A289484 -id:A289484 - cp's OEIS frontend

A291125 "Doubly A289484" numbers: numbers with prime factorization p1^e1 * p2^e2 * ... * pk^ek such that there exist i < j < k with p1^e1 * p2^e2 * ... pi^ei > p(i+1) and p1^e1 * p2^e2 * ... pj^ej > p(j+1).

60, 84, 120, 132, 168, 180, 210, 240, 252, 264, 280, 300, 312, 315, 330, 336, 360, 390, 396, 408, 420, 440, 456, 468, 480, 495, 504, 510, 520, 528, 540, 552, 560, 570, 585, 588, 600, 612, 616, 624, 630, 660, 672, 680, 684, 690, 693, 720, 728, 756, 760, 765, 770, 780

Offset: 1

Richard Locke Peterson, Aug 17 2017

nonn

These form a subsemigroup and a subsequence of the sequence A289484.

Density: Only 4.3% of the integers between 1 and 400 are doubly A289484.divisible by at least 3 primes. If a term in the sequence is squarefree, it must be divisible by at least 4 primes. If a number n is in the sequence, then every multiple is also in it. Using Wolfram Alpha, about 48% of the integers between 10^40+1 to 10^40+62 were found to be doubly A289484.

			60=2^2*3*5 is a term because 2^2 > 3 and 2^2*3 > 5.
315=3^2*5*7 is a term because 3^2 > 5 and 3^2*5 > 7.

Cf. A289484.

isA291125 := proc(n)
    local pset,p,pprodidx,pprod,nu,falls ;
    pset := sort(convert(numtheory[factorset](n),list)) ;
    pprod := 1;
    falls := 0 ;
    for pprodidx from 1 to nops(pset)-1 do
        p := pset[pprodidx] ;
        nu := padic[ordp](n,p) ;
        pprod := pprod*p^nu ;
        if pprod > pset[pprodidx+1] then
            falls := falls+1 ;
            if falls >= 2 then
                return true;
            end if;
        end if;
    end do:
    return false ;
end proc:
for n from 1 to 3000 do
    if isA291125(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 20 2017

is(n,f=factor(n))=my(p=1,t,s); for(i=1,#f~, t=f[i,1]^f[i,2]; if(p>t,s++); p*=t); s>1 \\ Charles R Greathouse IV, Jun 10 2020

New name from Charles R Greathouse IV, Jun 11 2020

A292480 p-INVERT of the odd positive integers, where p(S) = 1 - S^2.

Original entry on oeis.org

0, 1, 6, 20, 56, 160, 480, 1456, 4384, 13136, 39360, 118064, 354272, 1062928, 3188736, 9565936, 28697632, 86093264, 258280512, 774841520, 2324523104, 6973567888, 20920705152, 62762119792, 188286360736, 564859074896, 1694577214656, 5083731648560

Offset: 0

Clark Kimberling, Oct 02 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
In the following guide to p-INVERT sequences using s = (1,3,5,7,9,...) = A005408, in some cases t(1,3,5,7,9,...) is a shifted (or differently indexed) version of the cited sequence:
p(S) *********** t(1,3,5,7,9,...)
- S               A003946
- S^2             A292480
- S^3             (not yet in OEIS)
(1 - S)^2           (not yet in OEIS)
(1 - S)^3           (not yet in OEIS)
- S - S^2         A289786
+ S - S^2         A289484
- S - 2 S^2       A289785
- S - 3 S^2       A289786
- S - 4 S^2       A289787
- S - 5 S^2       A289788
- S - 6 S^2       A289789
- S - 7 S^2       A289790
+ S - 2 S^2       A289791
- S + S^2 - S^3   A289792
+ S - 3 S^2       A289793
- S - S^2 - S^3   A289794

			s = (1,3,5,7,9,...), S(x) = x + 3 x^2 + 5 x^3 + 7 x^4 + ...,
p(S(x)) = 1 - ( x + 3 x^2 + 5 x^3 + 7 x^4 + ...)^2,
1/p(S(x)) = 1 + x^2 + 6 x^3 + 20 x^4 + 56 x^5 + ...
T(x) = (-1 + 1/p(S(x)))/x = x + 6 x^2 + 20 x^3 + 56 x^4 + ...
t(s) = (0,1,2,20,56,...).

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,6)

Cf. A005408, A292479.

I:=[0,1,6,20]; [n le 4 select I[n] else 4*Self(n-1)- 5*Self(n-2)+6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 03 2017

z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292480 *)
Join[{0}, LinearRecurrence[{4, -5, 6}, {1, 6, 20}, 30]] (* Vincenzo Librandi, Oct 03 2017 *)

G.f.: x*(1 + x)^2/((1 - 3*x)*(1 - x + 2*x^2)).

a(n) = 4*a(n-1) - 5*a(n-2) + 6*a(n-3) for n >= 5.

A027854 Mutinous numbers: n > 1 such that n/p^k > p, where p is the largest prime dividing n and p^k is the highest power of p dividing n.

Original entry on oeis.org

12, 24, 30, 36, 40, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 96, 105, 108, 112, 120, 126, 132, 135, 140, 144, 150, 154, 160, 165, 168, 175, 176, 180, 182, 189, 192, 195, 198, 200, 208, 210, 216, 220, 224, 225, 231, 234, 240, 252, 260, 264, 270, 273, 275, 280

Offset: 1

Leroy Quet

nonn

Numbers n > 1 such that n/A053585(n) > A006530(n). - Michael De Vlieger, Jul 13 2017
If p = A006530(a(n)) then p * a(n) is in the sequence. E.g., as 12 is in the sequence with gpf(12) = A006530(12) = 3, 12*3^k is in the sequence for k > 0. Conjecture: if m is in the sequence then so is A003961(m). - David A. Corneth, Jul 13 2017
At present this and A027855 are complements in the set of integers >= 2. If a 1 were inserted at the start, then this and A027855 are complements in the set of positive integers. - Harry Richman, Sep 08 2019
The sequence is closed under multiplication (a semigroup). For, suppose x = p^i*m1, y = q^j*m2 are in the sequence, with p, q, p^i, p^j as given, with m1 > p and m2 > q, and suppose q >= p. If q = p then xy/q^(i+j) = m1*m2 > q. If q > p, then xy/q^j = p^i*m1*m2 > q (since q > p and p is greater than all primes in m1). - Richard Peterson, May 29 2022
There are subsequences that constitute subsemigroups: Consider as a subsequence all terms x such that x/p^k > a*p^b, with p,k as specified in the definition and a,b fixed real numbers greater than or equal to 1. Each pair (a,b) determines a subsequence that is also a subsemigroup of the original (1,1) semigroup that constitutes the whole sequence. The proof of closure is similar. To see that such proposed subsemigroups are nonempty, choose any prime p greater than 2 and multiply p by a sufficiently large power of 2. - Richard Peterson, May 29 2022
This sequence is a subsequence and subsemigroup of A289484. - Richard Peterson, Oct 29 2022

			From _Michael De Vlieger_, Jul 13 2017: (Start)
12 is a term since 12/A053585(12) = 12/3 = 4, A006530(12) = 3, and 4 > 3.
30 is a term since 30/A053585(30) = 30/5 = 6, A006530(30) = 5, and 6 > 5.
(End)

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A006530, A027855, A053585.

Select[Range@ 280, Function[n, (n/Apply[Power, Last@ #]) > #[[-1, 1]] &@ FactorInteger[n]]] (* Michael De Vlieger, Jul 13 2017 *)

isok(n) = {my(f = factor(n)); my(maxf = #f~); my(p = f[maxf, 1]); my(pk = f[maxf, 2]); (n/p^pk) > p;} \\ Michel Marcus, Jan 16 2014

from sympy import factorint, primefactors
def a053585(n):
    if n==1: return 1
    p = primefactors(n)[-1]
    return p**factorint(n)[p]
print([n for n in range(2, 301) if n>a053585(n)*primefactors(n)[-1]]) # Indranil Ghosh, Jul 13 2017

Extended by Ray Chandler, Nov 17 2008

Offset changed to 1 by Michel Marcus, Jan 16 2014

A290965 Let n = p1^e1...pj^ej...pm^em be the prime factorization of n > 1, with the primes in increasing order and ej > 0. If for some j < m the sum p1^2 + ... + pj^2 > p(j+1), then n is in the sequence.

Original entry on oeis.org

6, 12, 15, 18, 21, 24, 30, 35, 36, 42, 45, 48, 54, 55, 60, 63, 65, 66, 70, 72, 75, 77, 78, 84, 85, 90, 91, 95, 96, 102, 105, 108, 110, 114, 115, 119, 120, 126, 130, 132, 133, 135, 138, 140, 143, 144, 147, 150, 154, 156, 161, 162, 165, 168, 170, 174, 175, 180, 182, 186, 187, 189, 190, 192, 195, 198, 203

Offset: 1

Richard Locke Peterson, Aug 15 2017

nonn

Sequence is a semigroup, since it is closed under multiplication, an associative operation--in fact, it is provably superclosed, i.e., a product of a term in sequence and an arbitrary number is a term in the sequence since the preexisting primes will still be in the new number.
Density: There are 28 terms in the sequence less than 100. Using WolframAlpha, 72% of numbers from 10^20 + 1 through 10^20 + 50 were found to be in the sequence.
Other facts: No primes or prime powers are in the sequence.
Related sequences: Some other sequences that are superclosed semigroups are the counting numbers, the numbers that are not squarefree, and the numbers with initial product in factorization greater than a later prime in the factorization. (See crossrefs.)

			6 = 2*3 is a term since 2^2 > 3.
1095 = 3*5*73 is a term because 3^2 > 5.
10, 20, and 100 are not terms since 2^2 < 5.
66 = 2*3*11 and 78 = 2*3*13 are terms since 2^2 + 3^2 > 11 and 2^2 + 3^2 = 13.
975560 = 2^3*5*29^3 is a term since 2^2 + 5^2 = 29.

Cf. A000027, A013929, A289484.

Select[Range@ 203, AnyTrue[Partition[FactorInteger[#][[All, 1]], 2, 1], #1^2 > #2 & @@ # &] &] (* Michael De Vlieger, Aug 17 2017 *)

A079877 Numbers that are divisible by at least one pair of twin primes (A001097).

Original entry on oeis.org

15, 30, 35, 45, 60, 70, 75, 90, 105, 120, 135, 140, 143, 150, 165, 175, 180, 195, 210, 225, 240, 245, 255, 270, 280, 285, 286, 300, 315, 323, 330, 345, 350, 360, 375, 385, 390, 405, 420, 429, 435, 450, 455, 465, 480, 490, 495, 510, 525, 540, 555, 560, 570, 572

Offset: 1

Reinhard Zumkeller, Feb 20 2003

Numbers that are divisible by both primes of some twin-prime pair. Harvey P. Dale, Aug 18 2017

By definition, if k is in the sequence, then so is every positive multiple of k. - Richard Locke Peterson, Aug 17 2017

			429 = 3*11*13 = 3*A001359(3)*A006512(3), therefore 429 is a term.

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

Cf. A001097, A001359, A006512, A037074, A062505, A074480, A289484.

Select[Range[600],MemberQ[Differences[Transpose[FactorInteger[#]][[1]]], 2]&] (* Harvey P. Dale, Sep 19 2011 *)

Definition clarified by N. J. A. Sloane, Aug 18 2017

Definition further clarified by Sean A. Irvine, Aug 29 2025

A289691 List of composite numbers n such that the prime decomposition of n contains an initial product of prime powers greater than a later prime, but in which no proper divisor of n is in the sequence.

Original entry on oeis.org

12, 30, 40, 45, 56, 63, 70, 105, 154, 165, 175, 176, 182, 195, 198, 208, 220, 231, 234, 260, 273, 275, 286, 297, 306, 325, 340, 351, 357, 374, 380, 385, 399, 418, 425

Offset: 1

Richard Locke Peterson, Jul 09 2017

nonn

From Richard Locke Peterson, Aug 29 2017: (Start)
Definition: Let the prime factorization of n be n = p1^e1*...*pj^ej*p(j+1)^e(j+1)*...*pm^em, with the primes in ascending order and the ej > 0. If in n there exists a partial product p1^e1*...*pk^ek > p(k+1) for some p(k+1) in the factorization, and also such that no proper divisor of n is also in the sequence, then n is in the sequence.
Context: This sequence is a subsequence of A289484, and might be called "A289484 primes," although not primes in the usual sense (nor do they obey a unique factorization law). Every number in A289484 is a multiple of at least one number in this sequence, and if n is in this sequence, then n and every multiple of n is in A289484 although no multiple of n(except n itself) will be in this sequence.
Properties: If n is in the sequence, then no multiple of n is in it, except n itself. No primes or prime powers, nor any composite powers, are in the sequence. A number in the sequence that is squarefree must have at least three prime divisors. Rather than being closed under multiplication, this sequence is anticlosed: No product or power of numbers in it are in the sequence. This causes it to be the minimal sequence that generates A289484. (End)

			30 = 2*3*5 is in the sequence since 2*3 > 5 and no proper divisor of 30 satisfies having an initial product greater than a later prime. Counterexamples: 42 = 2*3*7 is not in the sequence because 2*3 < 7. 60 = 2^2*3*5 is also not in the sequence, because although 2^2 > 3 (and 2^2*3 > 5, for that matter), the proper divisor 30 is already in the sequence.

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

Cf. A289484.

With[{S = Select[Range[10^3], Function[f, AnyTrue[Range[Length@ f - 1], Times @@ Map[#1^#2 & @@ # &, #1] > #2[[1, 1]] & @@ TakeDrop[f, #] &]]@ FactorInteger@ # &]}, Select[#, # > 0 &] &@ Table[Function[{d, n}, First[n] Boole@ NoneTrue[d, Divisible[First@ n, #] &]] @@ TakeDrop[Take[S, n], n - 1], {n, Length@ S}]] (* Michael De Vlieger, Aug 17 2017 *)

New name from Richard Locke Peterson, Aug 29 2017

A291046 Minimal multiplicative semigroup of numbers n > 1 such that in the prime factorization of n an initial product of primes is greater than a later prime in the factorization.

Original entry on oeis.org

30, 60, 70, 90, 105, 120, 140, 150, 154, 165, 180, 182, 195, 210, 231, 240, 270, 273, 280, 286, 300, 308, 315, 330, 350, 357, 360, 364, 374, 385, 390, 399, 418, 420, 429, 442, 450, 455, 462, 480, 490, 494, 495, 510, 525, 540, 546, 560, 561, 570, 572, 585, 595, 598, 600, 616, 627

Offset: 1

Richard Locke Peterson, Aug 16 2017

nonn

Definition: Let a number n>1 have prime factorization n=p1^e1*...*pi^ei*..*pm^em, with the primes written in ascending order and the ei>0. If an initial product p1*..*pi is greater than some later prime p(i+1), then n is in the sequence. The definition contains a more restrictive requirement than A289484 does, so it is a proper subsemigroup of A289484.  It can be seen that if s and t are in the sequence, the so is s*t. More strongly, if n is in the sequence, so is every multiple of n. Any number in it is divisible by at least 3 primes, although that is not a sufficient condition.
Differs from A212666 first at a(93), because 930=2*3*5*31 is in this sequence but not in A212666. - R. J. Mathar, Sep 02 2018
Numbers whose squarefree kernel (A007947) is in A164596. - Peter Munn, Feb 05 2024

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

Cf. A007947, A164596, A212666, A289484.

filter:= proc(n) local S,p,i;
  S:= sort(convert(numtheory:-factorset(n),list));
  p:= 1;
  for i from 1 to nops(S)-1 do
    p:= p*S[i];
    if p > S[i+1] then return true fi;
  od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 26 2018

A291045 Numbers with prime factorization such that the cube of a lesser prime in the factorization is greater than the square of a greater prime in the factorization.

Original entry on oeis.org

15, 30, 35, 45, 55, 60, 70, 75, 77, 90, 91, 105, 110, 119, 120, 135, 140, 143, 150, 154, 165, 175, 180, 182, 187, 195, 209, 210, 220, 221, 225, 231, 238, 240, 245, 247, 253, 255, 270, 273, 275, 280, 285, 286, 299, 300, 308, 315, 319, 323, 330, 341, 345, 350, 357, 360, 364, 374, 375, 377

Offset: 1

Richard Locke Peterson, Aug 16 2017

nonn

Definition rephrased: if n is a number with prime divisors p and q with p < q but p^3 > q^2, then n will be in the sequence, otherwise, not.

Sequence is a superclosed semigroup; that is, if s is in the sequence and x is any number, then x*s is in the sequence: if s in the sequence, there are primes p,q dividing s with p < q, p^3 > q^2, so p and q would also divide x*s.

			6 = 2*3 is not in the sequence since 2^3 < 3^2.
15 = 3*5 is in the sequence because 3^3 > 5^2.

Cf. A289484.

isA291045 := proc(n)
    local pdivs,i,j;
    pdivs := sort(convert(numtheory[factorset](n),list)) ;
    for i from 1 to nops(pdivs)-1 do
    for j from i+1 to nops(pdivs) do
        if op(i,pdivs)^3 > op(j,pdivs)^2 then
            return true;
        end if;
    end do:
    end do:
    false;
end proc:
for n from 1 to 400 do
    if isA291045(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 04 2017

Select[Range@ 400, AnyTrue[Partition[FactorInteger[#][[All, 1]], 2, 1], #1^3 > #2^2 & @@ # &] &] (* Michael De Vlieger, Aug 17 2017 *)

cp's OEIS Frontend

A291125 "Doubly A289484" numbers: numbers with prime factorization p1^e1 * p2^e2 * ... * pk^ek such that there exist i < j < k with p1^e1 * p2^e2 * ... pi^ei > p(i+1) and p1^e1 * p2^e2 * ... pj^ej > p(j+1).

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A292480 p-INVERT of the odd positive integers, where p(S) = 1 - S^2.

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A027854 Mutinous numbers: n > 1 such that n/p^k > p, where p is the largest prime dividing n and p^k is the highest power of p dividing n.

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A290965 Let n = p1^e1*...*pj^ej*...*pm^em be the prime factorization of n > 1, with the primes in increasing order and ej > 0. If for some j < m the sum p1^2 + ... + pj^2 > p(j+1), then n is in the sequence.

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Mathematica

A079877 Numbers that are divisible by at least one pair of twin primes (A001097).

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Mathematica

Extensions

A289691 List of composite numbers n such that the prime decomposition of n contains an initial product of prime powers greater than a later prime, but in which no proper divisor of n is in the sequence.

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Mathematica

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A291046 Minimal multiplicative semigroup of numbers n > 1 such that in the prime factorization of n an initial product of primes is greater than a later prime in the factorization.

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Programs

Maple

A291045 Numbers with prime factorization such that the cube of a lesser prime in the factorization is greater than the square of a greater prime in the factorization.

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A290965 Let n = p1^e1...pj^ej...pm^em be the prime factorization of n > 1, with the primes in increasing order and ej > 0. If for some j < m the sum p1^2 + ... + pj^2 > p(j+1), then n is in the sequence.