A063989 -id:A063989 - cp's OEIS frontend

A116661 Integers in both sequences A114522 and A063989.

4, 6, 8, 9, 10, 12, 18, 20, 22, 25, 27, 32, 34, 44, 48, 49, 50, 58, 68, 72, 80, 82, 108, 116, 118, 121, 125, 128, 142, 162, 164, 165, 169, 176, 192, 200, 202, 214, 236, 242, 243, 272, 273, 274, 284, 288, 289, 298, 320, 343, 345, 358, 361, 382, 385, 394, 399, 404

Offset: 1

Leroy Quet, Feb 21 2006

nonn

			20 = 2^2 *5^1. Both the number of prime divisors (counted with multiplicity), 2+1 = 3 and the sum of the distinct prime divisors, 2+5 = 7, are primes. So 20 is in the sequence.

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

Cf. A114522, A063989, A116660.

f:=func; [k:k in [2..450]| IsPrime(f(k)) and IsPrime(&+PrimeDivisors(k))]; // Marius A. Burtea, Nov 14 2019

Select[Range[500],AllTrue[{PrimeOmega[#],Total[FactorInteger[#][[All, 1]]]}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 03 2019 *)

More terms from Robert Gerbicz, Jun 09 2007

A134333 Numbers n whose number of prime factors (counted with multiplicity) is a prime factor of n.

Original entry on oeis.org

4, 6, 10, 12, 14, 18, 22, 26, 27, 30, 34, 38, 42, 45, 46, 58, 62, 63, 66, 74, 75, 78, 80, 82, 86, 94, 99, 102, 105, 106, 114, 117, 118, 120, 122, 134, 138, 142, 146, 147, 153, 158, 165, 166, 171, 174, 178, 180, 186, 194, 195, 200, 202, 206, 207, 214, 218, 222, 226

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

			a(1) = 4, since 4 has 2 prime factors and 2 is a prime factor of 4.
a(4) = 12, since 12 = 2*2*3 has 3 prime factors, and 3 is a prime factor of 12.
a(21) = 75, since 75 = 3*3*5 has 3 prime factors. and 3 is a prime factor of 75.
9 = 3*3 is not a term, since the number of prime factors (=2) is not a divisor of 9.
28 = 2*2*7 is not a term, since the number of prime factors (=3) is not a divisor of 28.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134334, A134344, A134376, A063989.

fQ[n_] := Module[{d = Total[Transpose[FactorInteger[n]][[2]]]}, PrimeQ[d] && Mod[n, d] == 0]; Select[Range[2, 226], fQ] (* T. D. Noe, Apr 05 2013 *)

a(n)=my(t=bigomega(n)); n%t==0 && isprime(t) \\ Charles R Greathouse IV, Sep 14 2015

a(n) << n log n/(log log n)^k for any fixed k. - Charles R Greathouse IV, Sep 14 2015

Sequence definition corrected and examples added by Hieronymus Fischer, Apr 05 2013

A323300 Number of ways to fill a matrix with the parts of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 2, 2, 4, 1, 6, 1, 4, 4, 3, 1, 6, 1, 6, 4, 4, 1, 12, 2, 4, 2, 6, 1, 12, 1, 2, 4, 4, 4, 18, 1, 4, 4, 12, 1, 12, 1, 6, 6, 4, 1, 10, 2, 6, 4, 6, 1, 12, 4, 12, 4, 4, 1, 36, 1, 4, 6, 4, 4, 12, 1, 6, 4, 12, 1, 20, 1, 4, 6, 6, 4, 12, 1, 10, 3, 4

Offset: 1

Gus Wiseman, Jan 12 2019

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(24) = 12 matrices whose entries are (2,1,1,1):
  [1 1 1 2] [1 1 2 1] [1 2 1 1] [2 1 1 1]
.
  [1 1] [1 1] [1 2] [2 1]
  [1 2] [2 1] [1 1] [1 1]
.
  [1] [1] [1] [2]
  [1] [1] [2] [1]
  [1] [2] [1] [1]
  [2] [1] [1] [1]

Positions of 1's are one and prime numbers A008578.
Positions of 2's are primes to prime powers A053810.
Cf. A000005, A001222, A008480, A056239, A063989, A112798, A120733.
Cf. A323295, A323305, A323307, A323351.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Array[Length[ptnmats[#]]&,100]

a(n) = A008480(n) * A000005(A001222(n)).

A067028 Numbers with a composite number of prime factors (counted with multiplicity).

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 64, 81, 84, 88, 90, 96, 100, 104, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 184, 189, 196, 198, 204, 210, 216, 220, 224, 225, 228, 232, 234, 240, 248, 250, 256, 260, 276, 294, 296, 297, 306, 308, 315, 324, 328, 330, 336

Offset: 1

Shyam Sunder Gupta, Feb 16 2002

nonn

			a(1)=16, 16 has 4 prime factors (counted with multiplicity) and 4 is composite.

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

Cf. A063989.

is(n)=my(t=bigomega(n)); t>1 && !isprime(t) \\ Charles R Greathouse IV, Oct 15 2015

A167175 Numbers with a nonprime number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 16, 17, 19, 23, 24, 29, 31, 36, 37, 40, 41, 43, 47, 53, 54, 56, 59, 60, 61, 64, 67, 71, 73, 79, 81, 83, 84, 88, 89, 90, 96, 97, 100, 101, 103, 104, 107, 109, 113, 126, 127, 131, 132, 135, 136, 137, 139, 140, 144, 149, 150, 151, 152, 156, 157, 160

Offset: 1

Juri-Stepan Gerasimov, Oct 29 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A063989, A141468.

Select[Range[200],!PrimeQ[PrimeOmega[#]]&] (* Harvey P. Dale, Apr 14 2013 *)

is(n)=!isprime(bigomega(n)) \\ Charles R Greathouse IV, Jun 19 2016

86 replaced by 96, 112 and 124 removed, 147 replaced by 144, 153 removed - R. J. Mathar, Apr 14 2010

A168645 Numbers with 2 or 3 prime divisors (counted with multiplicity).

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 27, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 102, 105, 106, 110, 111, 114, 115, 116, 117, 118, 119

Offset: 1

Juri-Stepan Gerasimov, Dec 01 2009

nonn

Below 32 this sequence and A063989 are identical.

Cf. A001222, A063989.

Select[Range[150],MemberQ[{2,3},Total[Transpose[ FactorInteger[#]] [[2]]]]&]  (* Harvey P. Dale, Mar 28 2011 *)

is(n)=my(t=bigomega(n)); t==2 || t==3 \\ Charles R Greathouse IV, Jun 19 2016

A323305 Number of divisors of the number of prime factors of n counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 4, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 3, 2, 1, 3, 2, 2, 2

Offset: 1

Gus Wiseman, Jan 13 2019

nonn

a(1) = 1 by convention.

First differs from A036430 at a(64) = 4, A036430(64) = 3.

Positions of 1's are 1 and the prime numbers A008578.
Positions of 2's are A063989.
Cf. A000005, A001222, A067028, A167175, A323295, A323350.

Mathematica
```
Array[Length@*Divisors@*PrimeOmega,100]
```

a(n) = if (n==1, 1, numdiv(bigomega(n))); \\ Michel Marcus, Jan 13 2019

a(n) = A000005(A001222(n)).

A064040 Integers whose number of distinct prime divisors is prime.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Lior Manor, Aug 23 2001

For all terms below 210 this sequence and A024619 are identical.

			210 = 2*3*5*7 has 4 prime factors, hence it is not here, but it is part of A024619.

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

Cf. A001221, A024619, A000977, A033992, A007774, A033993, A051270, A063989.

q:= n-> isprime(nops(ifactors(n)[2])):
select(q, [$1..210])[];  # Alois P. Heinz, Apr 18 2024

Select[Range[200], PrimeQ[PrimeNu[#]] &] (* Paolo Xausa, Mar 28 2024 *)

n=0; for (m=1, 10^9, if (isprime(omega(m)), write("b064040.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Sep 06 2009

is(n)=isprime(omega(n)) \\ Charles R Greathouse IV, Sep 18 2015

Edited by Charles R Greathouse IV, Mar 18 2010

Name edited by Michel Marcus, Oct 16 2023

A323521 Numbers whose number of prime factors counted with multiplicity (A001222) is not a perfect square.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 27, 28, 30, 32, 33, 34, 35, 38, 39, 42, 44, 45, 46, 48, 49, 50, 51, 52, 55, 57, 58, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 96, 98, 99, 102, 105, 106

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

Cf. A000037, A001222, A000290, A026478, A063989, A323519, A323520, A323527.

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

A110893 Numbers with a semiprime number of prime divisors (counted with multiplicity).

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 64, 81, 84, 88, 90, 96, 100, 104, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 184, 189, 196, 198, 204, 210, 216, 220, 224, 225, 228, 232, 234, 240, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 324, 328, 330, 336, 340, 342

Offset: 1

Jonathan Vos Post, Sep 20 2005

Below 256 = 2^8 this is identical to A067028 (Numbers with a composite number of prime factors, counted with multiplicity).

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

Cf. A000040, A001222, A001358, A014613, A046306, A046312, A046314, A063989, A067028, A069276.

is(n)=n>9 && bigomega(bigomega(n))==2 \\ Charles R Greathouse IV, Jan 31 2017

a(n) such that A001222(a(n)) is an element of A001358. a(n) such that bigomega(a(n)) is an element of A001358. Union[4-almost primes(A014613), 6-almost primes(A046306), 9-almost primes(A046312), 10-almost primes(A046314), 14-almost primes(A069275), 15-almost primes(A069276), 21-almost primes, 22-almost primes, 25-almost primes, 26-almost primes, ...]

cp's OEIS Frontend

A116661 Integers in both sequences A114522 and A063989.

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A134333 Numbers n whose number of prime factors (counted with multiplicity) is a prime factor of n.

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A323300 Number of ways to fill a matrix with the parts of the integer partition with Heinz number n.

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A067028 Numbers with a composite number of prime factors (counted with multiplicity).

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A167175 Numbers with a nonprime number of prime divisors (counted with multiplicity).

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A168645 Numbers with 2 or 3 prime divisors (counted with multiplicity).

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A323305 Number of divisors of the number of prime factors of n counted with multiplicity.

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A064040 Integers whose number of distinct prime divisors is prime.

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