A229109 -id:A229109 - cp's OEIS frontend

A064800 n plus the number of its prime factors: a(n) = n + A001222(n).

1, 3, 4, 6, 6, 8, 8, 11, 11, 12, 12, 15, 14, 16, 17, 20, 18, 21, 20, 23, 23, 24, 24, 28, 27, 28, 30, 31, 30, 33, 32, 37, 35, 36, 37, 40, 38, 40, 41, 44, 42, 45, 44, 47, 48, 48, 48, 53, 51, 53, 53, 55, 54, 58, 57, 60, 59, 60, 60, 64, 62, 64, 66, 70, 67, 69, 68, 71, 71, 73, 72

Offset: 1

Reinhard Zumkeller, Oct 21 2001

nonn

Prime factors counted with multiplicity. - Harvey P. Dale, Aug 19 2015

			a(42) = 45 = 42 + 3 (as 42 = 2 * 3 * 7)

Harry J. Smith, Table of n, a(n) for n=1,...,1000

Cf. A001222, A229109.

a064800 n = a001222 n + n  -- Reinhard Zumkeller, Oct 27 2012

A064800 := proc(n)
    n+numtheory[bigomega](n) ;
end proc: # R. J. Mathar, Oct 21 2012

Table[n + PrimeOmega[n],{n,80}] (* Harvey P. Dale, Aug 19 2015 *)

{ for (n=1, 1000, write("b064800.txt", n, " ", n + bigomega(n)) ) } \\ Harry J. Smith, Sep 26 2009

A005236 Barriers for omega(n): numbers n such that, for all m < n, m + omega(m) <= n.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 17, 18, 20, 24, 26, 28, 30, 33, 38, 42, 48, 50, 54, 60, 65, 74, 82, 84, 90, 98, 102, 108, 110, 114, 126, 129, 138, 150, 164, 168, 174, 180, 194, 198, 228, 234, 244, 252, 258, 264, 270, 290, 294, 318, 348, 354, 360, 384, 390, 402

Offset: 1

N. J. A. Sloane

omega(m) is the number of distinct prime factors of m.

			1 + omega(1) = 1, 2 + omega(2) = 3, 3 + omega(3) = 4, 4 + omega(4) = 5, 5 + omega(5) = 6.
Thus we have verified that m + omega(m) < 6 for m < 6, so 6 is in the sequence.
But since 6 + omega(6) = 8 > 7, 7 is not in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, B8.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Paul Erdős, Some Unconventional Problems in Number Theory, Mathematics Magazine, Vol. 52, No. 2, Mar., 1979, pp. 67-70. See Problem 4, p. 68.
Paul Erdős, Some unconventional problems in number theory, Acta Mathematica Hungarica, 33(1):71-80, 1979.

Cf. A001221, A229109.

a005236 n = a005236_list !! (n-1)
a005236_list = filter (\x -> all (<= x) $ map a229109 [1..x-1]) [2..]
-- Reinhard Zumkeller, Sep 13 2013

omegaBarrierQ[n_] := (For[m = 1, m < n, m++, If[m + PrimeNu[m] > n, Return[False]]]; True); Select[Range[2, 500], omegaBarrierQ] (* Jean-François Alcover, Feb 03 2015 *)

is(n)=for(k=1,log(n)\log(5),if(omega(n-k)>k,return(0)));n>1 \\ Charles R Greathouse IV, Sep 19 2012

More terms from John W. Layman

A062509 a(n) = n^omega(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 36, 7, 8, 9, 100, 11, 144, 13, 196, 225, 16, 17, 324, 19, 400, 441, 484, 23, 576, 25, 676, 27, 784, 29, 27000, 31, 32, 1089, 1156, 1225, 1296, 37, 1444, 1521, 1600, 41, 74088, 43, 1936, 2025, 2116, 47, 2304, 49, 2500, 2601, 2704, 53, 2916

Offset: 1

Labos Elemer, Jul 13 2001

nonn

Not always equal to product of unitary divisors of n [compare with A061537]. This deviates from A061537 at 30, 42, 60, 66, etc.

			n=30: a(30) = 30^3 = 27000;
n=72: a(72) = 72^2 = 5184.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001221, A061537, A034444, A000005, A033992.

Cf. A176029, A229109.

a062509 n = n ^ a001221 n  -- Reinhard Zumkeller, Sep 13 2013

Table[n^PrimeNu[n],{n,60}] (* Harvey P. Dale, Jan 14 2013 *)

a(n)={n^omega(n)} \\ Harry J. Smith, Aug 08 2009

a(n) = Sum_{d|n} tau(d^n)*mu(n/d). - Ridouane Oudra, Sep 17 2022

A279436 Number of nonprimes less than or equal to n that do not divide n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 3, 4, 5, 3, 6, 6, 7, 6, 9, 7, 10, 8, 11, 12, 13, 9, 14, 15, 15, 15, 18, 15, 19, 16, 20, 21, 22, 18, 24, 24, 25, 22, 27, 24, 28, 26, 27, 30, 31, 25, 32, 31, 34, 33, 36, 32, 37, 34, 39, 40, 41, 34, 42, 42, 41, 40, 45, 43, 47, 45, 48, 46, 50, 42, 51, 51, 50, 51, 54, 52, 56, 50, 55, 58, 59, 52, 60, 61, 62, 59, 64, 57, 65, 64, 67, 68, 69, 62, 71, 69, 70, 68

Offset: 1

Ilya Gutkovskiy, Dec 12 2016

nonn

			a(10) = 4 because 10 has 4 divisors {1,2,5,10} therefore 6 non-divisors {3,4,6,7,8,9} out of which 4 are nonprimes {4,6,8,9}.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000005, A000040, A000720, A001221, A014689, A033273, A048865, A049820, A062298, A065890, A082514, A229109.

Table[n - PrimePi[n] - DivisorSigma[0, n] + PrimeNu[n], {n, 1, 100}]

for(n=1,50, print1(n - primepi(n) - numdiv(n) + omega(n), ", ")) \\ G. C. Greubel, May 22 2017

first(n)=my(v=vector(n),pp); forfactored(k=1,n, if(k[2][,2]==[1]~, pp++); v[k[1]]=k[1] - pp - numdiv(k) + omega(k)); v \\ Charles R Greathouse IV, May 23 2017

from sympy import primepi, divisor_count, primefactors
def a(n): return 0 if n==1 else n - primepi(n) - divisor_count(n) + len(primefactors(n)) # Indranil Ghosh, May 23 2017

G.f.: A(x) = B(x) + C(x) - D(x), where B(x) = Sum_{k>=1} x^(2*k+1)/((1 - x^k)*(1 - x^(k+1))), C(x) = Sum_{k>=1} x^prime(k)/(1 - x^prime(k)), D(x) = Sum_{k>=1} x^prime(k)/(1 - x).
a(n) = n - A000720(n) - A000005(n) + A001221(n).
a(n) = A062298(n) - A033273(n).
a(n) = A049820(n) - A048865(n).
a(n) = A229109(n) - A082514(n).
a(A000040(n)) = A065890(n).
a(A000040(n)) + 1 = A014689(n).
A000040(n) - a(A000040(n)) = n + 1.

A329717 a(n) is n (plus or minus) the number of distinct primes dividing n according to parity (even or odd).

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 6, 7, 8, 12, 10, 14, 12, 16, 17, 15, 16, 20, 18, 22, 23, 24, 22, 26, 24, 28, 26, 30, 28, 27, 30, 31, 35, 36, 37, 38, 36, 40, 41, 42, 40, 39, 42, 46, 47, 48, 46, 50, 48, 52, 53, 54, 52, 56, 57, 58, 59, 60, 58, 57, 60, 64, 65, 63, 67, 63, 66

Offset: 1

Lars Blomberg, Nov 20 2019

nonn

			A001221(3) = 1, so a(3) = 3-1 = 2. A001221(6) = 2, so a(6) = 6+2 = 8.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A001221, A229109.

Array[# + (om = PrimeNu[#]) * (-1)^om &, 67] (* Amiram Eldar, Nov 23 2019 *)

a(n) = my(om=omega(n)); n + (-1)^om*om; \\ Michel Marcus, Nov 20 2019

a(n) = n + A001221(n)*(-1)^A001221(n).

A378993 a(n) = n - omega(n), where omega = A001221.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 7, 8, 8, 10, 10, 12, 12, 13, 15, 16, 16, 18, 18, 19, 20, 22, 22, 24, 24, 26, 26, 28, 27, 30, 31, 31, 32, 33, 34, 36, 36, 37, 38, 40, 39, 42, 42, 43, 44, 46, 46, 48, 48, 49, 50, 52, 52, 53, 54, 55, 56, 58, 57, 60, 60, 61, 63, 63, 63, 66, 66

Offset: 1

Torlach Rush, Dec 17 2024

			a(40) = 38, since 40 has two distinct prime divisors (2 and 5), and so 40 - 2 = 38.
a(41) = 40 also, since 41 is prime and therefore 41 - 1 = 40.
a(42) = 39, since 42 has three distinct prime divisors (2, 3, 7), and so 42 - 3 = 39.

Cf. A001221, A229109.

a[n_]:=n-PrimeNu[n]; Array[a,68] (* Stefano Spezia, Dec 29 2024 *)

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

a(n) = n - A001221(n).

cp's OEIS Frontend

A064800 n plus the number of its prime factors: a(n) = n + A001222(n).

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A005236 Barriers for omega(n): numbers n such that, for all m < n, m + omega(m) <= n.

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A062509 a(n) = n^omega(n).

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A279436 Number of nonprimes less than or equal to n that do not divide n.

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A329717 a(n) is n (plus or minus) the number of distinct primes dividing n according to parity (even or odd).

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A378993 a(n) = n - omega(n), where omega = A001221.

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