A088381 -id:A088381 - cp's OEIS frontend

A080257 Numbers having at least two distinct or a total of at least three prime factors.

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Reinhard Zumkeller, Feb 10 2003

Complement of A000430; A080256(a(n)) > 3.
A084114(a(n)) > 0, see also A084110.
Also numbers greater than the square of their smallest prime-factor: a(n)>A020639(a(n))^2=A088377(a(n));
a(n)>A000430(k) for n<=13, a(n) < A000430(k) for n>13.
Numbers with at least 4 divisors. - Franklin T. Adams-Watters, Jul 28 2006
Union of A024619 and A033942; A211110(a(n)) > 2. - Reinhard Zumkeller, Apr 02 2012
Also numbers > 1 that are neither prime nor a square of a prime. Also numbers whose omega-sequence (A323023) has sum > 3. Numbers with omega-sequence summing to m are: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7). - Gus Wiseman, Jul 03 2019
Numbers n such that sigma_2(n)*tau(n) = A001157(n)*A000005(n) >= 4*n^2. Note that sigma_2(n)*tau(n) >= sigma(n)^2 = A072861 for all n. - Joshua Zelinsky, Jan 23 2025

			8=2*2*2 and 10=2*5 are terms; 4=2*2 is not a term.
From _Gus Wiseman_, Jul 03 2019: (Start)
The sequence of terms together with their prime indices begins:
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
(End)

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

Cf. A001248, A001221, A001222, A006881, A030078, A088381, A088383, A000005.

Cf. A060687, A118914, A323014, A323023, A325249.

a080257 n = a080257_list !! (n-1)
a080257_list = m a024619_list a033942_list where
   m xs'@(x:xs) ys'@(y:ys) | x < y  = x : m xs ys'
                           | x == y = x : m xs ys
                           | x > y  = y : m xs' ys
-- Reinhard Zumkeller, Apr 02 2012

Select[Range[100],PrimeNu[#]>1||PrimeOmega[#]>2&] (* Harvey P. Dale, Jul 23 2013 *)

PARI
```
is(n)=omega(n)>1 || isprimepower(n)>2
```

is(n)=my(k=isprimepower(n)); if(k, k>2, !isprime(n)) \\ Charles R Greathouse IV, Jan 23 2025

a(n) = n + O(n/log n). - Charles R Greathouse IV, Sep 14 2015

Definition clarified by Harvey P. Dale, Jul 23 2013

A138511 Semiprimes where the larger prime factor is greater than the square of the smaller prime factor, short: semiprimes p*q, p^2 < q.

Original entry on oeis.org

10, 14, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 134, 141, 142, 145, 146, 155, 158, 159, 166, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 215, 218, 219, 226, 235, 237, 249, 254, 262, 265, 267, 274, 278, 291, 295, 298, 302, 303, 305

Offset: 1

Reinhard Zumkeller, Mar 21 2008

From Antti Karttunen, Dec 17 2014, further edited Jan 01 & 04 2014: (Start)
Semiprimes p*q, p < q, such that the smallest r for which r^k <= p and q < r^(k+1) [for some k >= 0] is q+1, and thus k = 0. In other words, semiprimes whose both prime factors do not fit (simultaneously) between any two consecutive powers of any natural number r less than or equal to the larger prime factor. This condition forces the larger prime factor q to be greater than the square of the smaller prime factor because otherwise the opposite condition given in A251728 would hold.
Assuming that A054272(n), the number of primes in interval [prime(n), prime(n)^2], is nondecreasing (implied for example if Legendre's or Brocard's conjecture is true), these are also "unsettled" semiprimes that occur in a square array A083221 constructed from the sieve of Eratosthenes, "above the line A251719", meaning that if and only if row < A251719(col) then a semiprime occurring at A083221(row, col) is in this sequence, and conversely, all the semiprimes that occur at any position A083221(row, col) where row >= A251719(col) are in the complementary sequence A251728.
(End)
Semiprimes p*q, p < q, such that b = q+1 is the minimal base with the property that p and q have equal length representations in base b. This was the original definition, which is based primarily on A138510: A138510(A174956(a(n))) = A084127(A174956(a(n))) + 1.

			See A138510.

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

Cf. A138510.
Complement of A251728 in A001358.
Subsequence of A088381.
An intersection of A001358 (semiprimes) and A251727.
Also an intersection of A001358 and A253569, from the latter which this sequence differs for the first time at n=60, where A253569(60) = 290, while here a(60) = 291.
Also an intersection A001358 and A245729.
Cf. also A006530, A020639, A054272, A083221, A083140, A084127, A138510, A162319, A174956, A251719.

a138511 n = a138511_list !! (n-1)
a138511_list = filter f [1..] where
                      f x = p ^ 2 < q && a010051' q == 1
                            where q = div x p; p = a020639 x
-- Reinhard Zumkeller, Jan 06 2015

Select[Range[305],PrimeOmega[#]==2&&Divisors[#][[-2]]>Divisors[#][[-3]]^2&&SquareFreeQ[#]&] (* James C. McMahon, Jun 07 2025 *)

isok(s) = my(f=factor(s)); (bigomega(f) == 2) && (#f~ == 2) && (f[1,1]^2 < f[2, 1]); \\ Michel Marcus, Sep 15 2020

;; Scheme with Antti Karttunen's IntSeq-library.
(define A138511 (MATCHING-POS 1 2 (lambda (n) (and (= 2 (A001222 n)) (= (A252375 n) (+ 1 (A006530 n)))))))
(define A138511 (COMPOSE A001358 (MATCHING-POS 1 1 (lambda (n) (= (A138510 n) (+ 1 (A006530 (A001358 n))))))))
;; Antti Karttunen, Dec 16-17 2014

;; Scheme with Antti Karttunen's IntSeq-library.
(define A138511 (MATCHING-POS 1 2 (lambda (n) (and (= 2 (A001222 n)) (> (A006530 n) (A000290 (A020639 n))))))) ;; Based on the new alternative definition - Antti Karttunen, Jan 01 2015

Other identities. For all n >= 1 it holds that:

A138510(A174956(a(n))) = A084127(A174956(a(n))) + 1.

Wrong comment corrected by Reinhard Zumkeller, Dec 16 2014
New definition by Antti Karttunen, Jan 01 2015; old definition moved to comment.
More terms from Antti Karttunen, Jan 09 2015

A253569 Composite numbers n = p_i * p_j * p_k * ... * p_u, p_i <= p_j <= p_k <= ... <= p_u, where each successive prime factor (when sorted into a nondecreasing order) is greater than the square of the previous: (p_i)^2 < p_j, (p_j)^2 < p_k, etc.

Original entry on oeis.org

10, 14, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 134, 141, 142, 145, 146, 155, 158, 159, 166, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 215, 218, 219, 226, 235, 237, 249, 254, 262, 265, 267, 274, 278, 290

Offset: 1

Antti Karttunen & Reinhard Zumkeller, Jan 09 2015

nonn

Numbers n = A020639(n) * A014673(n) * A054576(n), for which A020639(n)^2 < A014673(n) and either A054576(n) = 1 or A032742(n) satisfies the same condition (is the term of this sequence).

			290 = 2*5*29 is a member, because 2^2 < 5 and 5^2 < 29.

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

Complement: A253567.
Subsequence of A002808, A005117, A088381, A251727, A245729 and A253785.
A138511 is a subsequence, from which this sequence differs for the first time at n=60, where A138511(60) = 291, while here a(60) = 290.
Cf. A000290, A001222, A020639, A032742, A014673, A054576.

a253569 n = a253569_list !! (n-1)
a253569_list = filter f [1..] where
                    f x = (p ^ 2 < a020639 q) && (a010051' q == 1 || f q)
                          where q = div x p; p = a020639 x
-- Antti Karttunen after Reinhard Zumkeller's code for A138511, Jan 09 2015
a253569 n = a253569_list !! (n-1)
a253569_list = filter (not . f''') a002808_list where
   f''' x = p ^ 2 > a020639 q || (a010051 q == 0 && f''' q)
            where q = div x p; p = a020639 x
-- Reinhard Zumkeller, Jan 12 2015
(Scheme, with Antti Karttunen's IntSeq-library)
(define A253569 (MATCHING-POS 1 1 (lambda (n) (and (> (A001222 n) 1) (numbers-sparsely-distributed? (ifactor n))))))
(define (numbers-sparsely-distributed? lista) (cond ((null? lista) #t) ((null? (cdr lista)) #t) ((> (A000290 (car lista)) (cadr lista)) #f) (else (numbers-sparsely-distributed? (cdr lista)))))
;; Antti Karttunen, Jan 16 2015

cnQ[n_]:=CompositeQ[n]&&Union[Boole[#[[2]]>#[[1]]^2&/@Partition[Flatten[Table[ #[[1]], #[[2]]]&/@FactorInteger[n]],2,1]]]=={1}; Select[Range[300],cnQ] (* Harvey P. Dale, Jul 10 2023 *)

A088380 Numbers not exceeding the cube of their smallest prime factor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) <= A020639(a(n))^3 = A088378(a(n)); complement of A088381;

a(n) < A088381(k) for n <= 28, a(n) > A088381(k) for n > 28.

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

Cf. A000430, A088382, A020639, A088378, A088381.

Positions of numbers less than 4 in A307908.

a088380 n = a088382_list !! (n-1)
a088380_list = [x | x <- [1..], x <= a020639 x ^ 3]
-- Reinhard Zumkeller, Feb 06 2015

Select[Range[200],#<=FactorInteger[#][[1,1]]^3&] (* Harvey P. Dale, Apr 28 2022 *)

A088383 Numbers greater than the 4th power of their smallest prime factor.

Original entry on oeis.org

18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118, 120, 122, 123, 124, 126

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) > A020639(a(n))^4 = A088379(a(n)); complement of A088382.

a(n) > A088382(k) for n <= 67, a(n) < A088382(k) for n > 67.

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

Cf. A088381, A080257, A020639, A088379, A088382.

Positions of numbers greater than 4 in A307908.

a088383 n = a088383_list !! (n-1)
a088383_list = [x | x <- [1..], x  a020639 x ^ 4]
-- Reinhard Zumkeller, Feb 06 2015

Select[Range[200],#>(FactorInteger[#][[1,1]])^4&] (* Harvey P. Dale, Aug 15 2015 *)

A245729 Composite numbers n = A020639(n) * A032742(n) where the greatest proper divisor A032742(n) is greater than the square of the smallest prime factor A020639(n), and that greatest proper divisor A032742(n) is either a prime or satisfies the same condition (i.e., is itself the term of this sequence).

Original entry on oeis.org

10, 14, 20, 22, 26, 28, 33, 34, 38, 39, 40, 44, 46, 51, 52, 56, 57, 58, 62, 66, 68, 69, 74, 76, 78, 80, 82, 86, 87, 88, 92, 93, 94, 99, 102, 104, 106, 111, 112, 114, 116, 117, 118, 122, 123, 124, 129, 132, 134, 136, 138, 141, 142, 145, 146, 148, 152, 153, 155, 156, 158, 159, 160, 164, 166, 171, 172, 174, 176, 177

Offset: 1

Antti Karttunen and Reinhard Zumkeller, Jan 09 2015

nonn

If n is present, then so is also 2*n.

If n = p_1^e_1*p_2^e_2*... with p_1 > p_2 > ..., then n is in this sequence iff p_1^2 > p_2 and e_1 = 1. - Charlie Neder, Jun 13 2019

			10 = 2*5 is present, because 2^2 < 5 and 5 is a prime.
20 = 2*10 is present, because 2^2 < 10, and 10 itself is present in the sequence.

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

Subsequence of A088381 and A251727.
Subsequences: A138511, A253569.
Cf. A001222, A000290, A010051, A020639, A032742.

a245729 n = a245729_list !! (n-1)
a245729_list = filter f [1..] where
                      f x = p ^ 2 < q && (a010051' q == 1 || f q)
                            where q = div x p; p = a020639 x
-- Antti Karttunen after Reinhard Zumkeller's code for A138511, Jan 09 2015

;; With Antti Karttunen's IntSeq-library.
(define (charfun_for_A245729 n) (if (and (> (A001222 n) 1) (> (A032742 n) (A000290 (A020639 n)))) (+ (A010051 (A032742 n)) (charfun_for_A245729 (A032742 n))) 0))
(define A245729 (NONZERO-POS 1 1 charfun_for_A245729))
;; Antti Karttunen, Jan 16 2015

A087719 Least number m such that the number of numbers k <= m with k > spf(k)^n exceeds the number of numbers with k <= spf(k)^n.

Original entry on oeis.org

15, 27, 57, 135, 345, 927, 2577, 7335, 21225, 62127, 183297, 543735, 1618905, 4832127, 14447217, 43243335, 129533385, 388206927, 1163834337, 3489930135, 10466644665, 31393642527, 94168344657, 282479868135

Offset: 1

Reinhard Zumkeller, Sep 29 2003

nonn

mspf(k)^n & 1<=k<=m} <= m/2;

m>=a(n): #{k: k>spf(k)^n & 1<=k<=m} > m/2.

Cf. A088382, A088382, A088380, A088381, A000430, A080257.

Numbers so far satisfy a(n) = 3^n + 3*2^n + 6. - Ralf Stephan, May 10 2004

Empirical G.f.: 3*x*(5-21*x+20*x^2)/(1-x)/(1-2*x)/(1-3*x). - Colin Barker, Feb 22 2012

a(14)-a(24) from Giovanni Resta, May 23 2013

cp's OEIS Frontend

A080257 Numbers having at least two distinct or a total of at least three prime factors.

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A138511 Semiprimes where the larger prime factor is greater than the square of the smaller prime factor, short: semiprimes p*q, p^2 < q.

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A253569 Composite numbers n = p_i * p_j * p_k * ... * p_u, p_i <= p_j <= p_k <= ... <= p_u, where each successive prime factor (when sorted into a nondecreasing order) is greater than the square of the previous: (p_i)^2 < p_j, (p_j)^2 < p_k, etc.

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A088380 Numbers not exceeding the cube of their smallest prime factor.

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A088383 Numbers greater than the 4th power of their smallest prime factor.

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A087719 Least number m such that the number of numbers k <= m with k > spf(k)^n exceeds the number of numbers with k <= spf(k)^n.

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