A143201 -id:A143201 - cp's OEIS frontend

A000961 Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1).

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

N. J. A. Sloane

The term "prime power" is ambiguous. To a mathematician it means any number p^k, p prime, k >= 0, including p^0 = 1.
Any nonzero integer is a product of primes and units, where the units are +1 and -1. This is tied to the Fundamental Theorem of Arithmetic which proves that the factorizations are unique up to order and units. (So, since 1 = p^0 does not have a well defined prime base p, it is sometimes not regarded as a prime power. See A246655 for the sequence without 1.)
These numbers are (apart from 1) the numbers of elements in finite fields. - Franz Vrabec, Aug 11 2004
Numbers whose divisors form a geometrical progression. The divisors of p^k are 1, p, p^2, p^3, ..., p^k. - Amarnath Murthy, Jan 09 2002
These are also precisely the orders of those finite affine planes that are known to exist as of today. (The order of a finite affine plane is the number of points in an arbitrarily chosen line of that plane. This number is unique for all lines comprise the same number of points.) - Peter C. Heinig (algorithms(AT)gmx.de), Aug 09 2006
Except for first term, the index of the second number divisible by n in A002378, if the index equals n. - Mats Granvik, Nov 18 2007
These are precisely the numbers such that lcm(1,...,m-1) < lcm(1,...,m) (=A003418(m) for m>0; here for m=1, the l.h.s. is taken to be 0). We have a(n+1)=a(n)+1 if a(n) is a Mersenne prime or a(n)+1 is a Fermat prime; the converse is true except for n=7 (from Catalan's conjecture) and n=1, since 2^1-1 and 2^0+1 are not considered as Mersenne resp. Fermat prime. - M. F. Hasler, Jan 18 2007, Apr 18 2010
The sequence is A000015 without repetitions, or more formally, A000961=Union[A000015]. - Zak Seidov, Feb 06 2008
Except for a(1)=1, indices for which the cyclotomic polynomial Phi[k] yields a prime at x=1, cf. A020500. - M. F. Hasler, Apr 04 2008
Also, {A138929(k) ; k>1} = {2*A000961(k) ; k>1} = {4,6,8,10,14,16,18,22,26,32,34,38,46,50,54,58,62,64,74,82,86,94,98,...} are exactly the indices for which Phi[k](-1) is prime. - M. F. Hasler, Apr 04 2008
A143201(a(n)) = 1. - Reinhard Zumkeller, Aug 12 2008
Number of distinct primes dividing n=omega(n) < 2. - Juri-Stepan Gerasimov, Oct 30 2009
Numbers n such that Sum_{p-1|p is prime and divisor of n} = Product_{p-1|p is prime and divisor of n}. A055631(n) = A173557(n-1). - Juri-Stepan Gerasimov, Dec 09 2009, Mar 10 2010
Numbers n such that A028236(n) = 1. Klaus Brockhaus, Nov 06 2010
A188666(k) = a(k+1) for k: 2*a(k) <= k < 2*a(k+1), k > 0; notably a(n+1) = A188666(2*a(n)). - Reinhard Zumkeller, Apr 25 2011
A003415(a(n)) = A192015(n); A068346(a(n)) = A192016(n); a(n)=A192134(n) + A192015(n). - Reinhard Zumkeller, Jun 26 2011
A089233(a(n)) = 0. - Reinhard Zumkeller, Sep 04 2013
The positive integers n such that every element of the symmetric group S_n which has order n is an n-cycle. - W. Edwin Clark, Aug 05 2014
Conjecture: these are numbers m such that Sum_{k=0..m-1} k^phi(m) == phi(m) (mod m), where phi(m) = A000010(m). - Thomas Ordowski and Giovanni Resta, Jul 25 2018
Numbers whose (increasingly ordered) divisors are alternatingly squares and nonsquares. - Michel Marcus, Jan 16 2019
Possible numbers of elements in a finite vector space. - Jianing Song, Apr 22 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
M. Koecher and A. Krieg, Ebene Geometrie, Springer, 1993.
R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge 1986, Theorem 2.5, p. 45.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Brady Haran and Günter Ziegler, Cannons and Sparrows, Numberphile video (2018).
Laurentiu Panaitopol, Some of the properties of the sequence of powers of prime numbers, Rocky Mountain Journal of Mathematics, Volume 31, Number 4, Winter 2001.
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Projective Plane
Index entries for "core" sequences

There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2). When you refer to "prime powers", be sure to specify which of these you mean. Also A001597 is the sequence of nontrivial powers n^k, n >= 1, k >= 2. - N. J. A. Sloane, Mar 24 2018
Cf. A008480, A010055, A065515, A095874, A025473.
Cf. indices of record values of A003418; A000668 and A019434 give a member of twin pairs a(n+1)=a(n)+1.
A138929(n) = 2*a(n).
Cf. A000040, A001221, A001477. - Juri-Stepan Gerasimov, Dec 09 2009
A028236 (if n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j). - Klaus Brockhaus, Nov 06 2010
A000015(n) = Min{term : >= n}; A031218(n) = Max{term : <= n}.
Complementary (in the positive integers) to sequence A024619. - Jason Kimberley, Nov 10 2015

import Data.Set (singleton, deleteFindMin, insert)
a000961 n = a000961_list !! (n-1)
a000961_list = 1 : g (singleton 2) (tail a000040_list) where
g s (p:ps) = m : g (insert (m * a020639 m) $ insert p s') ps
where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2012, Apr 25 2011

[1] cat [ n : n in [2..250] | IsPrimePower(n) ]; // corrected by Arkadiusz Wesolowski, Jul 20 2012

readlib(ifactors): for n from 1 to 250 do if nops(ifactors(n)[2])=1 then printf(`%d,`,n) fi: od:
# second Maple program:
a:= proc(n) option remember; local k; for k from
      1+a(n-1) while nops(ifactors(k)[2])>1 do od; k
    end: a(1):=1: A000961:= a:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 08 2013

Select[ Range[ 2, 250 ], Mod[ #, # - EulerPhi[ # ] ] == 0 & ]
Select[ Range[ 2, 250 ], Length[FactorInteger[ # ] ] == 1 & ]
max = 0; a = {}; Do[m = FactorInteger[n]; w = Sum[m[[k]][[1]]^m[[k]][[2]], {k, 1, Length[m]}]; If[w > max, AppendTo[a, n]; max = w], {n, 1, 1000}]; a (* Artur Jasinski *)
Join[{1}, Select[Range[2, 250], PrimePowerQ]] (* Jean-François Alcover, Jul 07 2015 *)

A000961(n,l=-1,k=0)=until(n--<1,until(lA000961(lim=999,l=-1)=for(k=1,lim, l==lcm(l,k) && next; l=lcm(l,k); print1(k,",")) \\ M. F. Hasler, Jan 18 2007

isA000961(n) = (omega(n) == 1 || n == 1) \\ Michael B. Porter, Sep 23 2009

nextA000961(n)=my(m,r,p);m=2*n;for(e=1,ceil(log(n+0.01)/log(2)),r=(n+0.01)^(1/e);p=prime(primepi(r)+1);m=min(m,p^e));m \\ Michael B. Porter, Nov 02 2009

is(n)=isprimepower(n) || n==1 \\ Charles R Greathouse IV, Nov 20 2012

list(lim)=my(v=primes(primepi(lim)),u=List([1])); forprime(p=2,sqrtint(lim\1),for(e=2,log(lim+.5)\log(p),listput(u,p^e))); vecsort(concat(v,Vec(u))) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import primerange
def A000961_list(limit): # following Python style, list terms < limit
    L = [1]
    for p in primerange(1, limit):
        pe = p
        while pe < limit:
            L.append(pe)
            pe *= p
    return sorted(L) # Chai Wah Wu, Sep 08 2014, edited by M. F. Hasler, Jun 16 2022

from sympy import primepi
from sympy.ntheory.primetest import integer_nthroot
def A000961(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 23 2024

def A000961_list(n):
    R = [1]
    for i in (2..n):
        if i.is_prime_power(): R.append(i)
    return R
A000961_list(227) # Peter Luschny, Feb 07 2012

a(n) = A025473(n)^A025474(n). - David Wasserman, Feb 16 2006
a(n) = A117331(A117333(n)). - Reinhard Zumkeller, Mar 08 2006
Panaitopol (2001) gives many properties, inequalities and asymptotics, including a(n) ~ prime(n). - N. J. A. Sloane, Oct 31 2014, corrected by M. F. Hasler, Jun 12 2023 [The reference gives pi*(x) = pi(x) + pi(sqrt(x)) + ... where pi*(x) counts the terms up to x, so it is the inverse function to a(n).]
m=a(n) for some n <=> lcm(1,...,m-1) < lcm(1,...,m), where lcm(1...0):=0 as to include a(1)=1. a(n+1)=a(n)+1 <=> a(n+1)=A019434(k) or a(n)=A000668(k) for some k (by Catalan's conjecture), except for n=1 and n=7. - M. F. Hasler, Jan 18 2007, Apr 18 2010
A001221(a(n)) < 2. - Juri-Stepan Gerasimov, Oct 30 2009
A008480(a(n)) = 1 for all n >= 1. - Alois P. Heinz, May 26 2018
Sum_{k=1..n} 1/a(k) ~ log(log(a(n))) + 1 + A077761 + A136141. - François Huppé, Jul 31 2024

Description modified by Ralf Stephan, Aug 29 2014

A006093 a(n) = prime(n) - 1.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270

Offset: 1

N. J. A. Sloane

These are also the numbers that cannot be written as i*j + i + j (i,j >= 1). - Rainer Rosenthal, Jun 24 2001; Henry Bottomley, Jul 06 2002
The values of k for which Sum_{j=0..n} (-1)^j*binomial(k, j)*binomial(k-1-j, n-j)/(j+1) produces an integer for all n such that n < k. Setting k=10 yields [0, 1, 4, 11, 19, 23, 19, 11, 4, 1, 0] for n = [-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], so 10 is in the sequence. Setting k=3 yields [0, 1, 1/2, 1/2] for n = [-1, 0, 1, 2], so 3 is not in the sequence. - Dug Eichelberger (dug(AT)mit.edu), May 14 2001
n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible. - Robert G. Wilson v, Jun 22 2002
Records for Euler totient function phi.
Together with 0, n such that (n+1) divides (n!+1). - Benoit Cloitre, Aug 20 2002; corrected by Charles R Greathouse IV, Apr 20 2010
n such that phi(n^2) = phi(n^2 + n). - Jon Perry, Feb 19 2004
Numbers having only the trivial perfect partition consisting of a(n) 1's. - Lekraj Beedassy, Jul 23 2006
Numbers n such that the sequence {binomial coefficient C(k,n), k >= n } contains exactly one prime. - Artur Jasinski, Dec 02 2007
Record values of A143201: a(n) = A143201(A001747(n+1)) for n > 1. - Reinhard Zumkeller, Aug 12 2008
From Reinhard Zumkeller, Jul 10 2009: (Start)
The first N terms can be generated by the following sieving process:
  start with {1, 2, 3, 4, ..., N - 1, N};
  for i := 1 until SQRT(N) do
  (if (i is not striked out) then
  (for j := 2 * i + 1 step i + 1 until N do
  (strike j from the list)));
  remaining numbers = {a(n): a(n) <= N}. (End)
a(n) = partial sums of A075526(n-1) = Sum_{1..n} A075526(n-1) = Sum_{1..n} (A008578(n+1) - A008578(n)) = Sum_{1..n} (A158611(n+2) - A158611(n+1)) for n >= 1. - Jaroslav Krizek, Aug 04 2009
A006093 U A072668 = A000027. - Juri-Stepan Gerasimov, Oct 22 2009
A171400(a(n)) = 1 for n <> 2: subsequence of A171401, except for a(2) = 2. - Reinhard Zumkeller, Dec 08 2009
Numerator of (1 - 1/prime(n)). - Juri-Stepan Gerasimov, Jun 05 2010
Numbers n such that A002322(n+1) = n. This statement is stronger than repeating the property of the entries in A002322, because it also says in reciprocity that this sequence here contains no numbers beyond the Carmichael numbers with that property. - Michel Lagneau, Dec 12 2010
a(n) = A192134(A095874(A000040(n))); subsequence of A192133. - Reinhard Zumkeller, Jun 26 2011
prime(a(n)) + prime(k) < prime(a(k) + k) for at least one k <= a(n): A212210(a(n),k) < 0. - Reinhard Zumkeller, May 05 2012
Except for the first term, numbers n such that the sum of first n natural numbers does not divide the product of first n natural numbers; that is, n*(n + 1)/2 does not divide n!. - Jayanta Basu, Apr 24 2013
BigOmega(a(n)) equals BigOmega(a(n)*(a(n) + 1)/2), where BigOmega = A001222. Rationale: BigOmega of the product on the right hand side factorizes as BigOmega(a/2) + Bigomega(a+1) = BigOmega(a/2) + 1 because a/2 and a + 1 are coprime, because BigOmega is additive, and because a + 1 is prime. Furthermore Bigomega(a/2) = Bigomega(a) - 1 because essentially all 'a' are even. - Irina Gerasimova, Jun 06 2013
Record values of A060681. - Omar E. Pol, Oct 26 2013
Deficiency of n-th prime. - Omar E. Pol, Jan 30 2014
Conjecture: All the sums Sum_{k=s..t} 1/a(k) with 1 <= s <= t are pairwise distinct. In general, for any integers d >= -1 and m > 0, if Sum_{k=i..j} 1/(prime(k)+d)^m = Sum_{k=s..t} 1/(prime(k)+d)^m with 0 < i <= j and 0 < s <= t then we must have (i,j) = (s,t), unless d = m = 1 and {(i,j),(s,t)} = {(4,4),(8,10)} or {(4,7),(5,10)}. (Note that 1/(prime(8)+1)+1/(prime(9)+1)+1/(prime(10)+1) = 1/(prime(4)+1) and Sum_{k=5..10} 1/(prime(k)+1) = 1/(prime(4)+1) + Sum_{k=5..7} 1/(prime(k)+1).) - Zhi-Wei Sun, Sep 09 2015
Numbers n such that (prime(i)^n + n) is divisible by (n+1), for all i >= 1, except when prime(i) = n+1. - Richard R. Forberg, Aug 11 2016
a(n) is the period of Fubini numbers (A000670) over the n-th prime. - Federico Provvedi, Nov 28 2020

Archimedeans Problems Drive, Eureka, 40 (1979), 28.
Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
M. Gardner, The Colossal Book of Mathematics, pp. 31, W. W. Norton & Co., NY, 2001.
M. Gardner, Mathematical Circus, pp. 251-2, Alfred A. Knopf, NY, 1979.
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..10000
Thomas F. Bloom, Unit fractions with shifted prime denominators, arXiv:2305.02689 [math.NT], 2023.
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Harvey Dubner, Generalized Fermat primes, J. Recreational Math. 18.4 (1985-1986), 279. (Annotated scanned copy)
Armel Mercier, Problem E 3065, American Mathematical Monthly, 1984, p. 649.
Armel Mercier, S. K. Rangarajan, J. C. Binz and Dan Marcus, Problem E 3065, American Mathematical Monthly, No. 4, 1987, pp. 378.
Poo-Sung Park, Additive uniqueness of PRIMES-1 for multiplicative functions, arXiv:1708.03037 [math.NT], 2017.
J. R. Rickard and J. J. Hitchcock, Problem Drive 4, Archimedeans Problems Drive, Eureka, 40 (1979), 28-29, 40. (Annotated scanned copy)
Index entries for sequences generated by sieves

a(n) = K(n, 1) and A034693(K(n, 1)) = 1 for all n. The subscript n refers to this sequence and K(n, 1) is the index in A034693. - Labos Elemer
Cf. A000040, A034694. Different from A075728.
Complement of A072668 (composite numbers minus 1), A072670(a(n))=0.
Essentially the same as A039915.
Cf. A084920, A006093, A050997, A008864, A060800, A131991, A131992, A131993.
Cf. A101301 (partial sums), A005867 (partial products).
Column 1 of the following arrays/triangles: A087738, A249741, A352707, A378979, A379010.
The last diagonal of A162619, and of A174996, the first diagonal in A131424.
Row lengths of irregular triangles A086145, A124223, A212157.

Filtered([1..280],IsPrime)-1; # Muniru A Asiru, Nov 25 2018

a006093 = (subtract 1) . a000040  -- Reinhard Zumkeller, Mar 06 2012

[NthPrime(n)-1: n in [1..100]]; // Vincenzo Librandi, Nov 17 2015

for n from 2 to 271 do if (n! mod n^2 = n*(n-1) and (n<>4) then print(n-1) fi od; # Gary Detlefs, Sep 10 2010
# alternative
A006093 := proc(n)
    ithprime(n)-1 ;
end proc:
seq(A006093(n),n=1..100) ; # R. J. Mathar, Feb 06 2019

Table[Prime[n] - 1, {n, 1, 30}] (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)
a[ n_] := If[ n < 1, 0, -1 + Prime @ n] (* Michael Somos, Jul 17 2011 *)
Prime[Range[60]] - 1 (* Alonso del Arte, Oct 26 2013 *)

isA006093(n) = isprime(n+1) \\ Michael B. Porter, Apr 09 2010

A006093(n) = prime(n)-1 \\ Michael B. Porter, Apr 09 2010

\\ Sieve as described in Rainer Rosenthal's comment.
m=270;s=vector(m);for(i=1,m,for(j=i,m,k=i*j+i+j;if(k<=m,s[k]=1)));for(k=1,m,if(s[k]==0,print1(k,", "))); \\ Hugo Pfoertner, May 14 2019

from sympy import prime
for n in range(1,100): print(prime(n)-1, end=', ') # Stefano Spezia, Nov 30 2018

a(n) = (p-1)! mod p where p is the n-th prime, by Wilson's theorem. - Jonathan Sondow, Jul 13 2010
a(n) = A000010(prime(n)) = A000010(A006005(n)). - Antti Karttunen, Dec 16 2012
a(n) = A005867(n+1)/A005867(n). - Eric Desbiaux, May 07 2013
a(n) = A000040(n) - 1. - Omar E. Pol, Oct 26 2013
a(n) = A033879(A000040(n)). - Omar E. Pol, Jan 30 2014

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010
Obfuscating comments removed by Joerg Arndt, Mar 11 2010
Edited by Charles R Greathouse IV, Apr 20 2010

A033845 Numbers k of the form 2^i*3^j, where i and j >= 1.

Original entry on oeis.org

6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162, 192, 216, 288, 324, 384, 432, 486, 576, 648, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8748, 9216, 10368, 11664

Offset: 1

Jeff Burch

This sequence is easily confused with A003586, which gives numbers of the form 2^i*3^j with i, j >= 0, and is one-sixth of the present sequence. . Don't simply say "numbers of the form 2^i*3^j", but specify which sequence you mean. - N. J. A. Sloane, May 26 2024
Solutions to phi(n)=n/3 [See J-M. de Koninck & A. Mercier, problème 733].
Numbers n such that Sum_{d prime divisor of n} 1/d = 5/6. - Benoit Cloitre, Apr 13 2002
Also n such that Sum_{d|n} mu(d)^2/d = 2. - Benoit Cloitre, Apr 15 2002
Complement of A006899 with respect to A003586. - Reinhard Zumkeller, Sep 25 2008
In the sieve of Eratosthenes, if one crosses numbers off multiple times, these numbers are crossed off twice, first for 2 and then for 3. - Alonso del Arte, Aug 22 2011
Subsequence of A051037. - Reinhard Zumkeller, Sep 13 2011
Numbers n such that Sum_{d|n} A008683(d)*A000041(d) = 7. - Carl Najafi, Oct 19 2011
Numbers n such that Sum_{d|n} A008683(d)*A000700(d) = 2. - Carl Najafi, Oct 20 2011
Solutions to the equation A001615(x) = 2x. - Enrique Pérez Herrero, Jan 02 2012
So these numbers are called Psi-perfect numbers [see J-M. de Koninck & A. Mercier, problème 654]. - Bernard Schott, Nov 20 2020

J-M. de Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, 2004, Problème 733, page 94.
J-M. de Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, 2004, Problème 654, page 85.

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

Subsequence of A000423, A003586, A051037, A256617.
Cf. A001615, A006899, A086410, A086411, A008683, A143201.
Cf. A191475, A191476.

import Data.Set (singleton, deleteFindMin, insert)
a033845 n = a033845_list !! (n-1)
a033845_list = f (singleton (2*3)) where
   f s = m : f (insert (2*m) $ insert (3*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

mx = 12000; Sort@ Flatten@ Table[2^i*3^j, {i, Log[2, mx]}, {j, Log[3, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

list(lim)=my(v=List(), N); for(n=0, log(lim\2)\log(3), N=6*3^n; while(N<=lim, listput(v, N); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 02 2012

from sympy import integer_log
def A033845(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))
    return 6*bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

# faster for initial segment of sequence
import heapq
from itertools import islice
def A033845gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield 6*v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A033845gen(), 50))) # Michael S. Branicky, Sep 18 2024

Six times the 3-smooth numbers (A003586). - Ralf Stephan, Apr 16 2004
A086411(a(n)) - A086410(a(n)) = 1. - Reinhard Zumkeller, Sep 25 2008
A143201(a(n)) = 2. - Reinhard Zumkeller, Sep 13 2011
a(n) = 2^A191475(n) * 3^A191476(n). - Zak Seidov, Nov 01 2013
Sum_{n>=1} 1/a(n) = 1/2. - Amiram Eldar, Oct 13 2020

Edited by N. J. A. Sloane, Jan 31 2010 and May 26 2024.

A001747 2 together with primes multiplied by 2.

Original entry on oeis.org

2, 4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502

Offset: 1

N. J. A. Sloane

When supplemented with 8, may be considered the "even primes", since these are the even numbers n = 2k which are divisible just by 1, 2, k and 2k. - Louis Zuckerman (louis(AT)trapezoid.com), Sep 12 2000
Sequence gives solutions of sigma(n) - phi(n) = n + tau(n) where tau(n) is the number of divisors of n.
Numbers n such that sigma(n) = 3*(n - phi(n)).
Except for 2, orders of non-cyclic groups k (in A060679(n)) such that x^k==1 (mod k) has only 1 solution 2<=x<=k. - Benoit Cloitre, May 10 2002
Numbers n such that A092673(n) = 2. - Jon Perry, Mar 02 2004
Except for initial terms, this sequence = A073582 = A074845 = A077017. Starting with the term 10, they are identical. - Robert G. Wilson v, Jun 15 2004
Together with 8 and 16, even numbers n such that n^2 does not divide (n/2)!. - Arkadiusz Wesolowski, Jul 16 2011
Twice noncomposite numbers. - Omar E. Pol, Jan 30 2012

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

Cf. A060679, A009530, A098764.

Equals {2} UNION {A100484}.

Concatenation([2], List([1..60], n-> 2*Primes[n])); # G. C. Greubel, May 18 2019

[2] cat [2*NthPrime(n): n in [1..60]]; // G. C. Greubel, May 18 2019

Join[{2},2*Prime[Range[60]]] (* Harvey P. Dale, Jul 23 2013 *)

print1(2);forprime(p=2,97,print1(", "2*p)) \\ Charles R Greathouse IV, Jan 31 2012

[2]+[2*nth_prime(n) for n in (1..60)] # G. C. Greubel, May 18 2019

a(n) = A001043(n) - A001223(n+1), except for initial term.
a(n) = A116366(n-2,n-2) for n>2. - Reinhard Zumkeller, Feb 06 2006
A006093(n) = A143201(a(n+1)) for n>1. - Reinhard Zumkeller, Aug 12 2008
a(n) = 2*A008578(n). - Omar E. Pol, Jan 30 2012, and Reinhard Zumkeller, Feb 16 2012

A143207 Numbers with distinct prime factors 2, 3, and 5.

Original entry on oeis.org

30, 60, 90, 120, 150, 180, 240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810, 900, 960, 1080, 1200, 1350, 1440, 1500, 1620, 1800, 1920, 2160, 2250, 2400, 2430, 2700, 2880, 3000, 3240, 3600, 3750, 3840, 4050, 4320, 4500, 4800, 4860

Offset: 1

Reinhard Zumkeller, Aug 12 2008

Numbers of the form 2^i * 3^j * 5^k with i, j, k > 0. - Reinhard Zumkeller, Sep 13 2011

Integers k such that phi(k)/k = 4/15. - Artur Jasinski, Nov 07 2008

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A069819.

Subsequence of A143204 and of A051037.

import Data.Set (singleton, deleteFindMin, insert)
a143207 n = a143207_list !! (n-1)
a143207_list = f (singleton (2*3*5)) where
   f s = m : f (insert (2*m) $ insert (3*m) $ insert (5*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

[n: n in [1..5000] | PrimeDivisors(n) eq [2,3,5]]; // Bruno Berselli, Sep 14 2015

a = {}; Do[If[EulerPhi[x]/x == 4/15, AppendTo[a, x]], {x, 1, 11664}]; a (* Artur Jasinski, Nov 07 2008 *)
n = 10^4; Table[2^i*3^j*5^k, {i, 1, Log[2, n]}, {j, 1, Log[3, n/2^i]}, {k, 1, Log[5, n/(2^i*3^j)]}] // Flatten // Sort (* Amiram Eldar, Sep 24 2020 *)

list(lim)=my(v=List(),s,t); for(i=1,logint(lim\6,5), t=5^i; for(j=1,logint(lim\t\2,3), s=t*3^j; while((s<<=1)<=lim, listput(v,s)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015

is(n) = if(n%30,return(0)); my(f=factor(n,6)[,1]); f[#f]<6 \\ David A. Corneth, Sep 22 2020

from sympy import integer_log
def A143207(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 i in range(integer_log(x,5)[0]+1):
            for j in range(integer_log(m:=x//5**i,3)[0]+1):
                c -= (m//3**j).bit_length()
        return c
    return bisection(f,n,n)*30 # Chai Wah Wu, Sep 16 2024

A001221(a(n)) = 3; A020639(a(n)) = 2; A006530(a(n)) = 5; A143201(a(n)) = 6.
a(n) = 30*A051037(n); A007947(a(n)) = A010869(n). - Reinhard Zumkeller, Sep 13 2011
a(n) ~ sqrt(30) * exp((6*log(2)*log(3)*log(5)*n)^(1/3)). - Vaclav Kotesovec, Sep 22 2020
Sum_{n>=1} 1/a(n) = 1/8. - Amiram Eldar, Sep 24 2020

New name from Charles R Greathouse IV, Sep 14 2015

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

cp's OEIS Frontend

A000961 Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1).

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A006093 a(n) = prime(n) - 1.

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A033845 Numbers k of the form 2^i*3^j, where i and j >= 1.

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A001747 2 together with primes multiplied by 2.

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GAP

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A143207 Numbers with distinct prime factors 2, 3, and 5.

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