A363122 -id:A363122 - cp's OEIS frontend

A363123 Primitive terms of A363122: terms k of A363122 such that k/2 is not a term of A363122.

2, 12, 40, 56, 120, 144, 168, 176, 208, 280, 528, 544, 608, 624, 720, 736, 800, 840, 864, 880, 928, 992, 1008, 1040, 1232, 1456, 1584, 1632, 1824, 1872, 2208, 2288, 2368, 2400, 2624, 2640, 2720, 2752, 2784, 2976, 3008, 3040, 3120, 3136, 3392, 3680, 3696, 3776

Offset: 1

Amiram Eldar, May 16 2023

If k is a term of this sequence then k*2^m is a term of A363122 for any m >= 0.

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

Cf. A363122.

filter:= proc(n) local F2,Fp,v2,vp, t;
  F2, Fp:= selectremove(t -> t[1]=2, ifactors(n)[2]);
  if Fp = [] then return (n=2) fi;
  v2:= 2^F2[1,2];
  vp:= max(map(t -> t[1]^t[2],Fp));
  v2 > vp and v2/2 <= vp;
end proc:
select(filter, [seq(i,i=2.10000,2)]); # Robert Israel, May 18 2023

q[n_] := Module[{e = IntegerExponent[n, 2]}, 2^e > Max[Power @@@ FactorInteger[n/2^e]]]; Select[Range[2, 10000, 2], q[#] && ! q[#/2] &]

is1(n) = {my(e = valuation(n, 2), m = n>>e); if(m == 1, n>1, f = factor(m); 2^e > vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); } \\ A363122
is(n) = !(n%2) && is1(n) && !is1(n/2)

from itertools import count, islice
from sympy import factorint
def A363123_gen(startvalue=2): # generator of terms
    return filter(lambda n:(m:=n&-n)>max((p**e for p, e in factorint(n>>(~n&n-1).bit_length()).items()),default=1)>=m>>1,count(max(startvalue,2)))
A363123_list = list(islice(A363123_gen(),20)) # Chai Wah Wu, May 17 2023

A116882 A number k is included if (highest odd divisor of k)^2 <= k.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 40, 48, 56, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 288, 320, 352, 384, 416, 448, 480, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800, 832, 864, 896, 928, 960, 992, 1024, 1088, 1152, 1216, 1280, 1344, 1408

Offset: 1

Leroy Quet, Feb 24 2006

Also k is included if (and only if) the greatest power of 2 dividing k is >= the highest odd divisor of k. All terms of the sequence are even besides the 1.
Equivalently, positive integers of the form k*2^m, where odd k <= 2^m. - Thomas Ordowski, Oct 19 2014
If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence consists of 1 and all numbers without a superior odd divisor. - Gus Wiseman, Feb 18 2021
Numbers k such that A006519(k) >= A000265(k), with equality only when k = 1. - Amiram Eldar, Jan 24 2023

			40 = 8 * 5, where 8 is highest power of 2 dividing 40 and 5 is the highest odd dividing 40. 8 is >= 5 (so 5^2 <= 40), so 40 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 38.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Amelia Carolina Sparavigna, Discussion of the groupoid of Proth numbers (OEIS A080075), Politecnico di Torino, Italy (2019).

Cf. A000265, A006519, A080075, A112714.
The complement is A116883.
Positions of zeros (and 1) in A341675.
A051283 = numbers without a superior prime-power divisor (zeros of A341593).
A059172 = numbers without a superior squarefree divisor (zeros of A341592).
A063539 = numbers without a superior prime divisor (zeros of A341591).
A333805 counts strictly inferior odd divisors.
A341594 counts strictly superior odd divisors.
- Inferior: A033676, A038548, A063962, A066839, A069288, A161906, A217581.
- Superior: A033677, A063538, A070038, A072500, A161908, A341676.
- Strictly Inferior: A056924, A060775, A070039, A333806, A341596, A341674.
- Strictly Superior: A056924, A064052/A048098, A140271, A238535, A341642, A341643, A341673.
Cf. A000005, A000203, A001248, A006530, A020639, A026804, A027193, A340101, A340854 (zeros of A340832), A001008, A002805.
Subsequence of A082662, {1} U A363122.

f[n_] := Select[Divisors[n], OddQ[ # ] &][[ -1]]; Insert[Select[Range[2, 1500], 2^FactorInteger[ # ][[1]][[2]] > f[ # ] &], 1, 1] (* Stefan Steinerberger, Apr 10 2006 *)
q[n_] := 2^(2*IntegerExponent[n, 2]) >= n; Select[Range[1500], q] (* Amiram Eldar, Jan 24 2023 *)

isok(n) = vecmax(select(x->((x % 2)==1), divisors(n)))^2 <= n; \\ Michel Marcus, Sep 06 2016

isok(n) = 2^(valuation(n,2)*2) >= n \\ Jeppe Stig Nielsen, Feb 19 2019

from itertools import count, islice
def A116882_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(n&-n)**2>=n,count(max(startvalue,1)))
A116882_list = list(islice(A116882_gen(),20)) # Chai Wah Wu, May 17 2023

a(n) = A080075(n-1)-1. - Klaus Brockhaus, Georgi Guninski and M. F. Hasler, Aug 16 2010
a(n) ~ n^2/2. - Thomas Ordowski, Oct 19 2014
Sum_{n>=1} 1/a(n) = 1 + (3/4) * Sum_{k>=1} H(2^k-1)/2^k = 2.3388865091..., where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jan 24 2023

More terms from Stefan Steinerberger, Apr 10 2006

A363063 Positive integers k such that the largest power of p dividing k is larger than or equal to the largest power of q dividing k (i.e., A305720(k,p) >= A305720(k,q)) for all primes p and q with p < q.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 720, 768, 864, 1024, 1152, 1440, 1536, 1728, 2048, 2304, 2880, 3072, 3456, 4096, 4320, 4608, 5760, 6144, 6912, 8192, 8640, 9216, 10368, 11520, 12288, 13824, 16384, 17280, 18432

Offset: 1

Pontus von Brömssen, May 16 2023

nonn

Includes all products of terms in A347284, but there are also other terms such as 4320.

Closed under multiplication. - Peter Munn, May 21 2023

			151200 = 2^5 * 3^3 * 5^2 * 7 is a term, because 2^5 >= 3^3 >= 5^2 >= 7.
72 = 2^3 * 3^2 is not a term, because 2^3 < 3^2.
40 = 2^3 * 3^0 * 5 is not a term, because 3^0 < 5.
From _Michael De Vlieger_, May 19 2023: (Start)
Sequence read as an irregular triangle delimited by appearance of 2^m:
     1
     2
     4
     8     12
    16     24
    32     48
    64     96
   128    144    192
   256    288    384
   512    576    720    768    864
  1024   1152   1440   1536   1728
  2048   2304   2880   3072   3456
  4096   4320   4608   5760   6144   6912
  8192   8640   9216  10368  11520  12288  13824
  ... (End)

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot p^e | a(n) at (x,y) = (n, pi(p)), n = 1..1024, showing multiplicity e with a color function such that e = 1 is black, e = 2 is red, e = 3 is orange, etc., 12X vertical exaggeration. On the bottom, a color code represents a(n) is empty product (black), prime (red), composite prime power (gold), neither squarefree nor prime power (blue).
Michael De Vlieger, Plot multiplicities e in a(n) = Product p^e at (x,y) = (e, -n) for n = 1..1024, 8X horizontal exaggeration.

Cf. A181818, A305720, A347284, A363098 (primitive terms).

Subsequence of: A087980, {1} U A363122.

Select[Range[20000], # == 1 || PrimePi[(f = FactorInteger[#])[[-1, 1]]] == Length[f] && Greater @@ (Power @@@ f) &] (* Amiram Eldar, May 16 2023 *)

from sympy import nextprime
primes = [2] # global list of first primes
def f(kmax,pi,ppmax):
    # Generate numbers up to kmax with nonincreasing prime-powers <= ppmax, starting at the (pi+1)-st prime.
    if len(primes) <= pi: primes.append(nextprime(primes[-1]))
    p0 = primes[pi]
    ppmax = min(ppmax,kmax)
    if ppmax < p0:
        yield 1
        return
    pp = 1
    while pp <= ppmax:
        for x in f(kmax//pp,pi+1,pp):
            yield pp*x
        pp *= p0
def A363063_list(kmax):
    return sorted(f(kmax,0,kmax))

A085231 Numbers k in whose canonical factorization the power of the smallest prime factor is greater than the power of the greatest prime factor.

Original entry on oeis.org

12, 24, 40, 45, 48, 56, 63, 80, 96, 112, 120, 135, 144, 160, 168, 175, 176, 189, 192, 208, 224, 240, 275, 280, 288, 297, 315, 320, 325, 336, 351, 352, 360, 384, 405, 416, 425, 448, 459, 475, 480, 504, 513, 528, 539, 544, 560, 567, 575, 576, 608, 621, 624

Offset: 1

Reinhard Zumkeller, Jun 22 2003

p*a(n) is a term for all primes p with A020639(a(n)) < p < A006530(a(n)).

			The canonical factorization of 240 is 2^4 * 3 * 5. 2^4 = 16 > 5, therefore 240 is a term.

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

Cf. A006530, A020639, A028233, A053585, A085232, A363122.
A085233 is a subsequence.
Subsequence of A102749.

pfgQ[n_]:=Module[{fe=#[[1]]^#[[2]]&/@FactorInteger[n]},fe[[1]]>fe[[-1]]]; Select[Range[700],pfgQ] (* Harvey P. Dale, Dec 11 2017 *)

A028233(a(n)) > A053585(a(n)).

Edited by Peter Munn, Jun 01 2025

cp's OEIS Frontend

A363123 Primitive terms of A363122: terms k of A363122 such that k/2 is not a term of A363122.

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A116882 A number k is included if (highest odd divisor of k)^2 <= k.

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A363063 Positive integers k such that the largest power of p dividing k is larger than or equal to the largest power of q dividing k (i.e., A305720(k,p) >= A305720(k,q)) for all primes p and q with p < q.

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A085231 Numbers k in whose canonical factorization the power of the smallest prime factor is greater than the power of the greatest prime factor.

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Mathematica

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Extensions