A003586 -id:A003586 - cp's OEIS frontend

A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 4, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 4, 4, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 4, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Oct 25 2022

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. This sequence gives the number of ways to choose a strict partition of each prime index of n.

The indices i, where a(i) = 1, form A003586, and the indices j, where a(j) > 1, form A059485. - Ivan N. Ianakiev, Oct 27 2022

			The a(121) = 9 twice-partitions are: (5)(5), (5)(41), (5)(32), (41)(5), (41)(41), (41)(32), (32)(5), (32)(41), (32)(32).

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
The non-strict version is A299200.
A horizontal version is A357978, non-strict A357977.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A273873, A296150, A299201-A299203, A357975, A357979, A357983.
Cf. A003586, A059485.

Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsQ[PrimePi[p]]^k],{n,100}]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f9(primepi(f[k,1]))); factorback(f); \\ Michel Marcus, Oct 26 2022

A006899 Numbers of the form 2^i or 3^j.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 16, 27, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 16384, 19683, 32768, 59049, 65536, 131072, 177147, 262144, 524288, 531441, 1048576, 1594323, 2097152, 4194304, 4782969, 8388608, 14348907, 16777216, 33554432

Offset: 1

Simon Plouffe and N. J. A. Sloane

Complement of A033845 with respect to A003586. - Reinhard Zumkeller, Sep 25 2008
In the 14th century, Levi Ben Gerson proved that the only pairs of terms which differ by 1 are (1, 2), (2, 3), (3, 4), and (8, 9); see A235365, A235366, A236210. - Jonathan Sondow, Jan 20 2014
Numbers n such that absolute value of the greatest prime factor of n minus the smallest prime not dividing n is 1 (that is, abs(A006530(n)-A053669(n)) = 1). - Anthony Browne, Jun 26 2016
Deficient 3-smooth numbers, i.e., intersection of A005100 and A003586. - Amiram Eldar, Jun 03 2022

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..500
Boris Alexeev, Minimal DFAs for testing divisibility, arXiv:cs/0309052 [cs.CC], 2003.
Jung-Chao Ban, Wen-Guei Hu, and Song-Sun Lin, Pattern generation problems arising in multiplicative integer systems, arXiv preprint arXiv:1207.7154 [math.DS], 2012.
Lukas Spiegelhofer, Collisions of the binary and ternary sum-of-digits functions, arXiv:2105.11173 [math.NT], 2021.
Eric Weisstein's World of Mathematics, Pillai's Theorem.

Union of A000079 and A000244. - Reinhard Zumkeller, Sep 25 2008
Cf. A085238, A085239.
Cf. A003586, A005100, A006530, A053669.
Cf. A170803, A235365, A235366, A236210.
A186927 and A186928 are subsequences.
Cf. A108906 (first differences), A006895, A227928.

a006899 n = a006899_list !! (n-1)
a006899_list = 1 : m (tail a000079_list) (tail a000244_list) where
   m us'@(u:us) vs'@(v:vs) = if u < v then u : m us vs' else v : m us' vs
-- Reinhard Zumkeller, Oct 09 2013

A:={seq(2^n,n=0..63)}: B:={seq(3^n,n=0..40)}: C:=sort(convert(A union B,list)): seq(C[j],j=1..39); # Emeric Deutsch, Aug 03 2005

seqMax = 10^20; Union[2^Range[0, Floor[Log[2, seqMax]]], 3^Range[0, Floor[Log[3, seqMax]]]] (* Stefan Steinerberger, Apr 08 2006 *)

is(n)=n>>valuation(n,2)==1 || n==3^valuation(n,3) \\ Charles R Greathouse IV, Aug 29 2016

upto(n) = my(res = vector(logint(n, 2) + logint(n, 3) + 1), t = 1); res[1] = 1; for(i = 2, 3, for(j = 1, logint(n, i), t++; res[t] = i^j)); vecsort(res) \\ David A. Corneth, Oct 26 2017

a(n) = my(i0= logint(3^(n-1),6), i= logint(3^n,6)); if(i > i0, 2^i, my(j=logint(2^n,6)); 3^j) \\ Ruud H.G. van Tol, Nov 10 2022

from sympy import integer_log
def A006899(n): return 1<Chai Wah Wu, Oct 01 2024

a(n) = A085239(n)^A085238(n). - Reinhard Zumkeller, Jun 22 2003
A086411(a(n)) = A086410(a(n)). - Reinhard Zumkeller, Sep 25 2008
A053669(a(n)) - A006530(a(n)) = (-1)^a(n) n > 1. - Anthony Browne, Jun 26 2016
Sum_{n>=1} 1/a(n) = 5/2. - Amiram Eldar, Jun 03 2022
a(n)^(1/n) tends to 3^(log(2)/log(6)) = 2^(log(3)/log(6)) = 1.529592328491883538... - Vaclav Kotesovec, Sep 19 2024

More terms from Reinhard Zumkeller, Jun 22 2003

A368109 Number of ways to choose a binary index of each binary index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 4, 4, 4, 4, 8, 8, 8, 8, 3, 3, 3, 3, 6, 6, 6, 6, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12

Offset: 0

Gus Wiseman, Dec 12 2023

nonn

First differs from A367912 at a(52) = 8, A367912(52) = 7.
A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.
Run-lengths are all 4 or 8.

			The binary indices of binary indices of 20 are {{1,2},{1,3}}, with choices (1,1), (1,3), (2,1), (2,3), so a(20) = 4.
The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with choices (1,1,1), (1,1,3), (1,3,2), (1,3,3), (2,1,2), (2,1,3), (2,3,2), (2,3,3), so a(52) = 8.

All entries appear to belong to A003586.
Positions of ones are A253317.
The version for prime indices is A355741, for multisets A355744.
Choosing a multiset (not sequence) gives A367912, firsts A367913.
Positions of first appearances are A368111, sorted A368112.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A355731, A355739, A367771, A367905, A367906, A367915.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
Table[Length[Tuples[bpe/@bpe[n]]], {n,0,100}]

a(n) = Product_{k in A048793(n)} A000120(k).

A190803 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x-1 and 3x-1 are in a.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 14, 15, 17, 23, 26, 27, 29, 33, 41, 44, 45, 50, 51, 53, 57, 65, 68, 77, 80, 81, 86, 87, 89, 98, 99, 101, 105, 113, 122, 129, 131, 134, 135, 149, 152, 153, 158, 159, 161, 170, 171, 173, 177, 194, 195, 197, 201, 203, 209, 225, 230, 239, 242

Offset: 1

Clark Kimberling, May 25 2011

nonn

This sequence represents a class of sequences generated by rules of the form "a(1)=1, and if x is in a then hx+i and jx+k are in a, where h,i,j,k are integers." If m>1, at least one of the numbers b(m)=(a(m)-i)/h and c(m)=(a(m)-k)/j is in the set N of natural numbers. Let d(n) be the n-th b(m) in N, and let e(n) be the n-th c(m) in N. Note that a is a subsequence of both d and e.
Examples, where [A......] indicates a conjecture:
A190803: (h,i,j,k)=(2,-1,3,-1); d=A190841, e=A190842
A190804: (h,i,j,k)=(2,-1,3,0); d=[A190803], e=A190844
A190805: (h,i,j,k)=(2,-1,3,1); d=A190845, e=[A190808]
A190806: (h,i,j,k)=(2,-1,3,2); d=[A190804], e=A190848
...
A190807: (h,i,j,k)=(2,0,3,-1); d=A190849, e=A190850
A003586: (h,i,j,k)=(2,0,3,0); d=e=A003586
A190808: (h,i,j,k)=(2,0,3,1); d=A190851, e=A190852
A190809: (h,i,j,k)=(2,0,3,2); d=A190853, e=A190854
...
A190810: (h,i,j,k)=(2,1,3,-1); d=A190855, e=A190856
A190811: (h,i,j,k)=(2,1,3,0); d=A002977, e=A190857
A002977: (h,i,j,k)=(2,1,3,1); d=A190858, e=A190859
A190812: (h,i,j,k)=(2,1,3,2); d=A069353, e=[A190812]
...
For h=j=3, see A191106; for h=3 and j=4, see A191113.

			1 -> 2 -> 3,5 -> 8,9,14 -> 15,17,23,26,27,41 -> ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, 10 (2007) 1-13.

Cf. A002977, A003586, A190804-A190812, A190841-A190860.

import Data.Set (singleton, deleteFindMin, insert)
a190803 n = a190803_list !! (n-1)
a190803_list = 1 : f (singleton 2)
   where f s = m : (f $ insert (2*m-1) $ insert (3*m-1) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 2; i = -1; j = 3; k = -1; f = 1; g = 10;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A190803 *)
b = (a + 1)/2; c = (a + 1)/3; r = Range[1, 300];
d = Intersection[b, r] (* A190841 *)
e = Intersection[c, r] (* A190842 *)
(* Regarding this program - useful for many choices of h,i,j,k,f,g - the depth g must be chosen with care - that is, large enough.  Assuming that h<=j, the least new terms in successive nests a are typically iterates of hx+i, starting from x=1.  If, for example, h=2 and i=0, the least terms are 2,4,8,...,2^g, so that g>=9 ensures inclusion of all the desired terms <=256. *)

a(34)=225 inserted by Reinhard Zumkeller, Jun 01 2011

A003591 Numbers of form 2^i*7^j, with i, j >= 0.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 28, 32, 49, 56, 64, 98, 112, 128, 196, 224, 256, 343, 392, 448, 512, 686, 784, 896, 1024, 1372, 1568, 1792, 2048, 2401, 2744, 3136, 3584, 4096, 4802, 5488, 6272, 7168, 8192, 9604, 10976, 12544, 14336, 16384, 16807, 19208, 21952, 25088

Offset: 1

N. J. A. Sloane

nonn

A204455(7*a(n)) = 7, and only for these numbers. - Wolfdieter Lang, Feb 04 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 100 terms from Vincenzo Librandi)
Vaclav Kotesovec, Graph - the asymptotic ratio (250000 terms)

Cf. A003586, A003592, A003593, A003594, A003595.

Cf. A025637, A025664.

Filtered([1..30000],n->PowerMod(14,n,n)=0); # Muniru A Asiru, Mar 19 2019

import Data.Set (singleton, deleteFindMin, insert)
a003591 n = a003591_list !! (n-1)
a003591_list = f $ singleton 1 where
   f s = y : f (insert (2 * y) $ insert (7 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015

[n: n in [1..26000] | PrimeDivisors(n) subset [2,7]]; // Bruno Berselli, Sep 24 2012

fQ[n_] := PowerMod[14,n,n]==0; Select[Range[30000], fQ] (* Vincenzo Librandi, Feb 04 2012 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(7),N=7^n;while(N<=lim,listput(v,N);N<<=1));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

isA003591(n)=n>>=valuation(n,2);ispower(n,,&n);n==1||n==7 \\ Charles R Greathouse IV, Jun 28 2011

from sympy import integer_log
def A003591(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//7**i).bit_length() for i in range(integer_log(x,7)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(14*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019
Sum_{n>=1} 1/a(n) = (2*7)/((2-1)*(7-1)) = 7/3. - Amiram Eldar, Sep 22 2020
a(n) ~ exp(sqrt(2*log(2)*log(7)*n)) / sqrt(14). - Vaclav Kotesovec, Sep 22 2020
a(n) = 2^A025637(n) *7^A025664(n). - R. J. Mathar, Jul 06 2025

A071364 Smallest number with same sequence of exponents in canonical prime factorization as n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 18, 2, 12, 6, 6, 2, 24, 4, 6, 8, 12, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 24, 2, 30, 2, 12, 12, 6, 2, 48, 4, 18, 6, 12, 2, 54, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, 2, 12, 6, 30, 2, 72, 2, 6, 18, 12, 6, 30, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24

Offset: 1

Reinhard Zumkeller, May 21 2002

nonn

A046523(a(n))=A046523(n); A046523(n)<=a(n)<=n; A001221(a(n))=A001221(n), A001222(a(n))=A001222(n); A020639(a(n))=2, A006530(a(n))=A000040(A001221(n))<=A006530(n); A000005(a(n))=A000005(n);
a(a(n))=a(n); a(n)=2^k iff n=p^k, p prime, k>0 (A000961); if n>1 is not a prime power, then a(n) mod 6 = 0; range of values = A055932, as distinct prime factors of a(n) are consecutive: a(n)=n iff n=A055932(k) for some k;
a(A003586(n))=A003586(n).

			a(105875) = a(5*5*5*7*11*11) = 2*2*2*3*5*5 = 600.

Daniel Forgues, Table of n, a(n) for n=1..100000

Cf. A071365, A071366.
Cf. A000040.
Cf. A085079, A089247.
The range is A055932.
The reversed version is A331580.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
Cf. A056239, A112798, A233249, A333217, A333219, A334032, A334033.

a071364 = product . zipWith (^) a000040_list . a124010_row
-- Reinhard Zumkeller, Feb 19 2012

Table[ e = Last /@ FactorInteger[n]; Product[Prime[i]^e[[i]], {i, Length[e]}], {n, 88}] (* Ray Chandler, Sep 23 2005 *)

a(n) = f = factor(n); for (i=1, #f~, f[i,1] = prime(i)); factorback(f); \\ Michel Marcus, Jun 13 2014

from math import prod
from sympy import prime, factorint
def A071364(n): return prod(prime(i+1)**p[1] for i,p in enumerate(sorted(factorint(n).items()))) # Chai Wah Wu, Sep 16 2022

In prime factorization of n, replace least prime by 2, next least by 3, etc.

a(n) = product(A000040(k)^A124010(k): k=1..A001221(n)). - Reinhard Zumkeller, Apr 27 2013

Extended by Ray Chandler, Sep 23 2005

A080197 13-smooth numbers: numbers whose prime divisors are all <= 13.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 56, 60, 63, 64, 65, 66, 70, 72, 75, 77, 78, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 104, 105, 108, 110, 112, 117, 120

Offset: 1

Klaus Brockhaus, Feb 10 2003

Numbers of the form 2^r*3^s*5^t*7^u*11^v*13^w with r, s, t, u, v, w >= 0.

			33 = 3*11 and 39 = 3*13 are terms but 34 = 2*17 is not.

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

Cf. A000079, A080196. For p-smooth numbers with other values of p, see A003586, A051037, A002473, A051038, A080681, A080682, A080683.

[n: n in [1..150] | PrimeDivisors(n) subset PrimesUpTo(13)]; // Bruno Berselli, Sep 24 2012

mx = 120; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l*11^m*13^n, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[5, mx/(2^i*3^j)]}, {l, 0, Log[7, mx/(2^i*3^j*5^k)]}, {m, 0, Log[11, mx/(2^i*3^j*5^k*7^l)]}, {n, 0, Log[13, mx/(2^i*3^j*5^k*7^l*11^m)]}] (* Robert G. Wilson v, Aug 17 2012 *)

test(n)=m=n; forprime(p=2,13, while(m%p==0,m=m/p)); return(m==1)
for(n=1,200,if(test(n),print1(n",")))

is_A080197(n,p=13)=n<=p||vecmax(factor(n,p+1)[,1])<=p \\ M. F. Hasler, Jan 16 2015

list(lim,p=13)=if(p==2, return(powers(2, logint(lim\1,2)))); my(v=[],q=precprime(p-1),t=1); for(e=0,logint(lim\=1,p), v=concat(v, list(lim\t,q)*t); t*=p); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

import heapq
from itertools import islice
from sympy import primerange
def agen(p=13): # generate all p-smooth terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(agen(), 69))) # Michael S. Branicky, Nov 20 2022

from sympy import integer_log, prevprime
def A080197(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,13)
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = Product_{primes p <= 13} p/(p-1) = (2*3*5*7*11*13)/(1*2*4*6*10*12) = 1001/192. - Amiram Eldar, Sep 22 2020

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

Original entry on oeis.org

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

A085398 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.

Original entry on oeis.org

3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 6, 2, 4, 3, 2, 10, 2, 22, 2, 2, 4, 6, 2, 2, 2, 2, 2, 14, 3, 61, 2, 10, 2, 14, 2, 15, 25, 11, 2, 5, 5, 2, 6, 30, 11, 24, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 30, 2, 9, 46, 85, 2, 3, 3, 3, 11, 16, 59, 7, 2, 2, 22, 2, 21, 61, 41, 7, 2, 2, 8, 5, 2, 2

Offset: 1

Don Reble, Jun 28 2003

nonn

Conjecture: a(n) is defined for all n. - Eric Chen, Nov 14 2014

Existence of a(n) is implied by Bunyakovsky's conjecture. - Robert Israel, Nov 13 2014

			a(11) = 5 because C11(k) is composite for k = 2, 3, 4 and prime for k = 5.
a(37) = 61 because C37(k) is composite for k = 2, 3, 4, ..., 60 and prime for k = 61.

Jinyuan Wang, Table of n, a(n) for n = 1..5000 (terms 1..1500 from Eric Chen)
Wikipedia, Bunyakowsky conjecture

Cf. A117544, A066180, A085399, A103795, A056993, A153438, A246119, A246120, A246121, A206418, A205506, A181980.

Cf. A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A006314, A217071, A164989, A217072, A217073, A153440, A217074, A217075, A006313, A097475.

f:= proc(n) local k;
for k from 2 do if isprime(numtheory:-cyclotomic(n,k)) then return k fi od
end proc:
seq(f(n), n = 1 .. 100); # Robert Israel, Nov 13 2014

Table[k = 2; While[!PrimeQ[Cyclotomic[n, k]], k++]; k, {n, 300}] (* Eric Chen, Nov 14 2014 *)

a(n) = k=2; while(!isprime(polcyclo(n, k)), k++); k; \\ Michel Marcus, Nov 13 2014

a(A072226(n)) = 2. - Eric Chen, Nov 14 2014
a(n) = A117544(n) except when n is a prime power, since if n is a prime power, then A117544(n) = 1. - Eric Chen, Nov 14 2014
a(prime(n)) = A066180(n), a(2*prime(n)) = A103795(n), a(2^n) = A056993(n-1), a(3^n) = A153438(n-1), a(2*3^n) = A246120(n-1), a(3*2^n) = A246119(n-1), a(6^n) = A246121(n-1), a(5^n) = A206418(n-1), a(6*A003586(n)) = A205506(n), a(10*A003592(n)) = A181980(n).

A018256 Divisors of 36.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 18, 36

Offset: 1

N. J. A. Sloane

36 is a highly composite number: A002182(7)=36. - Reinhard Zumkeller, Jun 21 2010

Numbers with all prime indices and exponents <= 2. Reversing inequalities gives A062739, strict A353502. - Gus Wiseman, Jun 28 2022

Index entries for sequences related to divisors of numbers

Cf. A018253, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Cf. A003586, A004709, A018338, A018710, A062739, A353502.

Divisors[36] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

divisors(36) \\ Charles R Greathouse IV, Jun 21 2017

Intersection of A003586 (3-smooth) and A004709 (cubefree). - Gus Wiseman, Jun 28 2022

cp's OEIS Frontend

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