2019

Select[Range[920000],FactorInteger[#][[All,1]]=={2,13}&] (* Harvey P. Dale, Jun 18 2021 *)

is(n) = factor(n)[, 1]~==[2, 13] \\ Felix Fröhlich, Jun 06 2017

list(lim)=my(v=List(),t); for(n=1,logint(lim\2,13), t=13^n; while((t<<=1)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jun 11 2017

a(n) = 26 * A107326(n). - David A. Corneth, Jun 06 2017

Sum_{n>=1} 1/a(n) = 1/12. - Amiram Eldar, Dec 22 2020

cp's OEIS Frontend

A033848 Numbers whose prime factors are 2 and 11.

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A033851 Numbers whose prime factors are 5 and 7.

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A147576 Numbers with exactly 3 distinct odd prime divisors {3,5,7}.

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A147580 Numbers with exactly 7 distinct odd prime divisors {3,5,7,11,13,17,19}.

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A284311 Array T(n,k) read by antidiagonals (downward): T(1,k) = A005117(k+1) (squarefree numbers > 1); for n > 1, columns are nonsquarefree numbers (in ascending order) with exactly the same prime factors as T(1,k).

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A147577 Numbers with exactly 4 distinct odd prime divisors {3,5,7,11}.

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A147578 Numbers with exactly 5 distinct odd prime divisors {3,5,7,11,13}.

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