A070011 -id:A070011 - cp's OEIS frontend

A360554 Numbers > 1 whose unordered prime signature has non-integer median.

12, 18, 20, 28, 44, 45, 48, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 124, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 192, 200, 207, 208, 212, 236, 242, 244, 245, 261, 268, 272, 275, 279, 284, 288, 292, 304, 316, 320, 325, 332, 333

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A187039 in having 2520 and lacking 1 and 12600.
A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The unordered prime signature of 2520 is {3,2,1,1}, with median 3/2, so 2520 is in the sequence.
The unordered prime signature of 12600 is {3,2,2,1}, with median 2, so 12600 is not in the sequence.

A subset of A030231.
For mean instead of median we have A070011.
Positions of odd terms in A360460.
The complement is A360553 (without 1), counted by A360687.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551 complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A304038, A316413, A326621, A348551, A360006, A360009, A360248, A360453.

Select[Range[2,100],!IntegerQ[Median[Last/@FactorInteger[#]]]&]

A360553 Numbers > 1 whose unordered prime signature has integer median.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A067340 in having 60.
A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The unordered prime signature of 60 is {1,1,2}, with median 1, so 60 is in the sequence.
The unordered prime signature of 1260 is {1,1,2,2}, with median 3/2, so 1260 is not in the sequence.

For mean instead of median we have A067340, complement A070011.
Positions of even terms in A360460.
The complement is A360554 (without 1).
These partitions are counted by A360687.
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
A112798 lists prime indices, length A001222, sum A056239.
A124010 lists prime signature.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360454 = numbers whose prime indices and signature have the same median.
Cf. A000975, A026424, A078174, A078175, A316413, A326621, A360453.

Select[Range[2,100],IntegerQ[Median[Last/@FactorInteger[#]]]&]

A070012 Floor of number of prime factors of n divided by the number of n's distinct prime factors.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1

Offset: 2

Rick L. Shepherd, Apr 11 2002

nonn

a(n) is the integer part of the average of the exponents in the prime factorization of n.

			a(12)=1 because 12=2^2 * 3^1 and floor(bigomega(12)/omega(12)) = floor((2+1)/2) = 1.
a(36)=2 because 36=2^2 * 3^2 and floor(bigomega(36)/omega(36)) = floor((2+2)/2) = 2.
a(60)=1 because 60=2^2 * 3^1 * 5^1 and floor(bigomega(60)/omega(60)) = floor((2+1+1)/3) = 1.
36 is in A067340. 12 and 60 are in A070011.

G. C. Greubel, Table of n, a(n) for n = 2..5000

Cf. A001221 (omega(n)), A001222 (bigomega(n)), A067340 (ratio is an integer before floor applied), A070011 (ratio is not an integer), A070013 (ratio rounded), A070014 (ceiling of ratio), A046660 (bigomega(n)-omega(n)).

A070012[n_]:=Floor[PrimeOmega[n]/PrimeNu[n]];Array[A070012,100]

v=[]; for(n=2,150,v=concat(v,(bigomega(n)\omega(n)))); v

a(n) = floor(bigomega(n)/omega(n)) for n>=2.

A070013 Number of prime factors of n divided by the number of n's distinct prime factors (rounded).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1

Offset: 2

Rick L. Shepherd, Apr 11 2002

nonn

a(n) is the rounded average of the exponents in the prime factorization of n.

			a(12)=2 because 12=2^2 * 3^1 and round(bigomega(12)/omega(12))=round((2+1)/2)=2.
a(36)=2 because 36=2^2 * 3^2 and round(bigomega(36)/omega(36))=round((2+2)/2)=2.
a(60)=1 because 60=2^2 * 3^1 * 5^1 and round(bigomega(60)/omega(60))= round((2+1+1)/3)=1.
36 is in A067340. 12 and 60 are in A070011.

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A001221 (omega(n)), A001222 (bigomega(n)), A067340 (ratio is an integer before rounding), A070011 (ratio is not an integer), A070012 (floor of ratio), A070014 (ceiling of ratio), A046660 (bigomega(n)-omega(n)).

Table[Round[PrimeOmega[n]/PrimeNu[n]], {n, 2, 50}] (* G. C. Greubel, May 08 2017 *)

v=[]; for(n=2,150,v=concat(v,round(bigomega(n)/omega(n)))); v

a(n) = round(bigomega(n)/omega(n)) for n>=2.

A070014 Ceiling of number of prime factors of n divided by the number of n's distinct prime factors.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 4, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1

Offset: 2

Rick L. Shepherd, Apr 11 2002

nonn

a(n) is the ceiling of the average of the exponents in the prime factorization of n.

			a(12) = 2 because 12 = 2^2 * 3^1 and ceiling(bigomega(12)/omega(12)) = ceiling((2+1)/2) = 2. a(36) = 2 because 36 = 2^2 * 3^2 and ceiling(bigomega(36)/omega(36)) = ceiling((2+2)/2) = 2. a(60) = 2 because 60 = 2^2 * 3^1 * 5^1 and ceiling(bigomega(60)/omega(60)) = ceiling((2+1+1)/3) = 2. 36 is in A067340. 12 and 60 are in A070011.

Antti Karttunen, Table of n, a(n) for n = 2..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A001221 (omega(n)), A001222 (bigomega(n)), A067340 (ratio is an integer before ceil is applied), A070011 (ratio is not an integer), A070012 (floor of ratio), A070013 (ratio rounded), A046660 (bigomega(n)-omega(n)), A088529, A088530.

Table[Ceiling[PrimeOmega[n]/PrimeNu[n]], {n, 2, 106}] (* Michael De Vlieger, Jul 12 2017 *)

v=[]; for(n=2,150,v=concat(v,ceil(bigomega(n)/omega(n)))); v

from sympy import primefactors, ceiling
def bigomega(n): return 0 if n==1 else bigomega(n//primefactors(n)[0]) + 1
def omega(n): return len(primefactors(n))
def a(n): return ceiling(bigomega(n)/omega(n))
print([a(n) for n in range(2, 51)]) # Indranil Ghosh, Jul 13 2017

(define (A070014 n) (let ((a (A001222 n)) (b (A001221 n))) (if (zero? (modulo a b)) (/ a b) (+ 1 (/ (- a (modulo a b)) b))))) ;; Antti Karttunen, Jul 12 2017

a(n) = ceiling(bigomega(n)/omega(n)) for n>=2.

A072588 Numbers having at least one prime factor with an odd and one with an even exponent.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 48, 50, 52, 60, 63, 68, 72, 75, 76, 80, 84, 90, 92, 98, 99, 108, 112, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 171, 172, 175, 176, 180, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228, 234, 236, 240, 242, 244

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

= Complement(Union(A002035, A000290)) = Intersection(A000037, A072587);
a(k)=A070011(k) for 0A070011(26)=120 is not a term, as 120=5*3*2^3 having only odd exponents (A002035(85)=120), and a(54)=240 is not a term of A070011, as from 240=5*3*2^4 follows that A001222(240)/A001221(240)=6/3=2 is an integer.
The asymptotic density of this sequence is 1 - A065463 = 0.2955577990... - Amiram Eldar, Sep 18 2022
Numbers k such that A007913(k) properly divides A007947(k). (Same as A072587 without square terms). A number k is in this sequence iff 1 < A007913(k) < A007947(k) < k, and A007913(k)|A007947(k), equivalently iff k is not in A000037 and A336643(k) is squarefree. - David James Sycamore, Sep 20 2023

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

Cf. A000037, A000290, A001221, A001222, A002035, A065463, A070011, A072587, A124010.

Cf. A007913, A007947, A336643.

a072588 n = a072588_list !! (n-1)
a072588_list = filter f [1..] where
   f x = any odd es && any even es  where es = a124010_row x
-- Reinhard Zumkeller, Nov 15 2012

oeeQ[n_]:=Module[{fi=Transpose[FactorInteger[n]][[2]]},Count[fi,?OddQ]>0  && Count[fi,?EvenQ]>0]; Select[Range[250],oeeQ] (* Harvey P. Dale, Jun 21 2015 *)

is(n)=#Set(factor(n)[,2]%2)==2 \\ Charles R Greathouse IV, Oct 16 2015

A084679 Composite numbers with coprime numbers of prime factors with and without repetition.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 48, 50, 52, 60, 63, 68, 72, 75, 76, 80, 84, 90, 92, 98, 99, 108, 112, 116, 117, 120, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 168, 171, 172, 175, 176, 180, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228, 234, 236, 242

Offset: 1

Reinhard Zumkeller, Jun 28 2003

nonn

A001221(a(n))>1 and GreatestCommonDivisor(A001221(a(n)), A001222(a(n)))=1;

a(n)=A070011(n) for n<228, but A070011(228)=840=7*5*3*2^3 with omega=4 and bigOmega=6, as GCD(4,6)=2>1 840 is not a term.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Different from A070011.

is(n)=my(f=factor(n)[,2]); #f>1 && gcd(vecsum(f),#f)==1 \\ Charles R Greathouse IV, Oct 19 2015

A245080 Numbers such that omega(a(n)) is a proper divisor of bigomega(a(n)).

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 36, 40, 49, 54, 56, 64, 81, 88, 96, 100, 104, 121, 125, 128, 135, 136, 144, 152, 160, 169, 184, 189, 196, 216, 224, 225, 232, 240, 243, 248, 250, 256, 289, 296, 297, 324, 328, 336, 343, 344, 351, 352, 360, 361, 375, 376, 384, 400, 416, 424, 441, 459

Offset: 1

Stanislav Sykora, Jul 11 2014

nonn

All proper powers of any number greater than 1 (A001597(n), n>1) are a subset of this sequence. On the other hand, this is a subset of A067340 which admits also numbers k for which bigomega(k) = omega(k). In particular, prime numbers are excluded.
The density of these numbers, i.e., the ratio n/a(n), apparently decreases with n, reaching 0.04420... for n = 10000000. Conjecture: n/a(n) might have a nonzero limit below 0.0427 (the density found in the interval 9500000 < n <= 10000000).
There are 40134838 terms in the range 10^9 <= k <= 2*10^9. - Hugo Pfoertner, Oct 28 2024

			240 is in the sequence because 240=5^1*3^1*2^4. Hence omega(240)=3 (three distinct prime divisors) is a proper divisor of bigomega(240)=6 (six prime divisors with multiplicity).

Stanislav Sykora, Table of n, a(n) for n = 1..20000
Wikipedia, Arithmetic functions

Cf. A000040, A001597, A067340, A070011.

Select[Range[500], Divisible[PrimeOmega[#], PrimeNu[#]] && PrimeNu[#] != PrimeOmega[#] &] (* Kritsada Moomuang, Oct 27 2024 *)

OmegaTest(n)=(bigomega(n)>omega(n))&&(bigomega(n)%omega(n)==0);
Ntest(nmax,test)={my(k=1,n=0,v);v=vector(nmax);while(1,n++;if(test(n),v[k]=n;k++;if(k>nmax,break)););return(v);}
Ntest(20000,OmegaTest)

is_a245080(n) = my(F=factor(n), o=omega(F), O=bigomega(F)); O>o && O%o==0; \\ Hugo Pfoertner, Oct 28 2024

cp's OEIS Frontend

A360554 Numbers > 1 whose unordered prime signature has non-integer median.

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A360553 Numbers > 1 whose unordered prime signature has integer median.

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A070012 Floor of number of prime factors of n divided by the number of n's distinct prime factors.

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A070013 Number of prime factors of n divided by the number of n's distinct prime factors (rounded).

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A070014 Ceiling of number of prime factors of n divided by the number of n's distinct prime factors.

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A072588 Numbers having at least one prime factor with an odd and one with an even exponent.

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A084679 Composite numbers with coprime numbers of prime factors with and without repetition.

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A245080 Numbers such that omega(a(n)) is a proper divisor of bigomega(a(n)).

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