A033845 -id:A033845 - cp's OEIS frontend

A027856 Dan numbers: numbers m of the form 2^j * 3^k such that m +- 1 are twin primes.

4, 6, 12, 18, 72, 108, 192, 432, 1152, 2592, 139968, 472392, 786432, 995328, 57395628, 63700992, 169869312, 4076863488, 10871635968, 2348273369088, 56358560858112, 79164837199872, 84537841287168, 150289495621632, 578415690713088, 1141260857376768

Offset: 1

Richard C. Schroeppel

nonn

Special twin prime averages (A014574).

Intersection of A014574 and A003586. - Jeppe Stig Nielsen, Sep 05 2017

			a(14) = 243*4096 = 995328 and {995327, 995329} are twin primes.

Ray Chandler, Table of n, a(n) for n = 1..62 (terms < 10^1000, first 55 terms from Donovan Johnson)

Cf. A014574, A060211, A078883, A078884.

Cf. also A002822, A033845, A058383, A003586.

Select[#, Total@ Boole@ Map[PrimeQ, # + {-1, 1}] == 2 &] &@ Select[Range[10^7], PowerMod[6, #, #] == 0 &] (* Michael De Vlieger, Dec 31 2016 *)
seq[max_] := Select[Sort[Flatten[Table[2^i*3^j, {i, 1, Floor[Log2[max]]}, {j, 0, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {-1, 1}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)

a(n) = A078883(n) + 1 = A078884(n) - 1. - Amiram Eldar, Aug 27 2024

Offset corrected by Donovan Johnson, Dec 02 2011

Entry revised by N. J. A. Sloane, Jan 01 2017

A109395 Denominator of phi(n)/n = Product_{p|n} (1 - 1/p); phi(n)=A000010(n), the Euler totient function.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 15, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 15, 31, 2, 33, 17, 35, 3, 37, 19, 13, 5, 41, 7, 43, 11, 15, 23, 47, 3, 7, 5, 51, 13, 53, 3, 11, 7, 19, 29, 59, 15, 61, 31, 7, 2, 65, 33, 67, 17, 69, 35, 71, 3, 73, 37, 15, 19, 77, 13, 79, 5, 3

Offset: 1

Franz Vrabec, Aug 26 2005

a(n)=2 iff n=2^k (k>0); otherwise a(n) is odd. If p is prime, a(p)=p; the converse is false, e.g.: a(15)=15. It is remarkable that this sequence often coincides with A006530, the largest prime P dividing n. Theorem: a(n)=P if and only if for every prime p < P in n there is some prime q in n with p|(q-1). - Franz Vrabec, Aug 30 2005

			a(10) = 10/gcd(10,phi(10)) = 10/gcd(10,4) = 10/2 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms 1..1000 from T. D. Noe)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A076512 for the numerator.
Cf. A000010, A009195, A054741, A059956, A082695, A318304, A318305, A323170.
Phi(m)/m = k: A000079 \ {1} (k=1/2), A033845 (k=1/3), A000244 \ {1} (k=2/3), A033846 (k=2/5), A000351 \ {1} (k=4/5), A033847 (k=3/7), A033850 (k=4/7), A000420 \ {1} (k=6/7), A033848 (k=5/11), A001020 \ {1} (k=10/11), A288162 (k=6/13), A001022 \ {1} (12/13), A143207 (k=4/15), A033849 (k=8/15), A033851 (k=24/35).

Table[Denominator[EulerPhi[n]/n], {n, 81}] (* Alonso del Arte, Sep 03 2011 *)

a(n)=n/gcd(n,eulerphi(n)) \\ Charles R Greathouse IV, Feb 20 2013

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
A109395(n) = denominator(A173557(n)/A007947(n)); \\ Antti Karttunen, Feb 09 2019

a(n) = n/gcd(n, phi(n)) = n/A009195(n).
From Antti Karttunen, Feb 09 2019: (Start)
a(n) = denominator of A173557(n)/A007947(n).
a(2^n) = 2 for all n >= 1.
(End)
From Amiram Eldar, Jul 31 2020: (Start)
Asymptotic mean of phi(n)/n: lim_{m->oo} (1/m) * Sum_{n=1..m} A076512(n)/a(n) = 6/Pi^2 (A059956).
Asymptotic mean of n/phi(n): lim_{m->oo} (1/m) * Sum_{n=1..m} a(n)/A076512(n) = zeta(2)*zeta(3)/zeta(6) (A082695). (End)

A076512 Denominator of cototient(n)/totient(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 2, 10, 1, 12, 3, 8, 1, 16, 1, 18, 2, 4, 5, 22, 1, 4, 6, 2, 3, 28, 4, 30, 1, 20, 8, 24, 1, 36, 9, 8, 2, 40, 2, 42, 5, 8, 11, 46, 1, 6, 2, 32, 6, 52, 1, 8, 3, 12, 14, 58, 4, 60, 15, 4, 1, 48, 10, 66, 8, 44, 12, 70, 1, 72, 18, 8, 9, 60, 4, 78, 2, 2, 20, 82, 2, 64, 21

Offset: 1

Reinhard Zumkeller, Oct 15 2002

a(n)=1 iff n=A007694(k) for some k.
Numerator of phi(n)/n=Prod_{p|n} (1-1/p). - Franz Vrabec, Aug 26 2005
From Wolfdieter Lang, May 12 2011: (Start)
For n>=2, a(n)/A109395(n) = sum(((-1)^r)*sigma_r,r=0..M(n)) with the elementary symmetric functions (polynomials) sigma_r of the indeterminates {1/p_1,...,1/p_M(n)} if n = prod((p_j)^e(j),j=1..M(n)) where M(n)=A001221(n) and sigma_0=1.
This follows by expanding the above given product for phi(n)/n.
The n-th member of this rational sequence 1/2, 2/3, 1/2, 4/5, 1/3, 6/7, 1/2, 2/3, 2/5,... is also (2/n^2)*sum(k,with 1<=k=2.
Therefore, this scaled sum depends only on the distinct prime factors of n.
See also A023896. Proof via PIE (principle of inclusion and exclusion). (End)
In the sequence of rationals r(n)=eulerphi(n)/n: 1, 1/2, 2/3, 1/2, 4/5, 1/3, 6/7, 1/2, 2/3, 2/5, 10/11, 1/3, ... one can observe that new values are obtained for squarefree indices (A005117); while for a nonsquarefree number n (A013929), r(n) = r(A007947(n)), where A007947(n) is the squarefree kernel of n. - Michel Marcus, Jul 04 2015

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A076511 (numerator of cototient(n)/totient(n)), A051953.

Phi(m)/m = k: A000079 \ {1} (k=1/2), A033845 (k=1/3), A000244 \ {1} (k=2/3), A033846 (k=2/5), A000351 \ {1} (k=4/5), A033847 (k=3/7), A033850 (k=4/7), A000420 \ {1} (k=6/7), A033848 (k=5/11), A001020 \ {1} (k=10/11), A288162 (k=6/13), A001022 \ {1} (12/13), A143207 (k=4/15), A033849 (k=8/15), A033851 (k=24/35).

[Numerator(EulerPhi(n)/n): n in [1..100]]; // Vincenzo Librandi, Jul 04 2015

Table[Denominator[(n - EulerPhi[n])/EulerPhi[n]], {n, 80}] (* Alonso del Arte, May 12 2011 *)

vector(80, n, numerator(eulerphi(n)/n)) \\ Michel Marcus, Jul 04 2015

a(n) = A000010(n)/A009195(n).

A033847 Numbers whose prime factors are 2 and 7.

Original entry on oeis.org

14, 28, 56, 98, 112, 196, 224, 392, 448, 686, 784, 896, 1372, 1568, 1792, 2744, 3136, 3584, 4802, 5488, 6272, 7168, 9604, 10976, 12544, 14336, 19208, 21952, 25088, 28672, 33614, 38416, 43904, 50176, 57344, 67228, 76832, 87808, 100352, 114688

Offset: 1

Jeff Burch

nonn

Numbers k such that phi(k) = (3/7)*k - Benoit Cloitre, Apr 19 2002

Subsequence of A143204. - Reinhard Zumkeller, Sep 13 2011

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

Cf. A033845, A033846, A033848, A033849, A033850, A033851, A143204.

import Data.Set (singleton, deleteFindMin, insert)
a033847 n = a033847_list !! (n-1)
a033847_list = f (singleton (2*7)) where
   f s = m : f (insert (2*m) $ insert (7*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

With[{nn=20},Select[Union[Flatten[Table[2^n 7^k,{n,nn},{k,nn}]]],#<=2^nn 7&]] (* Harvey P. Dale, Nov 25 2020 *)

A143201(a(n)) = 6. - Reinhard Zumkeller, Sep 13 2011

Sum_{n>=1} 1/a(n) = 1/6. - Amiram Eldar, Dec 22 2020

Offset fixed by Reinhard Zumkeller, Sep 13 2011

A058383 Primes of form 1+(2^a)*(3^b), a>0, b>0.

Original entry on oeis.org

7, 13, 19, 37, 73, 97, 109, 163, 193, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, 1179649, 1492993

Offset: 1

Labos Elemer, Dec 20 2000

nonn

Prime numbers n such that cos(2*Pi/n) is an algebraic number of a 3-smooth degree, but not a 2-smooth degree. - Artur Jasinski, Dec 13 2006
From Antonio M. Oller-Marcén, Sep 24 2009: (Start)
In this case gcd(a,b) is a power of 2.
A regular polygon of n sides is constructible by paper folding if and only if n=2^r3^sp_1...p_t with p_i being distinct primes of this kind. (End)
Primes in A005109 but not in A092506. - R. J. Mathar, Sep 28 2012
Conjecture: these are the only solutions >=7 to the equation A000010(x) + A000010(x-1) = floor((4*x-3)/3). - Benoit Cloitre, Mar 02 2018
These are also called Pierpont primes. - Harvey P. Dale, Apr 13 2019

Ray Chandler, Table of n, a(n) for n = 1..8378 (terms < 10^1000, first 1000 terms from T. D. Noe)

Cf. A033845, A000423, A125866, A217035.

N:= 10^10: # to get all terms <= N+1
sort(select(isprime, [seq(seq(1+2^a*3^b, a=1..ilog2(N/3^b)), b=1..floor(log[3](N)))])); # Robert Israel, Mar 02 2018

Do[If[Take[FactorInteger[EulerPhi[2n + 1]][[ -1]],1] == {3} && PrimeQ[2n + 1], Print[2n + 1]], {n, 1, 10000}] (* Artur Jasinski, Dec 13 2006 *)
mx = 1500000; s = Sort@ Flatten@ Table[1 + 2^j*3^k, {j, Log[2, mx]}, {k, Log[3, mx/2^j]}]; Select[s, PrimeQ] (* Robert G. Wilson v, Sep 28 2012 *)
Select[Prime[Range[114000]],FactorInteger[#-1][[All,1]]=={2,3}&] (* Harvey P. Dale, Apr 13 2019 *)

Primes of the form 1 + A033845(n).

A065119 Numbers k such that the k-th cyclotomic polynomial is a trinomial.

Original entry on oeis.org

3, 6, 9, 12, 18, 24, 27, 36, 48, 54, 72, 81, 96, 108, 144, 162, 192, 216, 243, 288, 324, 384, 432, 486, 576, 648, 729, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2187, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6561, 6912, 7776, 8748, 9216

Offset: 1

Len Smiley, Nov 12 2001

nonn

Appears to be numbers of form 2^a * 3^b, a >= 0, b > 0. - Lekraj Beedassy, Sep 10 2004
This is true: see link "Cyclotomic trinomials". - Robert Israel, Jul 14 2015
3-smooth numbers (A003586) which are not powers of 2 (A000079). - Amiram Eldar, Nov 10 2020
These are the conjugates of semiprimes, where conjugation is A122111; or Heinz numbers of conjugates of length-2 partitions. - Gus Wiseman, Nov 09 2023
A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 13 2024

			The 54th cyclotomic polynomial is x^18 - x^9 + 1 which is trinomial, so 54 is in the sequence.
From _Gus Wiseman_, Nov 09 2023: (Start)
The terms and conjugate semiprimes, showing their respective Heinz partitions, begin:
    3: (2)              4: (1,1)
    6: (2,1)            6: (2,1)
    9: (2,2)            9: (2,2)
   12: (2,1,1)         10: (3,1)
   18: (2,2,1)         15: (3,2)
   24: (2,1,1,1)       14: (4,1)
   27: (2,2,2)         25: (3,3)
   36: (2,2,1,1)       21: (4,2)
   48: (2,1,1,1,1)     22: (5,1)
   54: (2,2,2,1)       35: (4,3)
   72: (2,2,1,1,1)     33: (5,2)
   81: (2,2,2,2)       49: (4,4)
   96: (2,1,1,1,1,1)   26: (6,1)
(End)

Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 733, pp. 74 and 310, Ellipses Paris, 2004.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Cyclotomic trinomials

Differs at the 18th term from A063996.
Cf. A003586, A033845, A206787.
For primes (A008578) we have conjugates A000079.
For triprimes (A014612) we have conjugates A080193.
A001358 lists semiprimes, squarefree A006881, complement A100959.
Cf. A000040, A000720, A001248, A046682, A056239, A086971, A122111, A220264.

with(numtheory): a := []; for m from 1 to 3000 do if nops([coeffs(cyclotomic(m,x))])=3 then a := [op(a),m] fi od; print(a);

max = 5000; Sort[Flatten[Table[2^a 3^b, {a, 0, Floor[Log[2, max]]}, {b, Floor[Log[3, max/2^a]]}]]] (* Alonso del Arte, May 19 2016 *)

isok(n)=my(vp = Vec(polcyclo(n))); sum(k=1, #vp, vp[k] != 0) == 3; \\ Michel Marcus, Jul 11 2015

list(lim)=my(v=List(),N); for(n=1,logint(lim\1,3), N=3^n; while(N<=lim, listput(v,N); N<<=1)); Set(v) \\ Charles R Greathouse IV, Aug 07 2015

A206787(a(n)) = 4. - Reinhard Zumkeller, Feb 12 2012
a(n) = A033845(n)/2 = 3 * A003586(n). - Robert Israel, Jul 14 2015
Sum_{n>=1} 1/a(n) = 1. - Amiram Eldar, Nov 10 2020

Offset set to 1 and more terms from Michel Marcus, Jul 11 2015

A059960 Smaller term of a pair of twin primes such that prime factors of their average are only 2 and 3.

Original entry on oeis.org

5, 11, 17, 71, 107, 191, 431, 1151, 2591, 139967, 472391, 786431, 995327, 57395627, 63700991, 169869311, 4076863487, 10871635967, 2348273369087, 56358560858111, 79164837199871, 84537841287167, 150289495621631, 578415690713087, 1141260857376767

Offset: 1

Labos Elemer, Mar 02 2001

nonn

Lesser of twin primes p such that p+1 = (2^u)*(3^w), u,w >= 1.

Primes p(k) such that the number of distinct prime divisors of all composite numbers between p(k) and p(k+1) is 2. - Amarnath Murthy, Sep 26 2002

			a(11)+1 = 2*2*2*3*3*3*3*3*3*3*3*3*3 = 472392.

Ray Chandler, Table of n, a(n) for n = 1..61 (terms < 10^1000, first 49 terms from T. D. Noe)

Cf. A014574, A002822, A033845, A058383, A027856.
Cf. A052297, A075581, A075580, A075583, A075584, A075585, A075586, A075587, A075588, A075589.
Apart from initial terms, same as A078883.

nn=10^15; Sort[Reap[Do[n=2^i 3^j; If[n<=nn && PrimeQ[n-1] && PrimeQ[n+1], Sow[n-1]], {i, Log[2, nn]}, {j, Log[3, nn]}]][[2, 1]]]
Select[Select[Partition[Prime[Range[38*10^5]],2,1],#[[2]]-#[[1]]==2&][[All,1]],FactorInteger[#+1][[All,1]]=={2,3}&] (* The program generates the first 15 terms of the sequence. *)
seq[max_] := Select[Sort[Flatten[Table[2^i*3^j - 1, {i, 1, Floor[Log2[max]]}, {j, 1, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {0, 2}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)

a(n) = A027856(n+1) - 1. - Amiram Eldar, Mar 17 2025

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

Original entry on oeis.org

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

A285332 a(0) = 1, a(1) = 2, a(2n) = A019565(a(n)), a(2n+1) = A065642(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 8, 15, 12, 14, 27, 10, 25, 7, 16, 210, 45, 35, 18, 105, 28, 462, 81, 21, 20, 154, 125, 30, 49, 11, 32, 10659, 420, 910, 75, 78, 175, 33, 24, 3094, 315, 385, 56, 780045, 924, 374, 243, 110, 63, 55, 40, 4389, 308, 170170, 625, 1155, 60, 286, 343, 42, 121, 13, 64, 54230826, 31977, 28405, 630, 1330665, 1820, 714

Offset: 0

Antti Karttunen, Apr 17 2017

Note the indexing: the domain starts from 0, while the range excludes zero.
This sequence can be represented as a binary tree. Each left hand child is produced as A019565(n), and each right hand child as A065642(n), when the parent node contains n >= 2:
                                     1
                                     |
                  ...................2...................
                 3                                       4
       6......../ \........9                   5......../ \........8
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  15       12         14       27         10       25          7       16
210 45   35  18    105  28  462  81     21  20  154  125     30 49   11  32
etc.
Where will 38 appear in this tree? It is a reasonable assumption that by iterating A087207 starting from 38, as A087207(38) = 129, A087207(129) = 8194, A087207(8194) = 1501199875790187, ..., we will eventually hit a prime A000040(k), most likely with a largish index k. This prime occurs at the penultimate edge at right, as a(A000918(k)) = a((2^k)-2), and thus 38 occurs somewhere below it as a(m) = 38, m > k. All the numbers that share prime factors with 38, namely 76, 152, 304, 608, 722, ..., occur similarly late in this tree, as they form the rightward branch starting from 38. Alternatively, by iterating A285330 (each iteration moves one step towards the root) starting from 38, we might instead first hit some power of 3, or say, one of the terms of A033845 (the rightward branch starting from 6), in which case the first prime encountered would be a(2)=3 and 38 would appear on the left-hand side instead of the right-hand side subtree.
As long as it remains conjecture that A019565 has no cycles, it is certainly also an open question whether this is a permutation of the natural numbers: If A019565 has any cycles, then neither any of the terms in those cycles nor any A065642-trajectories starting from those terms (that is, numbers sharing same prime factors) may occur in this tree.
Sequence exhibits some outrageous swings, for example, a(703) = 224, but a(704) is 1427 decimal digits (4739 binary digits) long, thus it no longer fits into a b-file.
However, the scatter plot of A286543 gives some flavor of the behavior of this sequence even after that point. - Antti Karttunen, Dec 25 2017

Antti Karttunen, Table of n, a(n) for n = 0..703
Michael De Vlieger, Diagram of the binary tree of a(n) showing 1 <= n <= 2^8.
Antti Karttunen and David J. Seal, Discussion on SeqFan mailing list
Index entries for sequences that are permutations of the natural numbers

Inverse: A285331.
Cf. A007947, A019565, A033845, A048675, A065642, A087207, A109162, A285319, A285320, A285328, A285330, A285333, A286542, A286543.
Compare also to permutation A285112 and array A285321.

Block[{a = {1, 2}}, Do[AppendTo[a, If[EvenQ[i], Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[a[[i/2 + 1]], 2], If[# == 1, 1, Function[{n, c}, SelectFirst[Range[n + 1, n^2], Times @@ FactorInteger[#][[All, 1]] == c &]] @@ {#, Times @@ FactorInteger[#][[All, 1]]}] &[a[[(i - 1)/2 + 1]] ] ]], {i, 2, 70}]; a] (* Michael De Vlieger, Mar 12 2021 *)

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014
A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };
A285332(n) = { if(n<=1,n+1,if(!(n%2),A019565(A285332(n/2)),A065642(A285332((n-1)/2)))); };
for(n=0, 4095, write("b285332.txt", n, " ", A285332(n)));

from operator import mul
from sympy import prime, primefactors
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # This function from Chai Wah Wu
def a065642(n):
    if n==1: return 1
    r=a007947(n)
    n = n + r
    while a007947(n)!=r:
        n+=r
    return n
def a(n):
    if n<2: return n + 1
    if n%2==0: return a019565(a(n//2))
    else: return a065642(a((n - 1)//2))
print([a(n) for n in range(51)]) # Indranil Ghosh, Apr 18 2017

;; With memoization-macro definec.
(definec (A285332 n) (cond ((<= n 1) (+ n 1)) ((even? n) (A019565 (A285332 (/ n 2)))) (else (A065642 (A285332 (/ (- n 1) 2))))))

a(0) = 1, a(1) = 2, a(2n) = A019565(a(n)), a(2n+1) = A065642(a(n)).
For n >= 0, a(2^n) = A109162(2+n). [The left edge of the tree.]
For n >= 0, a(A000225(n)) = A000079(n). [Powers of 2 occur at the right edge of the tree.]
For n >= 2, a(A000918(n)) = A000040(n). [And the next vertices inwards contain primes.]
For n >= 2, a(A036563(1+n)) = A001248(n). [Whose right children are their squares.]
For n >= 0, a(A055010(n)) = A000244(n). [Powers of 3 are at the rightmost edge of the left subtree.]
For n >= 2, a(A129868(n-1)) = A062457(n).
A048675(a(n)) = A285333(n).
A046523(a(n)) = A286542(n).

A256617 Numbers having exactly two distinct prime factors, which are also adjacent prime numbers.

Original entry on oeis.org

6, 12, 15, 18, 24, 35, 36, 45, 48, 54, 72, 75, 77, 96, 108, 135, 143, 144, 162, 175, 192, 216, 221, 225, 245, 288, 323, 324, 375, 384, 405, 432, 437, 486, 539, 576, 648, 667, 675, 768, 847, 864, 875, 899, 972, 1125, 1147, 1152, 1215, 1225, 1296, 1458, 1517, 1536, 1573, 1715, 1728, 1763, 1859, 1875, 1944

Offset: 1

Reinhard Zumkeller, Apr 05 2015

nonn

			.   n | a(n)                      n | a(n)
. ----+------------------       ----+------------------
.   1 |   6 = 2 * 3              13 |  77 = 7 * 11
.   2 |  12 = 2^2 * 3            14 |  96 = 2^5 * 3
.   3 |  15 = 3 * 5              15 | 108 = 2^2 * 3^3
.   4 |  18 = 2 * 3^2            16 | 135 = 3^3 * 5
.   5 |  24 = 2^3 * 3            17 | 143 = 11 * 13
.   6 |  35 = 5 * 7              18 | 144 = 2^4 * 3^2
.   7 |  36 = 2^2 * 3^2          19 | 162 = 2 * 3^4
.   8 |  45 = 3^2 * 5            20 | 175 = 5^2 * 7
.   9 |  48 = 2^4 * 3            21 | 192 = 2^6 * 3
.  10 |  54 = 2 * 3^3            22 | 216 = 2^3 * 3^3
.  11 |  72 = 2^3 * 3^2          23 | 221 = 13 * 17
.  12 |  75 = 3 * 5^2            24 | 225 = 3^2 * 5^2 .

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

Subsequence of A007774.
Subsequences: A006094, A033845, A033849, A033851.
Cf. A000040, A001222, A020639, A006530, A049084, A083553, A151800.

import Data.Set (singleton, deleteFindMin, insert)
a256617 n = a256617_list !! (n-1)
a256617_list = f (singleton (6, 2, 3)) $ tail a000040_list where
   f s ps@(p : ps'@(p':_))
     | m < p * p' = m : f (insert (m * q, q, q')
                          (insert (m * q', q, q') s')) ps
     | otherwise  = f (insert (p * p', p, p') s) ps'
     where ((m, q, q'), s') = deleteFindMin s

Select[Range[2000], MatchQ[FactorInteger[#], {{p_, }, {q, }} /; q == NextPrime[p]]&] (* _Jean-François Alcover, Dec 31 2017 *)

is(n) = if(omega(n)!=2, return(0), my(f=factor(n)[, 1]~); if(f[2]==nextprime(f[1]+1), return(1))); 0 \\ Felix Fröhlich, Dec 31 2017

list(lim)=my(v=List(),c=sqrtnint(lim\=1,3),d=nextprime(c+1),p=2); forprime(q=3,d, for(i=1,logint(lim\q,p), my(t=p^i); while((t*=q)<=lim, listput(v,t))); p=q); forprime(q=d+1,lim\precprime(sqrtint(lim)), listput(v,p*q); p=q); Set(v) \\ Charles R Greathouse IV, Apr 12 2020

from sympy import primefactors, nextprime
A256617_list = []
for n in range(1,10**5):
    plist = primefactors(n)
    if len(plist) == 2 and plist[1] == nextprime(plist[0]):
        A256617_list.append(n) # Chai Wah Wu, Aug 23 2021

A001222(a(n)) = 2.
A006530(a(n)) = A151800(A020639(n)) = A000040(A049084(A020639(a(n)))+1).
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/A083553(n) = Sum_{n>=1} 1/((prime(n)-1)*(prime(n+1)-1)) = 0.7126073495... - Amiram Eldar, Dec 23 2020

cp's OEIS Frontend

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A033847 Numbers whose prime factors are 2 and 7.

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A058383 Primes of form 1+(2^a)*(3^b), a>0, b>0.

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A065119 Numbers k such that the k-th cyclotomic polynomial is a trinomial.

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A059960 Smaller term of a pair of twin primes such that prime factors of their average are only 2 and 3.

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