A091377 -id:A091377 - cp's OEIS frontend

A091371 Smallest prime factor of n - number of prime factors of n with multiplicity.

1, 1, 2, 0, 4, 0, 6, -1, 1, 0, 10, -1, 12, 0, 1, -2, 16, -1, 18, -1, 1, 0, 22, -2, 3, 0, 0, -1, 28, -1, 30, -3, 1, 0, 3, -2, 36, 0, 1, -2, 40, -1, 42, -1, 0, 0, 46, -3, 5, -1, 1, -1, 52, -2, 3, -2, 1, 0, 58, -2, 60, 0, 0, -4, 3, -1, 66, -1, 1, -1, 70, -3, 72, 0, 0, -1, 5, -1, 78, -3

Offset: 1

Reinhard Zumkeller, Jan 04 2004

sign

a(n) = A020639(n) - A001222(n).

a(A091375(n)) < 0. a(A091376(n)) = 0. a(A091377(n)) > 0.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A020639 (lpf), A001222 (Omega).

Cf. A091372, A091373, A091374, A091375, A091376, A091377.

with(numtheory); A091371:=n->`if`(n=1,1,min(op(factorset(n)))-bigomega(n)); seq(A091371(k), k=1..100); # Wesley Ivan Hurt, Oct 27 2013

Array[FactorInteger[#][[1,1]]-PrimeOmega[#]&,80] (* Harvey P. Dale, May 25 2012 *)

Definition clarified by Harvey P. Dale, May 25 2012

A091375 Numbers k with property that the number of prime factors of k (counted with repetition) exceeds the smallest prime factor of k.

Original entry on oeis.org

8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 100, 102, 104, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 144, 148, 150, 152, 154, 156, 160, 162

Offset: 1

Reinhard Zumkeller, Jan 04 2004

nonn

A091371(a(n)) < 0: A001222(a(n))>A020639(a(n)).

Numbers of the form m*i + n*j = k*(i+j), where i and j are > 1. - Giovanni Teofilatto, Aug 29 2007

Michael De Vlieger, Table of n, a(n) for n = 1..10438

Cf. A091372, A091376, A091377.

Select[Range@ 162, Length@ Flatten[Table[#1, {#2}] & @@@ #] > #[[1, 1]] &@ FactorInteger@ # &] (* Michael De Vlieger, Jul 06 2016 *)

is(n)=n>7 && bigomega(n) > factor(n)[1,1] \\ Charles R Greathouse IV, Jul 06 2016

bigomegaAtLeast(n,k,startAt=2)=if(k<3, return(if(k==2,n>1&&!isprime(n),k==0||n>1))); forprime(p=startAt,logint(n,k), if(n%p==0, k-=valuation(n,p);n/=p^valuation(n,p); return(bigomegaAtLeast(n,k)))); 0
is(n)=if(n%2==0,return(bigomegaAtLeast(n/2,2))); if(n%3==0,return(bigomegaAtLeast(n/3,3,3))); if(n<9,return(0)); forprime(p=5,log(n)/lambertw(log(n)), if(n%p==0, return(bigomegaAtLeast(n/p,p,p)))); 0 \\ Charles R Greathouse IV, Jul 06 2016

Missing a(10001)-a(10424) inserted into b-file by Andrew Howroyd, Feb 25 2018

Definition clarified by N. J. A. Sloane, Jan 20 2020

A091376 Numbers k with property that the number of prime factors of k (counted with repetition) equals the smallest prime factor of k.

Original entry on oeis.org

4, 6, 10, 14, 22, 26, 27, 34, 38, 45, 46, 58, 62, 63, 74, 75, 82, 86, 94, 99, 105, 106, 117, 118, 122, 134, 142, 146, 147, 153, 158, 165, 166, 171, 178, 194, 195, 202, 206, 207, 214, 218, 226, 231, 254, 255, 261, 262, 273, 274, 278, 279, 285, 298, 302, 314

Offset: 1

Reinhard Zumkeller, Jan 04 2004

nonn

A091371(a(n)) = 0: A001222(a(n))=A020639(a(n)).

Prime factors counted with multiplicity. - Harvey P. Dale, Nov 11 2012

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

Cf. A091373, A091375, A091377.
Cf. A002808.
Cf. A100484 (subsequence).

a091376 n = a091376_list !! (n-1)
a091376_list = [x | x <- a002808_list, a001222 x == a020639 x]
-- Reinhard Zumkeller, Nov 11 2012

pfQ[n_]:=Module[{fi=Transpose[FactorInteger[n]]},fi[[1,1]] == Total[Last[fi]]]; Rest[Select[Range[400],pfQ]] (* Harvey P. Dale, Nov 11 2012 *)
Select[Range[400],PrimeOmega[#]==FactorInteger[#][[1,1]]&] (* Harvey P. Dale, Nov 26 2024 *)

Definition edited by N. J. A. Sloane, Jan 21 2020

A091374 Number of numbers <= n having fewer prime factors than the value of their smallest prime factor.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35

Offset: 1

Reinhard Zumkeller, Jan 04 2004

nonn

a(n) = #{m: A001222(m) < A020639(m), m<=n};

A091372(n) + A091373(n) + a(n) = n.

Cf. A091377, A091371.

isok(n) = {my(f=factor(n)); (#f~ == 0) || (bigomega(n) < f[1,1]);}
a(n) = sum(k=1, n, isok(k)); \\ Michel Marcus, Feb 05 2016

A277334 Numbers n, that apart from 2 are all odd and for which n/(largest prime dividing n) is squarefree.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145, 147, 149, 151, 155, 157, 159, 161, 163, 165, 167, 169

Offset: 1

Antti Karttunen, Oct 12 2016

nonn

In other words, after 1 and 2, such odd numbers that only the largest prime factor in their prime factorization may have exponent 1 or 2, while all lesser prime factors occur at most once.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Disjoint union of A056911 and A129598(A056911(n)).
Cf. A005117, A008683, A052126.
Cf. A277332 (permutation of this sequence).
Differs from A091377 for the first time at n=36, where a(36)=75, while A091377(36)=77.

with(numtheory): A277334_list := n -> seq(`if`(i=2 or (i::odd and issqrfree(i/ max(factorset(i)))),i,NULL),i=1..n): A277334_list(169); # Peter Luschny, Oct 23 2016

A377723 Numbers whose number of prime factors (counted with repetition) is greater than or equal to its smallest prime factor.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116

Offset: 1

James C. McMahon, Dec 28 2024

nonn

Numbers k such that A001222(k) >= A020639(k).
Complement of A091377.
A091371(a(n)) < 1: A001222(a(n)) => A020639(a(n)).

			4 is a term because bigomega(4) = spf(4) = 2.
12 is a term because bigomega(12) = 3 > spf(12) = 2.
3 is not a term because bigomega(3) = 1 < spf(3) = 3.

James C. McMahon, Table of n, a(n) for n = 1..10000

Cf. A001222, A020639, A091371, A091375, A091376, A091377.

Select[Range[116], PrimeOmega[#]>=FactorInteger[#][[1, 1]]&]

cp's OEIS Frontend

A091371 Smallest prime factor of n - number of prime factors of n with multiplicity.

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A091375 Numbers k with property that the number of prime factors of k (counted with repetition) exceeds the smallest prime factor of k.

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A091376 Numbers k with property that the number of prime factors of k (counted with repetition) equals the smallest prime factor of k.

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Haskell

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A091374 Number of numbers <= n having fewer prime factors than the value of their smallest prime factor.

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PARI

A277334 Numbers n, that apart from 2 are all odd and for which n/(largest prime dividing n) is squarefree.

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Crossrefs

Programs

Maple

A377723 Numbers whose number of prime factors (counted with repetition) is greater than or equal to its smallest prime factor.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica