A156542 -id:A156542 - cp's OEIS frontend

A001221 Number of distinct primes dividing n (also called omega(n)).

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 3, 2

Offset: 1

N. J. A. Sloane

From Peter C. Heinig (algorithms(AT)gmx.de), Mar 08 2008: (Start)
This is also the number of maximal ideals of the ring (Z/nZ,+,*). Since every finite integral domain must be a field, every prime ideal of Z/nZ is a maximal ideal and since in general each maximal ideal is prime, there are just as many prime ideals as maximal ones in Z/nZ, so the sequence gives the number of prime ideals of Z/nZ as well.
The reason why this number is given by the sequence is that the ideals of Z/nZ are precisely the subgroups of (Z/nZ,+). Hence for an ideal to be maximal it has form a maximal subgroup of (Z/nZ,+) and this is equivalent to having prime index in (Z/nZ) and this is equivalent to being generated by a single prime divisor of n.
Finally, all the groups arising in this way have different orders, hence are different, so the number of maximal ideals equals the number of distinct primes dividing n. (End)
Equals double inverse Mobius transform of A143519, where A051731 = the inverse Mobius transform. - Gary W. Adamson, Aug 22 2008
a(n) is the number of unitary prime power divisors of n (not including 1). - Jaroslav Krizek, May 04 2009 [corrected by Ilya Gutkovskiy, Oct 09 2019]
Sum_{d|n} 2^(-A001221(d) - A001222(n/d)) = Sum_{d|n} 2^(-A001222(d) - A001221(n/d)) = 1 (see Dressler and van de Lune link). - Michel Marcus, Dec 18 2012
Up to 2*3*5*7*11*13*17*19*23*29  - 1 = 6469693230 - 1, also the decimal expansion of the constant 0.01111211... = Sum_{k>=0} 1/(10 ^ A000040(k) - 1) (see A073668). - Eric Desbiaux, Jan 20 2014
The average order of a(n): Sum_{k=1..n} a(k) ~ Sum_{k=1..n} log log k. - Daniel Forgues, Aug 13-16 2015
From Peter Luschny, Jul 13 2023: (Start)
We can use A001221 and A001222 to classify the positive integers as follows.
   A001222(n) = A001221(n) = 0 singles out {1}.
Restricting to n > 1:
   A001222(n)^A001221(n) = 1: A000040, prime numbers.
   A001221(n)^A001222(n) = 1: A246655, prime powers.
   A001222(n)^A001221(n) > 1: A002808, the composite numbers.
   A001221(n)^A001222(n) > 1: A024619, complement of A246655.
   n^(A001222(n) - A001221(n)) = 1: A144338, products of distinct primes. (End)
Inverse Möbius transform of the characteristic function of primes (A010051). - Wesley Ivan Hurt, Jun 22 2024
Dirichlet convolution of A010051(n) and 1. - Wesley Ivan Hurt, Jul 15 2025

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 48-57.
J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) (British Association Mathematical Tables Vol.V), Burlington House/Cambridge University Press London 1935.
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).

N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 from NJAS)
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].
Henry Bottomley, Prime factors calculator
J. Brennen, Prime Factoring Applet
J. Britton, Prime Factorization Machine
A. Dendane, Prime Factors Calculator
Robert E. Dressler and Jan van de Lune, Some remarks concerning the number theoretic functions omega and Omega, Proc. Amer. Math. Soc. 41 (1973), 403-406
J. Flament, Decomposition d'un nombre en facteurs premiers
G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92. Also Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI (2000): 262-275.
A. Hodges, Java Applet for Factorisation
M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
S. O. S. Math, Prime factorization of the first 1000 integers
K. Matthews, Factorization and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)
J. Moyer, "Prime Factors of Integers" server for numbers up to 10^36
Primefan, The First 2500 Integers,Factored
Primefan, Factorer
F. Richman, Factoring into Primes
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Moebius Transform
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Wikipedia, Prime factor
Wikipedia, Table of prime factors
G. Xiao, WIMS server, Factoris
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

Cf. A001222 (primes counted with multiplicity), A046660, A285577, A346617. Partial sums give A013939.
Cf. also A125666, A069010, A087624, A091202, A091221, A143519, A144494, A158210, A156542, A156552, A000010, A008683.
Sum of the k-th powers of the primes dividing n for k=0..10: this sequence (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), A351196 (k=8), A351197 (k=9), A351198 (k=10).
Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k=0..10: this sequence (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

import Math.NumberTheory.Primes.Factorisation (factorise)
a001221 = length . snd . unzip . factorise
-- Reinhard Zumkeller, Nov 28 2015

using Nemo
function NumberOfPrimeFactors(n; distinct=true)
    distinct && return length(factor(ZZ(n)))
    sum(e for (p, e) in factor(ZZ(n)); init=0)
end
println([NumberOfPrimeFactors(n) for n in 1:60]) # Peter Luschny, Jan 02 2024

[#PrimeDivisors(n): n in [1..120]]; // Bruno Berselli, Oct 15 2021

A001221 := proc(n) local t1, i; if n = 1 then return 0 else t1 := 0; for i to n do if n mod ithprime(i) = 0 then t1 := t1 + 1 end if end do end if; t1 end proc;
A001221 := proc(n) nops(numtheory[factorset](n)) end proc: # Emeric Deutsch
omega := n -> NumberTheory:-NumberOfPrimeFactors(n, 'distinct'): # Peter Luschny, Jun 15 2025

Array[ Length[ FactorInteger[ # ] ]&, 100 ]
PrimeNu[Range[120]]  (* Harvey P. Dale, Apr 26 2011 *)

func(nops(numlib::primedivisors(n)), n):

numlib::omega(n)$ n=1..110 // Zerinvary Lajos, May 13 2008

PARI
```
a(n)=omega(n)
```

from sympy.ntheory import primefactors
print([len(primefactors(n)) for n in range(1, 1001)])  # Indranil Ghosh, Mar 19 2017

def A001221(n): return sum(1 for p in divisors(n) if is_prime(p))
[A001221(n) for n in (1..80)] # Peter Luschny, Feb 01 2012

[sloane.A001221(n) for n in (1..111)] # Giuseppe Coppoletta, Jan 19 2015

[gp.omega(n) for n in range(1,101)] # G. C. Greubel, Jul 13 2024

G.f.: Sum_{k>=1} x^prime(k)/(1-x^prime(k)). - Benoit Cloitre, Apr 21 2003; corrected by Franklin T. Adams-Watters, Sep 01 2009
Dirichlet generating function: zeta(s)*primezeta(s). - Franklin T. Adams-Watters, Sep 11 2005
Additive with a(p^e) = 1.
a(1) = 0, a(p) = 1, a(pq) = 2, a(pq...z) = k, a(p^k) = 1, where p, q, ..., z are k distinct primes and k natural numbers. - Jaroslav Krizek, May 04 2009
a(n) = log_2(Sum_{d|n} mu(d)^2). - Enrique Pérez Herrero, Jul 09 2012
a(A002110(n)) = n, i.e., a(prime(n)#) = n. - Jean-Marc Rebert, Jul 23 2015
a(n) = A091221(A091202(n)) = A069010(A156552(n)). - Antti Karttunen, circa 2004 & Mar 06 2017
L.g.f.: -log(Product_{k>=1} (1 - x^prime(k))^(1/prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = log_2(Sum_{k=1..n} mu(gcd(n,k))^2/phi(n/gcd(n,k))) = log_2(Sum_{k=1..n} mu(n/gcd(n,k))^2/phi(n/gcd(n,k))), where phi = A000010 and mu = A008683. - Richard L. Ollerton, May 13 2021
Sum_{k=1..n} 2^(-a(gcd(n,k)) - A001222(n/gcd(n,k)))/phi(n/gcd(n,k)) = Sum_{k=1..n} 2^(-A001222(gcd(n,k)) - a(n/gcd(n,k)))/phi(n/gcd(n,k)) = 1, where phi = A000010. - Richard L. Ollerton, May 13 2021
a(n) = A005089(n) + A005091(n) + A059841(n) = A005088(n) +A005090(n) +A079978(n). - R. J. Mathar, Jul 22 2021
From Wesley Ivan Hurt, Jun 22 2024: (Start)
a(n) = Sum_{p|n, p prime} 1.
a(n) = Sum_{d|n} c(d), where c = A010051. (End)

A005384 Sophie Germain primes p: 2p+1 is also prime.

Original entry on oeis.org

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559

Offset: 1

N. J. A. Sloane

Then 2p+1 is called a safe prime: see A005385.
Primes p such that the equation phi(x) = 2p has solutions, where phi is the totient function. See A087634 for another such collection of primes. - T. D. Noe, Oct 24 2003
Subsequence of A117360. - Reinhard Zumkeller, Mar 10 2006
Let q = 2n+1. For these n (and q), the difference of two cyclotomic polynomials can be written as a cyclotomic polynomial in x^2: Phi(q,x) - Phi(2q,x) = 2x Phi(n,x^2). - T. D. Noe, Jan 04 2008
A Sophie Germain prime p is 2, 3 or of the form 6k-1, k >= 1, i.e., p = 5 (mod 6). A prime p of the form 6k+1, k >= 1, i.e., p = 1 (mod 6), cannot be a Sophie Germain prime since 2p+1 is divisible by 3. - Daniel Forgues, Jul 31 2009
Also solutions to the equation: floor(4/A000005(2*n^2+n)) = 1. - Enrique Pérez Herrero, May 03 2012
In the spirit of the conjecture related to A217788, we conjecture that for any integers n >= m > 0 there are infinitely many integers b > a(n) such that the number Sum_{k=m..n} a(k)*b^(n-k) is prime. - Zhi-Wei Sun, Mar 26 2013
If k is the product of a Sophie Germain prime p and its corresponding safe prime 2p+1, then a(n) = (k-phi(k))/3, where phi is Euler's totient function. - Wesley Ivan Hurt, Oct 03 2013
Giovanni Resta found the first Sophie Germain prime which is also a Brazilian number (A125134), 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73. - _Bernard Schott, Mar 07 2019
For all Sophie Germain primes p >= 5, 2*p + 1 = min(A, B) where A is the smallest prime factor of 2^p - 1 and B the smallest prime factor of (2^p + 1) / 3. - Alain Rocchelli, Feb 01 2023
Consider a pair of numbers (p, 2*p+1), with p >= 3. Then p is a Sophie Germain prime iff (p-1)!^2 + 6*p == 1 (mod p*(2*p+1)). - Davide Rotondo, May 02 2024

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.
A. Peretti, The quantity of Sophie Germain primes less than x, Bull. Number Theory Related Topics, Vol. 11, No. 1-3 (1987), pp. 81-92.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 76, 227-230.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 83.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

J. S. Cheema, Table of n, a(n) for n = 1..100000. [This replaces an earlier b-file computed by T. D. Noe]
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].
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
P. Bruillard, S.-H. Ng, E. Rowell and Z. Wang, On modular categories, arXiv preprint arXiv:1310.7050 [math.QA], 2013.
Chris K. Caldwell, The Prime Glossary, Sophie Germain Prime
Andrea Del Centina, Letters of Sophie Germain preserved in Florence Historia Mathematica, Vol. 32 (2005), pp. 60-75.
Harvey Dubner, Large Sophie Germain Primes, Math. Comp., Vol. 65, No. 213 (1996), pp. 393-396.
Luis H. Gallardo and Olivier Rahavandrainy, There are finitely many even perfect polynomials over F_p with p+1 irreducible divisors, Acta Mathematica Universitatis Comenianae, Vol. 83, No. 2, 2016, 261-275.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Reikku Kulon, Sublinear arbitrary precision generator of Sophie Germain and safe primes in C (public domain).
Henri Lifchitz, A new and simpler primality test for Sophie-Germain numbers(q=2*p+1).
Victor Meally, Letter to N. J. A. Sloane, no date.
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.
Frans Oort, Prime numbers, 2013.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 31, 114.
Larry Riddle, Sophie Germain and Fermat's Last Theorem, Agnes Scott College, Math. Dept., Jul, 1999.
Carlos Rivera, Puzzle 1122. OEIS A005385, The Prime Puzzles & Problems Connection.
Carlos Rivera, Puzzle 1140. Test for Sophie Germain primes, The Prime Puzzles & Problems Connection.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Anthony G. Shannon, Peter J.-S. Shiue, Tian-Xiao He, and Christopher Saito, On the special cases of Carmichael's totient conjecture, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 504-534.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS, Vol. 13 (2013), Article A65.
Agoh Takashi, On Sophie Germain primes, Number theory (Liptovský Ján, 1999), Tatra Mt. Math. Publ., Vol. 20 (2000), pp. 65-73.
Terence Tao, Obstructions to uniformity and arithmetic patterns in the primes, arXiv:math/0505402 [math.NT], 2005.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Vmoraru, PlanetMath.org, Sophie Germain prime.
Samuel S. Wagstaff, Jr., Sum of Reciprocals of Germain Primes, Journal of Integer Sequences, Vol. 24, No. 2 (2021), Article 21.9.5.
Eric Weisstein's World of Mathematics, Sophie Germain Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Wikipedia, Sophie Germain prime.
Samuel Yates, Sophie Germain primes, in "The mathematical heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, 1991, pp. 882-886.

Cf. A005385, A057331, A005602, A087634.
Cf. also A000355, A156541, A156542, A156592, A161896, A156660, A156874, A092816, A023212, A007528 (primes of the form 6k-1).
For primes p that remains prime through k iterations of the function f(x) = 2x + 1: this sequence (k=1), A007700 (k=2), A023272 (k=3), A023302 (k=4), A023330 (k=5), A278932 (k=6), A138025 (k=7), A138030 (k=8).

Filtered([1..1600],p->IsPrime(p) and IsPrime(2*p+1)); # Muniru A Asiru, Mar 06 2019

[ p: p in PrimesUpTo(1560) | IsPrime(2*p+1) ]; // Klaus Brockhaus, Jan 01 2009

A:={}: for n from 1 to 246 do if isprime(2*ithprime(n)+1) then A:=A union {ithprime(n)} fi od: A:=A; # Emeric Deutsch, Dec 09 2004

Select[Prime[Range[1000]],PrimeQ[2#+1]&]
lst = {}; Do[If[PrimeQ[n + 1] && PrimeOmega[n] == 2, AppendTo[lst, n/2]], {n, 2, 10^4}]; lst (* Hilko Koning, Aug 17 2021 *)

select(p->isprime(2*p+1), primes(1000)) \\ In old PARI versions <= 2.4.2, use select(primes(1000), p->isprime(2*p+1)).

forprime(n=2, 10^3, if(ispseudoprime(2*n+1), print1(n, ", "))) \\ Felix Fröhlich, Jun 15 2014

is_A005384=(p->isprime(2*p+1)&&isprime(p));
  {A005384_vec(N=100,p=1)=vector(N,i,until(isprime(2*p+1),p=nextprime(p+1));p)} \\ M. F. Hasler, Mar 03 2020

from sympy import isprime, nextprime
def ok(p): return isprime(2*p+1)
def aupto(limit): # only test primes
  alst, p = [], 2
  while p <= limit:
    if ok(p): alst.append(p)
    p = nextprime(p)
  return alst
print(aupto(1559)) # Michael S. Branicky, Feb 03 2021

a(n) mod 10 <> 7. - Reinhard Zumkeller, Feb 12 2009
A156660(a(n)) = 1; A156874 gives numbers of Sophie Germain primes <= n. - Reinhard Zumkeller, Feb 18 2009
tau(4*a(n) + 2) = tau(4*a(n)) - 2, for n > 1. - Arkadiusz Wesolowski, Aug 25 2012
eulerphi(4*a(n) + 2) = eulerphi(4*a(n)) + 2, for n > 1. - Arkadiusz Wesolowski, Aug 26 2012
A005097 INTERSECT A000040. - R. J. Mathar, Mar 23 2017
Sum_{n>=1} 1/a(n) is in the interval (1.533944198, 1.8026367) (Wagstaff, 2021). - Amiram Eldar, Nov 04 2021
a(n) >> n log^2 n. - Charles R Greathouse IV, Jul 25 2024

A053176 Primes p such that 2p+1 is composite.

Original entry on oeis.org

7, 13, 17, 19, 31, 37, 43, 47, 59, 61, 67, 71, 73, 79, 97, 101, 103, 107, 109, 127, 137, 139, 149, 151, 157, 163, 167, 181, 193, 197, 199, 211, 223, 227, 229, 241, 257, 263, 269, 271, 277, 283, 307, 311, 313, 317, 331, 337, 347, 349, 353, 367, 373, 379, 383

Offset: 1

Enoch Haga, Feb 29 2000

Primes not in A005384 = non-Sophie Germain primes.
Also, numbers n such that odd part of A005277(n) is prime. Proof by John Renze, Sep 30 2004
Sequence gives primes p such that B(2p) has denominator 6, where B(2n) are the Bernoulli numbers. - Benoit Cloitre, Feb 06 2002
Sequence gives all n such that the equation phi(x)=2n has no solution. - Benoit Cloitre, Apr 07 2002
A010051(a(n))*(1-A156660(a(n))) = 1; subsequence of A138887. - Reinhard Zumkeller, Feb 18 2009
Mersenne prime exponents > 3 must be in the union of this sequence and (A002144). - Roderick MacPhee, Jan 12 2017

			17 is a term because 2*17 + 1 = 35 is composite.

T. D. Noe, Table of n, a(n) for n = 1..10000
Lambert A'Campo, Every 7-Dimensional Abelian Variety over the p-adic Numbers has a Reducible L-adic Galois Representation, arXiv:2006.06737 [math.NT], 2020.

Cf. A005384, A005385, A059452, A059453, A059454, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156543, A156542.

[p: p in PrimesUpTo(12200) | not IsPrime(2*p+1)]; // Vincenzo Librandi, Jun 18 2015

Select[Prime[Range[1000]], ! PrimeQ[2 # + 1] &] (* Vincenzo Librandi, Jun 18 2015 *)

list(lim)=select(p->!isprime(2*p+1),primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011

a(n) ~ n log n. - Charles R Greathouse IV, Feb 20 2012

A156543 Multiplicative closure of primes that are not Sophie Germain primes (A053176).

Original entry on oeis.org

1, 7, 13, 17, 19, 31, 37, 43, 47, 49, 59, 61, 67, 71, 73, 79, 91, 97, 101, 103, 107, 109, 119, 127, 133, 137, 139, 149, 151, 157, 163, 167, 169, 181, 193, 197, 199, 211, 217, 221, 223, 227, 229, 241, 247, 257, 259, 263, 269, 271, 277, 283, 289, 301, 307, 311

Offset: 1

Reinhard Zumkeller, Feb 09 2009

nonn

a(n) mod 6 = 1 or = 5; subsequence of A007310;
A156542(a(n)) = 0;
A045979 is a subsequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A007310, A045979, A053176, A156541, A156542.

A156541 Multiplicative closure of Sophie Germain primes (A005384).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 23, 24, 25, 27, 29, 30, 32, 33, 36, 40, 41, 44, 45, 46, 48, 50, 53, 54, 55, 58, 60, 64, 66, 69, 72, 75, 80, 81, 82, 83, 87, 88, 89, 90, 92, 96, 99, 100, 106, 108, 110, 113, 115, 116, 120, 121, 123, 125, 128, 131, 132

Offset: 1

Reinhard Zumkeller, Feb 10 2009

nonn

A156542(a(n)) = A001221(a(n));

Subsequence of A130176.

R. Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A156541, A156543.

Select[Range@132, And @@ PrimeQ[FactorInteger[#][[All, 1]]*2 + 1] &] (* Ivan Neretin, Aug 30 2015 *)

A178146 a(n) is the number of distinct prime factors <= 5 of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2

Offset: 1

Vladimir Shevelev, May 21 2010

The sequence is periodic with period {0 1 1 1 1 2 0 1 1 2 0 2 0 1 2 1 0 2 0 2 1 1 0 2 1 1 1 1 0 3} of length 30. There are 26 coincidences on the interval [1,30] with A156542.

Vladimir Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1,0,1,2,2,1).

Cf. A000005, A001221, A156542, A171182, A178143, A178144, A178142.
Cf. A059841, A079978, A079998, A355582, A356006.
Number of distinct prime factors <= p: A171182 (p=3), this sequence (p=5), A210679 (p=7).

Rest@ CoefficientList[Series[-x^2*(3*x^6 + 6*x^5 + 7*x^4 + 6*x^3 + 5*x^2 + 3*x + 1)/((x - 1)*(x + 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* G. C. Greubel, May 16 2017 *)
LinearRecurrence[{-2,-2,-1,0,1,2,2,1},{0,1,1,1,1,2,0,1},120] (* Harvey P. Dale, Sep 29 2021 *)
a[n_] := PrimeNu[GCD[n, 30]]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

my(x='x+O('x^50)); concat([0], Vec(-x^2*(3*x^6+6*x^5+7*x^4+6*x^3+5*x^2+3*x+1)/((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)))) \\ G. C. Greubel, May 16 2017

a(n) = omega(gcd(n, 30)); \\ Amiram Eldar, Sep 16 2023

a(n) = a(n-2) + a(n-3) - a(n-7) - a(n-8) + a(n-10), a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 2, a(7) = 0, a(8) = 1, a(9) = 1, a(10) = 2.
G.f.: -x^2*(3*x^6+6*x^5+7*x^4+6*x^3+5*x^2+3*x+1) / ((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)). - Colin Barker, Mar 13 2013
From Amiram Eldar, Sep 16 2023: (Start)
Additive with a(p^e) = 1 if p <= 5, and 0 otherwise.
a(n) = A059841(n) + A079978(n) + A079998(n).
a(n) = A001221(gcd(n, 30)).
a(n) = A001221(A355582(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 31/30. (End)

Name edited by Amiram Eldar, Sep 16 2023

cp's OEIS Frontend

A001221 Number of distinct primes dividing n (also called omega(n)).

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A005384 Sophie Germain primes p: 2p+1 is also prime.

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A053176 Primes p such that 2p+1 is composite.

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A156543 Multiplicative closure of primes that are not Sophie Germain primes (A053176).

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A156541 Multiplicative closure of Sophie Germain primes (A005384).

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A178146 a(n) is the number of distinct prime factors <= 5 of n.

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