A033845 -id:A033845 - cp's OEIS frontend

A143201 Product of distances between prime factors in factorization of n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 6, 3, 1, 1, 2, 1, 4, 5, 10, 1, 2, 1, 12, 1, 6, 1, 6, 1, 1, 9, 16, 3, 2, 1, 18, 11, 4, 1, 10, 1, 10, 3, 22, 1, 2, 1, 4, 15, 12, 1, 2, 7, 6, 17, 28, 1, 6, 1, 30, 5, 1, 9, 18, 1, 16, 21, 12, 1, 2, 1, 36, 3, 18, 5, 22, 1, 4, 1, 40, 1, 10, 13, 42, 27, 10, 1, 6, 7, 22

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

a(n) is the product of the sum of 1 and first differences of prime factors of n with multiplicity, with a(n) = 1 for n = 1 or prime n. - Michael De Vlieger, Nov 12 2023.
a(A007947(n)) = a(n);
A006093 and A001747 give record values and where they occur:
A006093(n)=a(A001747(n+1)) for n>1.
a(n) = 1 iff n is a prime power: a(A000961(n))=1;
a(n) = 2 iff n has exactly 2 and 3 as prime factors:
a(A033845(n))=2;
a(n) = 3 iff n is in A143202;
a(n) = 4 iff n has exactly 2 and 5 as prime factors:
a(A033846(n))=4;
a(n) = 5 iff n is in A143203;
a(n) = 6 iff n is in A143204;
a(n) = 7 iff n is in A143205;
a(n) <> A006512(k)+1 for k>1.
a(A033849(n))=3; a(A033851(n))=3; a(A033850(n))=5; a(A033847(n))=6; a(A033848(n))=10. [Reinhard Zumkeller, Sep 19 2011]

			a(86) = a(43*2) = 43-2+1 = 42;
a(138) = a(23*3*2) = (23-3+1)*(3-2+1) = 42;
a(172) = a(43*2*2) = (43-2+1)*(2-2+1) = 42;
a(182) = a(13*7*2) = (13-7+1)*(7-2+1) = 42;
a(276) = a(23*3*2*2) = (23-3+1)*(3-2+1)*(2-2+1) = 42;
a(330) = a(11*5*3*2) = (11-5+1)*(5-3+1)*(3-2+1) = 42.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A000961, A001747, A006093, A020639, A033845, A033846.
Cf. A033847, A033848, A033849, A033850.
Cf. A143203, A143204, A143205.
Cf. A027746.

a143201 1 = 1
a143201 n = product $ map (+ 1) $ zipWith (-) (tail pfs) pfs
   where pfs = a027748_row n
-- Reinhard Zumkeller, Sep 13 2011

Table[Times@@(Differences[Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[ n]]]+1),{n,100}] (* Harvey P. Dale, Dec 07 2011 *)

a(n) = f(n,1,1) where f(n,q,y) = if n=1 then y else if q=1 then f(n/p,p,1)) else f(n/p,p,y*(p-q+1)) with p = A020639(n) = smallest prime factor of n.

A033850 Numbers whose prime factors are 3 and 7.

Original entry on oeis.org

21, 63, 147, 189, 441, 567, 1029, 1323, 1701, 3087, 3969, 5103, 7203, 9261, 11907, 15309, 21609, 27783, 35721, 45927, 50421, 64827, 83349, 107163, 137781, 151263, 194481, 250047, 321489, 352947, 413343, 453789, 583443, 750141, 964467

Offset: 1

Jeff Burch

nonn

Numbers k such that phi(k)/k = 4/7, where phi is the Euler totient function A000010. - Lekraj Beedassy, Jul 18 2008

Subsequence of A143203. - Reinhard Zumkeller, Sep 13 2011

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 189, p. 57, Ellipses, Paris 2008.

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

Cf. A000010, A033845, A033846, A033847, A033848, A033849, A033851, A143203.

import Data.Set (singleton, deleteFindMin, insert)
a033850 n = a033850_list !! (n-1)
a033850_list = f (singleton (3*7)) where
   f s = m : f (insert (3*m) $ insert (7*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

Select[Range[10^6],Union[FactorInteger[#][[;;,1]]]=={3,7}&] (* Harvey P. Dale, Mar 01 2023 *)

A143201(a(n)) = 5. - Reinhard Zumkeller, Sep 13 2011

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

Offset fixed by Reinhard Zumkeller, Sep 13 2011

A033848 Numbers whose prime factors are 2 and 11.

Original entry on oeis.org

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

A069352 Total number of prime factors of 3-smooth numbers.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 3, 4, 3, 4, 3, 5, 4, 5, 4, 6, 5, 4, 6, 5, 7, 6, 5, 7, 6, 5, 8, 7, 6, 8, 7, 6, 9, 8, 7, 6, 9, 8, 7, 10, 9, 8, 7, 10, 9, 8, 11, 7, 10, 9, 8, 11, 10, 9, 12, 8, 11, 10, 9, 12, 8, 11, 10, 13, 9, 12, 11, 10, 13, 9, 12, 11, 14, 10, 13, 9, 12, 11, 14, 10, 13, 12, 15, 11

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

a(n) = A001222(A003586(n));
a(n) = A022328(n) + A022329(n);
A086414(n) <= A086415(n) <= a(n).

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A003586, A001222, A003586, A022328, A022329, A086414, A086415.

a069352 = a001222 . a003586  -- Reinhard Zumkeller, May 16 2015

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]; PrimeOmega /@ smoothNumbers[3, 10^5] (* Jean-François Alcover, Nov 11 2016 *)

a(n) = i+j for 3-smooth numbers n = 2^i*3^j (A003586).

a(n) = A001222(A033845(n))-2. - Enrique Pérez Herrero, Jan 04 2012

Edited by N. J. A. Sloane, Oct 27 2008 at the suggestion of R. J. Mathar.

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

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)

A147572 Numbers with exactly 5 distinct prime divisors {2,3,5,7,11}.

Original entry on oeis.org

2310, 4620, 6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720, 32340, 34650, 36960, 41580, 46200, 48510, 50820, 55440, 57750, 62370, 64680, 69300, 73920, 76230, 80850, 83160, 92400, 97020, 101640, 103950, 110880, 113190, 115500, 124740, 127050

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. A051038, A060735, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x]/x == 16/77, AppendTo[a, x]], {x, 1, 100000}]; a
Select[Range[130000],FactorInteger[#][[All,1]]=={2,3,5,7,11}&] (* Harvey P. Dale, Oct 04 2020 *)

from sympy import integer_log, prevprime
def A147572(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 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))
    def f(x): return n+x-g(x,11)
    return 2310*bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

a(n) = 2310 * A051038(n). - Amiram Eldar, Mar 10 2020

Sum_{n>=1} 1/a(n) = 1/480. - Amiram Eldar, Nov 12 2020

More terms from Amiram Eldar, Mar 10 2020

A205506 Least positive integer m > 1 such that 1 - m^k + m^(2*k) is prime, where k=A003586(n).

Original entry on oeis.org

2, 2, 6, 2, 3, 5, 7, 3, 4, 3, 6, 93, 2, 88, 5, 33, 5, 196, 15, 106, 174, 196, 14, 342, 207, 28, 372, 14, 47, 25, 569, 646, 141, 129, 278, 5, 421, 224, 629, 26, 424, 1081, 688, 246, 736, 4392, 124, 484, 759, 791, 4401, 863, 2854, 410, 1044, 22, 848, 1402, 2006

Offset: 1

Lei Zhou, Feb 01 2012

1 - m^k + m^(2*k) equals Phi(6*k,m) when k=2^p*3^q, p>=0, q>=0, which may be prime numbers for certain positive integer m>1.
The Mathematica program given here generates the first 33 terms.  Further terms were generated by OpenPFGW.
a(62)=7426, while A003586(62)=3^8=6561.

			n=1, A003586(1)=1, when m=2, 1-2^1+2^2=3 is prime, so a(1)=2;
n=2, A003586(2)=2, when m=2, 1-2^2+2^4=13 is prime, so a(2)=2;
...
n=7, A003586(7)=9, when m=7, 1-7^9+7^18=1628413557556843 is prime, so a(7)=7.

Cf. A003586, A033845, A056993, A085398, A153438.

fQ[n_] := n == 3 EulerPhi@n; a = Select[6 Range@500, fQ]/6; l =
Length[a]; Table[m = a[[j]]; i = 1;
While[i++; cp = 1 - i^m + i^(2*m); ! PrimeQ[cp]]; i, {j, 1, l}]

from itertools import count
from sympy import isprime, integer_log
def A205506(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): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    k = bisection(f,n,n)
    return next(filter(lambda m:isprime(1-m**k+m**(k<<1)),count(2))) # Chai Wah Wu, Oct 22 2024

a(n) = A085398(6*A003586(n)). - Jinyuan Wang, Jan 01 2023

a(n) is smallest positive m such that Phi(A033845(n),m) is prime. - Chai Wah Wu, Sep 16 2024

A086410 Smallest prime factor of 3-smooth numbers, with a(1)=1.

Original entry on oeis.org

1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jul 18 2003

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

Cf. A003586, A006899, A020639, A033845, A086411.

Reap[For[n = 1, n <= 2*10^5, n++, If[EulerPhi[6*n] == 2*n, Sow[ FactorInteger[n][[1, 1]]]]]][[2, 1]] (* Jean-François Alcover, Sep 02 2016 *)

a(n) = A020639(A003586(n));
a(n) <= A086411(n) <= 3.
a(A033845(n)) = A086411(A033845(n))-1; a(A006899(n)) = A086411(A006899(n)). - Reinhard Zumkeller, Sep 25 2008

cp's OEIS Frontend

A143201 Product of distances between prime factors in factorization of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A033850 Numbers whose prime factors are 3 and 7.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

A033848 Numbers whose prime factors are 2 and 11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A033851 Numbers whose prime factors are 5 and 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

A069352 Total number of prime factors of 3-smooth numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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).

Views

Author

Keywords

Comments

Examples

Links