A033851 -id:A033851 - cp's OEIS frontend

A087207 A binary representation of the primes that divide a number, shown in decimal.

0, 1, 2, 1, 4, 3, 8, 1, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 3, 4, 33, 2, 9, 512, 7, 1024, 1, 18, 65, 12, 3, 2048, 129, 34, 5, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 3, 20, 9, 130, 513, 65536, 7, 131072, 1025, 10, 1, 36, 19, 262144, 65, 258

Offset: 1

Mitch Cervinka (puritan(AT)planetkc.com), Oct 26 2003

The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.
For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017
From Antti Karttunen, Jun 18 & 20 2017: (Start)
A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n).
For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself.
Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565.
If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612.
(End)
Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024

			a(38) = 129 because 38 = 2*19 = prime(1)*prime(8) and 129 = 2^0 + 2^7 (in binary 10000001).
a(140) = 13, binary 1101 because 140 is divisible by the first, third and fourth primes and 2^(1-1) + 2^(3-1) + 2^(4-1) = 13.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 [First 1000 terms from _T. D. Noe_]
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

For partial sums see A288566.
Cf. A000040, A000120, A001221, A005117, A008479, A019565, A055396, A285320, A285321, A285329, A285330, A285332.
Sequences with related definitions: A007947, A008472, A027748, A048675, A248663, A276379 (same sequence shown in base 2), A288569, A289271, A297404.
Cf. A286608 (numbers n for which a(n) < n), A286609 (n for which a(n) > n), and also A286611, A286612.
A003986, A003961, A059896 are used to express relationship between terms of this sequence.
Related to A267116 via A225546.
Positions of particular values are: A000079\{1} (1), A000244\{1} (2), A033845 (3), A000351\{1} (4), A033846 (5), A033849 (6), A143207 (7), A000420\{1} (8), A033847 (9), A033850 (10), A033851 (12), A147576 (14), A147571 (15), A001020\{1} (16), A033848 (17).
A048675 gives binary rank of prime indices.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices (listed A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- sum A029931, product A096111
- max A029837 or A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000720, A005940, A018819, A023506, A071814, A225620, A277319, A277905, A304818, A372689, A372890.

a087207 = sum . map ((2 ^) . (subtract 1) . a049084) . a027748_row
-- Reinhard Zumkeller, Jul 16 2013

a[n_] := Total[ 2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]; a[1] = 0; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Dec 12 2011 *)

a(n) = {if (n==1, 0, my(f=factor(n), v = []); forprime(p=2, vecmax(f[,1]), v = concat(v, vecsearch(f[,1], p)!=0);); fromdigits(Vecrev(v), 2));} \\ Michel Marcus, Jun 05 2017

A087207(n)=vecsum(apply(p->1<M. F. Hasler, Jun 23 2017

from sympy import factorint, primepi
def a(n):
    return sum(2**primepi(i - 1) for i in factorint(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

(definec (A087207 n) (if (= 1 n) 0 (+ (A000079 (+ -1 (A055396 n))) (A087207 (A028234 n))))) ;; This uses memoization-macro definec
(define (A087207 n) (A048675 (A007947 n))) ;; Needs code from A007947 and A048675. - Antti Karttunen, Jun 19 2017

Additive with a(p^e) = 2^(i-1) where p is the i-th prime. - Vladeta Jovovic, Oct 29 2003
a(n) gives the m such that A019565(m) = A007947(n). - Naohiro Nomoto, Oct 30 2003
A000120(a(n)) = A001221(n); a(n) = Sum(2^(A049084(p)-1): p prime-factor of n). - Reinhard Zumkeller, Nov 30 2003
G.f.: Sum_{k>=1} 2^(k-1)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
From Antti Karttunen, Apr 17 2017, Jun 19 2017 & Dec 06 2018: (Start)
a(n) = A048675(A007947(n)).
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A028234(n)).
A000035(a(n)) = 1 - A000035(n). [a(n) and n are of opposite parity.]
A248663(n) <= a(n) <= A048675(n). [XOR-, OR- and +-variants.]
a(A293214(n)) = A218403(n).
a(A293442(n)) = A267116(n).
A069010(a(n)) = A287170(n).
A007088(a(n)) = A276379(n).
A038374(a(n)) = A300820(n) for n >= 1.
(End)
From Peter Munn, Jan 08 2020: (Start)
a(A059896(n,k)) = a(n) OR a(k) = A003986(a(n), a(k)).
a(A003961(n)) = 2*a(n).
a(n^2) = a(n).
a(n) = A267116(A225546(n)).
a(A225546(n)) = A267116(n).
(End)

More terms from Don Reble, Ray Chandler and Naohiro Nomoto, Oct 28 2003

Name clarified by Antti Karttunen, Jun 18 2017

A033846 Numbers whose prime factors are 2 and 5.

Original entry on oeis.org

10, 20, 40, 50, 80, 100, 160, 200, 250, 320, 400, 500, 640, 800, 1000, 1250, 1280, 1600, 2000, 2500, 2560, 3200, 4000, 5000, 5120, 6250, 6400, 8000, 10000, 10240, 12500, 12800, 16000, 20000, 20480, 25000, 25600, 31250, 32000, 40000, 40960

Offset: 1

Jeff Burch

Numbers k such that Sum_{d prime divisor of k} 1/d = 7/10. - Benoit Cloitre, Apr 13 2002
Numbers k such that phi(k) = (2/5)*k. - Benoit Cloitre, Apr 19 2002
Numbers k such that Sum_{d|k} A008683(d)*A000700(d) = 7. - Carl Najafi, Oct 20 2011

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

Cf. A033845, A033847, A033848, A033849, A033850, A033851, A003592.
Cf. A086780, A143201.
Cf. A000700, A008683.

import Data.Set (singleton, deleteFindMin, insert)
a033846 n = a033846_list !! (n-1)
a033846_list = f (singleton (2*5)) where
   f s = m : f (insert (2*m) $ insert (5*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

[n:n in [1..100000]| Set(PrimeDivisors(n)) eq {2,5}]; // Marius A. Burtea, May 10 2019

A033846 := proc(n)
if (numtheory[factorset](n) = {2,5}) then
   RETURN(n)
fi: end:  seq(A033846(n),n=1..50000); # Jani Melik, Feb 24 2011

Take[Union[Times@@@Select[Flatten[Table[Tuples[{2,5},n],{n,2,15}],1], Length[Union[#]]>1&]],45] (* Harvey P. Dale, Dec 15 2011 *)

isA033846(n)=factor(n)[,1]==[2,5]~ \\ Charles R Greathouse IV, Feb 24 2011

a(n) = 10*A003592(n).
A143201(a(n)) = 4. - Reinhard Zumkeller, Sep 13 2011
Sum_{n>=1} 1/a(n) = 1/4. - Amiram Eldar, Dec 22 2020

Offset fixed by Reinhard Zumkeller, Sep 13 2011

A033849 Numbers whose prime factors are 3 and 5.

Original entry on oeis.org

15, 45, 75, 135, 225, 375, 405, 675, 1125, 1215, 1875, 2025, 3375, 3645, 5625, 6075, 9375, 10125, 10935, 16875, 18225, 28125, 30375, 32805, 46875, 50625, 54675, 84375, 91125, 98415, 140625, 151875, 164025, 234375, 253125, 273375, 295245

Offset: 1

Jeff Burch

nonn

Numbers k such that phi(k) = (8/15)*k. - Benoit Cloitre, Apr 19 2002

Subsequence of A143202. - Reinhard Zumkeller, Sep 13 2011

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

Cf. A033845, A033846, A033847, A033848, A033850, A033851, A143201, A143202.

Subsequence of A256617.

import Data.Set (singleton, deleteFindMin, insert)
a033849 n = a033849_list !! (n-1)
a033849_list = f (singleton (3*5)) where
   f s = m : f (insert (3*m) $ insert (5*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

Sort[Flatten[Table[Table[3^j*5^k, {j, 1, 10}], {k, 1, 10}]]] (* Geoffrey Critzer, Dec 07 2014 *)
Select[Range[300000],FactorInteger[#][[All,1]]=={3,5}&] (* Harvey P. Dale, Oct 19 2022 *)

from sympy import integer_log
def A033849(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(integer_log(x//5**i,3)[0]+1 for i in range(integer_log(x,5)[0]+1))
    return 15*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

From Reinhard Zumkeller, Sep 13 2011: (Start)
A143201(a(n)) = 3.
a(n) = 15*A003593(n). (End)
Sum_{n>=1} 1/a(n) = 1/8. - Amiram Eldar, Dec 22 2020

Offset and typo in data fixed by Reinhard Zumkeller, Sep 13 2011

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).

A003595 Numbers of the form 5^i*7^j with i, j >= 0.

Original entry on oeis.org

1, 5, 7, 25, 35, 49, 125, 175, 245, 343, 625, 875, 1225, 1715, 2401, 3125, 4375, 6125, 8575, 12005, 15625, 16807, 21875, 30625, 42875, 60025, 78125, 84035, 109375, 117649, 153125, 214375, 300125, 390625, 420175, 546875, 588245, 765625, 823543, 1071875, 1500625

Offset: 1

N. J. A. Sloane

nonn

Successive k such that phi(35*k) = 24*k: 35*a(n) = A033851(n). - Artur Jasinski, Nov 09 2008

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Vaclav Kotesovec, Graph - the asymptotic ratio (400000 terms).
David Ryan, An algorithm to assign musical prime commas to every prime number and construct a universal and compact free Just Intonation musical notation, Preprint, arXiv:1612.01860 [cs.SD], 2016-2017.

Cf. A033851, A143207, A147571-A147580.
Cf. A003586, A003592, A003593, A003594.
Cf. A025652, A025667.

import Data.Set (singleton, deleteFindMin, insert)
a003595 n = a003595_list !! (n-1)
a003595_list = f $ singleton 1 where
   f s = y : f (insert (5 * y) $ insert (7 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015

[n: n in [1..600000] | PrimeDivisors(n) subset [5,7]]; // Bruno Berselli, Sep 24 2012

a = {}; Do[If[EulerPhi[35 k] == 24 k, AppendTo[a, k]], {k, 1, 10000}]; a (* Artur Jasinski, Nov 09 2008 *)
fQ[n_] := PowerMod[35, n, n] == 0; Select[Range[600000], fQ] (* Bruno Berselli, Sep 24 2012 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(7),N=7^n;while(N<=lim,listput(v,N);N*=5));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

from sympy import integer_log
def A003595(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(integer_log(x//7**i,5)[0]+1 for i in range(integer_log(x,7)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = (5*7)/((5-1)*(7-1)) = 35/24. - Amiram Eldar, Sep 22 2020
a(n) ~ exp(sqrt(2*log(5)*log(7)*n)) / sqrt(35). - Vaclav Kotesovec, Sep 22 2020
a(n) = 5^A025652(n) * 7^A025667(n). - R. J. Mathar, Jul 06 2025

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

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

A087207 A binary representation of the primes that divide a number, shown in decimal.

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A033846 Numbers whose prime factors are 2 and 5.

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A033849 Numbers whose prime factors are 3 and 5.

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

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

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A003595 Numbers of the form 5^i*7^j with i, j >= 0.

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