A091376 -id:A091376 - cp's OEIS frontend

A100484 The primes doubled; Even semiprimes.

4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 1

Reinhard Zumkeller, Nov 22 2004

Essentially the same as A001747.
Right edge of the triangle in A065342. - Reinhard Zumkeller, Jan 30 2012
A253046(a(n)) > a(n). - Reinhard Zumkeller, Dec 26 2014
Apart from first term, these are the tau2-primes as defined in [Anderson, Frazier] and [Lanterman]. - Michel Marcus, May 15 2019
For every positive integer b and each m in this sequence b^(m-1) == b (mod m). - Florian Baur, Nov 26 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. D. Anderson and Andrea M. Frazier, On a general theory of factorization in integral domains, Rocky Mountain J. Math., Volume 41, Number 3 (2011), 663-705. See pp. 698, 699, 702.
James Lanterman, Irreducibles in the Integers modulo n, arXiv:1210.2991 [math.NT], 2012.
Eric Weisstein's World of Mathematics, Semiprime

Subsequence of A091376. After the initial 4 also a subsequence of A039956.
Cf. A046315, A152099, A179740.
Cf. A001748, A253046, A353478 (characteristic function).
Row 3 of A286625, column 3 of A286623.

2*Filtered([1..300],IsPrime); # Muniru A Asiru, Oct 05 2018

List([1..70], n-> 2*Primes[n]); # G. C. Greubel, May 18 2019

a100484 n = a100484_list !! (n-1)
a100484_list = map (* 2) a000040_list
-- Reinhard Zumkeller, Jan 31 2012

[2*p: p in PrimesUpTo(350)]; // Vincenzo Librandi, Mar 27 2014

A100484:=n->2*ithprime(n); seq(A100484(n), n=1..70); # Wesley Ivan Hurt, Mar 27 2014

2*Prime[Range[70]] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

2*primes(70) \\ Charles R Greathouse IV, Aug 21 2011

[2*nth_prime(n) for n in (1..70)] # G. C. Greubel, May 18 2019

a(n) = 2 * A000040(n).
a(n) = A001747(n+1).
n>1: A000005(a(n)) = 4; A000203(a(n)) = 3*A008864(n); A000010(a(n)) = A006093(n); intersection of A001358 and A005843.
a(n) = A116366(n-1, n-1) for n>1. - Reinhard Zumkeller, Feb 06 2006
a(n) = A077017(n+1), n>1. - R. J. Mathar, Sep 02 2008
A078834(a(n)) = A000040(n). - Reinhard Zumkeller, Sep 19 2011
a(n) = A087112(n, 1). - Reinhard Zumkeller, Nov 25 2012
A000203(a(n)) = 3*n/2 + 3, n > 1. - Wesley Ivan Hurt, Sep 07 2013

Simpler definition.

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

Original entry on oeis.org

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

A091377 Numbers having fewer prime factors than the value of their smallest prime factor.

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, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 143

Offset: 1

Reinhard Zumkeller, Jan 04 2004

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

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

Cf. A091374, A091376, A091375.

Select[Range[143],PrimeOmega[#]James C. McMahon, Dec 28 2024 *)

is(n)=if(n%2==0, return(n==2)); if(n<27, return(1)); forprime(p=2,bigomega(n), if(n%p==0, return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

A091373 Number of numbers <= n having exactly as many prime factors as the value of their smallest prime factor.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17

Offset: 1

Reinhard Zumkeller, Jan 04 2004

nonn

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

A091372(n) + a(n) + A091374(n) = n.

Cf. A091376, A091371.

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]]&]

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A100484 The primes doubled; Even semiprimes.

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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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A091377 Numbers having fewer prime factors than the value of their smallest prime factor.

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PARI

A091373 Number of numbers <= n having exactly as many prime factors as the value of their smallest prime factor.

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

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