A067340 -id:A067340 - cp's OEIS frontend

A077480 Total number of prime factors of numbers m with bigomega(m) == 0 modulo omega(m) (counted with repetition).

1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 1, 2, 2, 4, 1, 1, 2, 2, 1, 4, 2, 2, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 2, 1, 2, 2, 1, 4, 2, 4, 2, 2, 1, 1, 2, 6, 2, 3, 1, 2, 3, 1, 1, 2, 2, 3, 1, 4, 2, 1, 2, 2, 2, 4, 1, 2, 2, 2, 2, 6, 1, 4, 1, 3, 1, 4, 3, 2, 1, 1, 3, 2, 1, 3, 2, 2, 2, 2, 2, 2, 3, 1, 7, 2, 3, 1, 2, 2, 4

Offset: 1

Reinhard Zumkeller, Nov 06 2002

nonn

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

Equals A001222(A067340(n)).

Cf. A077479, A077481, A001221.

PrimeOmega[Select[Range[2,500], Divisible[PrimeOmega[#], PrimeNu[#]] &]] (* G. C. Greubel, May 08 2017 *)

A077481 Quotient A077480(n) / A077479(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 7, 1, 1, 1, 1, 1, 2

Offset: 1

Reinhard Zumkeller, Nov 06 2002

nonn

Enrique Pérez Herrero, Table of n, a(n) for n=1..7000.

Cf. A067340, A001221, A001222, A077479, A077480.

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

A323350 Nonprime numbers > 1 whose number of prime factors counted with multiplicity is a perfect square.

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 210, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 330, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376

Offset: 1

Gus Wiseman, Jan 15 2019

nonn

First differs from A014613 in having 512.

			360 = 2*2*2*3*3*5 has 6 prime factors, and 6 is not a perfect square, so 360 does not belong to the sequence.
2160 = 2*2*2*2*3*3*3*5 has 8 prime factors, and 8 is not a perfect square, so 2160 does not belong to the sequence.
10800 = 2*2*2*2*3*3*3*5*5 has 9 prime factors, and 9 is a perfect square, so 10800 belongs to the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A001222, A063989, A067028, A067340, A070175, A071625, A118914, A167175, A323305.

filter:= proc(n) local t;
  t:= numtheory:-bigomega(n);
  t > 1 and issqr(t)
end proc:
select(filter, [$4..1000]); # Robert Israel, Jan 15 2019

Select[Range[100],#>1&&!PrimeQ[#]&&IntegerQ[Sqrt[PrimeOmega[#]]]&]

isok(n) = (n>1) && !isprime(n) && issquare(bigomega(n)); \\ Michel Marcus, Jan 15 2019

A360690 Number of integer partitions of n with non-integer median of multiplicities.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 5, 6, 8, 8, 14, 12, 21, 20, 31, 36, 57, 61, 94, 108, 157, 188, 261, 305, 409, 484, 632, 721, 942, 1083, 1376, 1585, 2004, 2302, 2860, 3304, 4103, 4742, 5849, 6745, 8281, 9599, 11706, 13605, 16481, 19176, 23078, 26838, 32145, 37387, 44465

Offset: 1

Gus Wiseman, Feb 22 2023

nonn

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 a(1) = 0 through a(9) = 8 partitions:
  .  .  .  (211)  (221)  (411)    (322)    (332)      (441)
                  (311)  (21111)  (331)    (422)      (522)
                                  (511)    (611)      (711)
                                  (22111)  (22211)    (22221)
                                  (31111)  (41111)    (33111)
                                           (2111111)  (51111)
                                                      (2211111)
                                                      (3111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is not counted under a(8).

These partitions have ranks A360554.
The complement is counted by A360687, ranks A360553.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360069 = partitions with integer mean of multiplicities, ranks A067340.
Cf. A000041, A000975, A090794, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

A067341 Number of prime factors divided by the number of distinct prime factors is an integer and n is neither squarefree, nor power of prime.

Original entry on oeis.org

24, 36, 40, 54, 56, 88, 96, 100, 104, 135, 136, 144, 152, 160, 184, 189, 196, 216, 224, 225, 232, 240, 248, 250, 296, 297, 324, 328, 336, 344, 351, 352, 360, 375, 376, 384, 400, 416, 424, 441, 459, 472, 484, 486, 488, 504, 513, 528, 536, 540, 544, 560, 568

Offset: 1

Labos Elemer, Jan 16 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A067340.

ff[x_] := Flatten[FactorInteger[x]] f1[x_] := Length[FactorInteger[x]] f2[x_] := Apply[Plus, Table[Part[ff[x], 2*w], {w, 1, f1[x]}]] Do[s=f2[n]/f1[n]; If[IntegerQ[s]&&Greater[s, 1]&&Greater[f1[n], 1], Print[n]], {n, 2, 10000}]

is(n)=my(f=factor(n)[, 2]); #f>1 && vecsum(f)%#f==0 && vecmax(f)>1 \\ Charles R Greathouse IV, Oct 15 2015

A001222(n)/A001221(n) is an integer, A001222(n)>A001221(n)>1.

A360070 Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 18, 20, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 121, 125, 128, 144, 147, 150, 160, 162, 169, 175, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 245, 250, 252, 256, 272

Offset: 1

Gus Wiseman, Jan 27 2023

nonn

Conjecture: No term > 1 is squarefree.

			A partition of 20 with the same mean as its multiplicities is (5,4,3,2,1,1,1,1,1,1), so 20 is in the sequence.

Positions of positive terms in A360068, ranked by A359903.
A000041 counts partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature (A124010).
A326567/A326568 gives mean of prime indices (A112798).
Cf. A005117, A067340, A067538, A240219, A316313, A349156, A359904, A359905, A360069.

Select[Range[30],Select[IntegerPartitions[#],Mean[#]==Mean[Length/@Split[#]]&]!={}&]

a(22)-a(58) from Alois P. Heinz, Jan 29 2023

A360668 Numbers > 1 whose greatest prime index is not divisible by their number of prime factors (bigomega).

Original entry on oeis.org

4, 8, 10, 12, 15, 16, 18, 22, 24, 25, 27, 28, 32, 33, 34, 36, 40, 42, 44, 46, 48, 51, 54, 55, 60, 62, 63, 64, 66, 68, 69, 70, 72, 76, 77, 80, 81, 82, 85, 88, 90, 93, 94, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 116, 118, 119, 120, 121, 123, 124

Offset: 1

Gus Wiseman, Feb 17 2023

nonn

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.

Also numbers > 1 whose first differences of 0-prepended prime indices have non-integer mean.

			The prime indices of 1617 are {2,4,4,5}, and 5 is not divisible by 4, so 1617 is in the sequence.

These partitions are counted by A200727.
The complement is A340610 (without 1), counted by A168659.
For median instead of mean we have A360557, counted by A360691.
Positions of terms > 1 in A360615 (numerator: A360614).
A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A334201 adds up all prime indices except the greatest.
A348551 = numbers w/ non-integer mean of prime indices, complement A316413.
Cf. A072978, A078175, A139710, A307683, A359912, A360551, A360554, A360556, A360669.

Select[Range[2,100],!Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

A382351 Numbers with an integer harmonic mean of the indices of distinct prime factors.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 53, 59, 61, 64, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 117, 121, 125, 127, 128, 130, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 195, 197, 199, 211

Offset: 1

Ilya Gutkovskiy, Mar 22 2025

nonn

Index entries for sequences computed from indices in prime factorization.

Cf. A067340, A078174, A326621.

Select[Range[2, 220], IntegerQ[HarmonicMean[PrimePi[#[[1]]] & /@ FactorInteger[#]]] &]

isok(k) = if (k>1, my(f=factor(k)); denominator(#f~/sum(i=1, #f~, 1/primepi(f[i,1]))) == 1); \\ Michel Marcus, Mar 22 2025

cp's OEIS Frontend

A077480 Total number of prime factors of numbers m with bigomega(m) == 0 modulo omega(m) (counted with repetition).

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A077481 Quotient A077480(n) / A077479(n).

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

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A323350 Nonprime numbers > 1 whose number of prime factors counted with multiplicity is a perfect square.

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A360690 Number of integer partitions of n with non-integer median of multiplicities.

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A067341 Number of prime factors divided by the number of distinct prime factors is an integer and n is neither squarefree, nor power of prime.

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A360070 Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.

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A360668 Numbers > 1 whose greatest prime index is not divisible by their number of prime factors (bigomega).

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A382351 Numbers with an integer harmonic mean of the indices of distinct prime factors.

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