A162643 -id:A162643 - cp's OEIS frontend

A282900 Least non-infinitary divisor of A162643(n).

2, 3, 2, 2, 3, 2, 5, 2, 4, 2, 2, 3, 2, 7, 5, 2, 2, 3, 2, 2, 3, 5, 2, 2, 3, 2, 3, 2, 4, 7, 3, 2, 2, 2, 2, 3, 11, 2, 3, 2, 2, 2, 7, 2, 5, 3, 2, 4, 3, 2, 13, 3, 2, 5, 2, 2, 2, 2, 2, 3

Offset: 1

Vladimir Shevelev, Feb 24 2017

nonn

Let n=q_1*...*q_t, where q_i are distinct increasing terms of A050376. This representation is unique (for n=1 the product is empty). Every subproduct is an infinitary divisor of n. All numbers having at least one non-infinitary divisor form A162643.

			For n=60=3*4*5, no subproduct is 2,6,10,30. They are all non-infinitary divisors of 60. Since 60=A162643(17) then a(17) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Infinitary Divisor

Cf. A000005, A036537, A037445, A050376, A162643.

Map[First@ Complement[Divisors@ #, If[# == 1, {1}, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[#] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]] &, Select[Range@ 198, ! IntegerQ@ Log2@ DivisorSigma[0, #] &]] (* Michael De Vlieger, Feb 24 2017, after Paul Abbott at A077609 *)

A282940 Largest non-infinitary divisor of A162643(n) having no non-infinitary divisors.

Original entry on oeis.org

2, 3, 6, 8, 6, 10, 5, 14, 8, 6, 22, 15, 24, 7, 10, 26, 30, 21, 8, 34, 24, 15, 38, 40, 27, 42, 30, 46, 24, 14, 33, 10, 54, 56, 58, 39, 11, 62, 42, 66, 70, 24, 21, 74, 30, 51, 78, 40, 54, 82, 13, 57, 86, 35, 88, 30, 94, 24, 14, 66, 40, 102, 69, 104, 106, 110, 56

Offset: 1

Vladimir Shevelev, Feb 25 2017

nonn

Or largest term of A036537 dividing A162643(n) (cf. our comment in A036537).

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

Cf. A000005, A036537, A162643, A282900, A327839, A372379.

f[p_, e_] := p^(2^Floor[Log2[e + 1]] - 1); s[1] = Nothing; s[n_] := Module[{v = Times @@ f @@@ FactorInteger[n]}, If[v == n, Nothing, v]]; Array[s, 300] (* Amiram Eldar, Apr 29 2024 *)

a(n) = A036537(m), where m = max{k: A036537(k)|A162643(n)}.
From Amiram Eldar, Apr 29 2024: (Start)
a(n) = A372379(A162643(n)).
Sum_{k=1..n} a(k) ~ ((c-d)/(1-d)^2) * n^2 / 2, where d = A327839 and c = 0.7907361848... is the constant in the asymptotic formula in A372379. (End)

More terms from Peter J. C. Moses, Feb 25 2017

A036537 Numbers whose number of divisors is a power of 2.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Labos Elemer

nonn

Primes and A030513(d(x)=4) are subsets; d(16k+4) and d(16k+12) have the form 3Q, so x=16k+4 or 16k-4 numbers are missing.
A number m is a term if and only if all its divisors are infinitary, or A000005(m) = A037445(m). - Vladimir Shevelev, Feb 23 2017
All exponents in the prime number factorization of a(n) have the form 2^k-1, k >= 1. So it is an S-exponential sequence (see Shevelev link) with S={2^k-1}. Using Theorem 1, we obtain that a(n) ~ C*n, where C = Product((1-1/p)*(1 + Sum_{i>=1} 1/p^(2^i-1))). - Vladimir Shevelev Feb 27 2017
This constant is C = 0.687827... . - Peter J. C. Moses, Feb 27 2017
From Peter Munn, Jun 18 2022: (Start)
1 and numbers j*m^2, j squarefree, m >= 1, such that all prime divisors of m divide j, and m is in the sequence.
Equivalently, the nonempty set of numbers whose squarefree part (A007913) and squarefree kernel (A007947) are equal, and whose square part's square root (A000188) is in the set.
(End)

			383, 384, 385, 386 have 2, 16, 8, 4 divisors, respectively, so they are consecutive terms of this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, S-exponential numbers, Acta Arithm., 175(2016), 385-395.
Eric Weisstein's World of Mathematics, Square Part, Squarefree Part

A005117, A030513, A058891, A175496, A336591 are subsequences.
Complement of A162643; subsequence of A002035. - Reinhard Zumkeller, Jul 08 2009
Subsequence of A162644, A337533.
Cf. A000005, A000188, A007913, A007947, A036538, A352780.
The closure of the squarefree numbers under application of A355038(.) and lcm.

a036537 n = a036537_list !! (n-1)
a036537_list = filter ((== 1) . a209229 . a000005) [1..]
-- Reinhard Zumkeller, Nov 15 2012

bi[ x_ ] := 1-Sign[ N[ Log[ 2, x ], 5 ]-Floor[ N[ Log[ 2, x ], 5 ] ] ]; ld[ x_ ] := Length[ Divisors[ x ] ]; Flatten[ Position[ Table[ bi[ ld[ x ] ], {x, 1, m} ], 1 ] ]
Select[Range[110],IntegerQ[Log[2,DivisorSigma[0,#]]]&] (* Harvey P. Dale, Nov 20 2016 *)

is(n)=n=numdiv(n);n>>valuation(n,2)==1 \\ Charles R Greathouse IV, Mar 27 2013

isok(m) = issquarefree(m) || (omega(m) == omega(core(m)) && isok(core(m,1)[2])); \\ Peter Munn, Jun 18 2022

from itertools import count, islice
from sympy import factorint
def A036537_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda m:not((k:=m+1)&-k)^k,factorint(n).values())),count(max(startvalue,1)))
A036537_list = list(islice(A036537_gen(),30)) # Chai Wah Wu, Jan 04 2023

A209229(A000005(a(n))) = 1. - Reinhard Zumkeller, Nov 15 2012
a(n) << n. - Charles R Greathouse IV, Feb 25 2017
m is in the sequence iff for k >= 0, A352780(m, k+1) | A352780(m, k)^2. - Peter Munn, Jun 18 2022

A072587 Numbers having at least one prime factor with an even exponent.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 108, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 200, 204

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

Complement of the union of {1} and A002035. [Correction, Nov 15 2012]
A162645 is a subsequence and this sequence is a subsequence of A162643. - Reinhard Zumkeller, Jul 08 2009
The asymptotic density of this sequence is 1 - A065463 = 0.2955577990... - Amiram Eldar, Jul 21 2020
A number k is a term iff its core (A007913) properly divides its kernel (A007947), that is iff A336643(k) > 1. - David James Sycamore, Sep 18 2023

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

Cf. A000037, A000203, A002035, A065463, A072588, A124010, A162643, A162645, A188999.

Cf. A007913, A007947, A336643.

a072587 n = a072587_list !! (n-1)
a072587_list = tail $ filter (any even . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 15 2012

Select[Range[210], MemberQ[EvenQ[Transpose[FactorInteger[#]][[2]]], True] &] (* Harvey P. Dale, Apr 03 2012 *)

is(n)=n>3 && Set(factor(n)[,2]%2)[1]==0 \\ Charles R Greathouse IV, Oct 16 2015

from itertools import count, islice
from sympy import factorint
def A072587_gen(startvalue=1): # generator of terms
    return filter(lambda n:not all(map(lambda m:m&1,factorint(n).values())),count(max(startvalue,1)))
A072587_list = list(islice(A072587_gen(),30)) # Chai Wah Wu, Jan 04 2023

Thanks to Zak Seidov, who noticed that 1 had to be removed. - Reinhard Zumkeller, Nov 15 2012

A348271 a(n) is the sum of noninfinitary divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 0, 14, 0, 9, 0, 12, 0, 0, 0, 0, 5, 0, 0, 16, 0, 0, 0, 12, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 0, 24, 18, 0, 0, 56, 7, 15, 0, 28, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 24, 42, 0, 0, 0, 36, 0, 0, 0, 45, 0, 0, 20, 40, 0, 0, 0, 84, 39

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

			a(12) = 8 since 12 has 2 noninfinitary divisors, 2 and 6, and 2 + 6 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A000203, A036537, A049417, A077609, A162643, A327633.

Similar sequences: A034448, A048146, A051377, A188999.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; a[n_]:= DivisorSigma[1,n] - isigma[n]; Array[a, 100]

a(n) = A000203(n) - A049417(n).
a(n) = 0 if and only if the number of divisors of n is a power of 2, (i.e., n is in A036537).
a(n) > 0 if and only if the number of divisors of n is not a power of 2, (i.e., n is in A162643).

A162511 Multiplicative function with a(p^e) = (-1)^(e-1).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -1

Offset: 1

Gerard P. Michon, Jul 05 2009

G. C. Greubel, Table of n, a(n) for n = 1..5000
Gérard P. Michon, Multiplicative functions.

Cf. A005117, A076479, A162510, A162512, A002035, A072587, A036537, A162643, A162644, A162645, A046660, A008836, A307868.

A162511 := proc(n)
    local a,f;
    a := 1;
    for f in ifactors(n)[2] do
        a := a*(-1)^(op(2,f)-1) ;
    end do:
    return a;
end proc: # R. J. Mathar, May 20 2017

a[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]); Array[a, 100] (* Jean-François Alcover, Apr 24 2017, after Reinhard Zumkeller *)

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,-(-1)^f[i]) \\ Charles R Greathouse IV, Mar 09 2015

from sympy import factorint
from operator import mul
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(-1)**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017

from functools import reduce
from sympy import factorint
def A162511(n): return -1 if reduce(lambda a,b:~(a^b), factorint(n).values(),0)&1 else 1 # Chai Wah Wu, Jan 01 2023

Multiplicative function with a(p^e)=(-1)^(e-1) for any prime p and any positive exponent e.
a(n) = 1 when n is a squarefree number (A005117).
From Reinhard Zumkeller, Jul 08 2009 (Start)
a(n) = (-1)^(A001222(n)-A001221(n)).
a(A162644(n)) = +1; a(A162645(n)) = -1. (End)
a(n) = A076479(n) * A008836(n). - R. J. Mathar, Mar 30 2011
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A307868. - Amiram Eldar, Sep 18 2022
Dirichlet g.f.: Product_{p prime} ((p^s + 2)/(p^s + 1)). - Amiram Eldar, Oct 26 2023

A348341 a(n) is the number of noninfinitary divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 6, 1, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 3, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 2, 2, 0, 0, 0, 6, 3, 0, 0, 4, 0, 0, 0

Offset: 1

Amiram Eldar, Oct 13 2021

nonn

			a(4) = 1 since 4 has one noninfinitary divisor, 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A000005, A036537, A037445, A077609, A162643, A327576, A348271.

Similar sequences: A034444, A048105, A286324, A049419.

a[1] = 0; a[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; Array[a, 100]

A348341(n) = (numdiv(n)-factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2]))); \\ Antti Karttunen, Oct 13 2021

a(n) = A000005(n) - A037445(n).
a(n) = 0 if and only if the number of divisors of n is a power of 2, (i.e., n is in A036537).
a(n) > 0 if and only if the number of divisors of n is not a power of 2, (i.e., n is in A162643).
Sum_{k=1..n} a(k) ~ c * n * log(n), where c = (1 - 2 * A327576) = 0.266749... . - Amiram Eldar, Dec 09 2022

A162645 Numbers m such that A162511(m) = -1.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 98, 99, 108, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228

Offset: 1

Reinhard Zumkeller, Jul 08 2009

nonn

Numbers n where A001222(n)-A001221(n) is odd. - Enrique Pérez Herrero, Jul 07 2012

This sequence has an asymptotic density (1 - A065472/zeta(2))/2 = 0.264159... (Mossinghoff and Trudgian, 2019). - Amiram Eldar, Jul 07 2020

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.

Complement of A162644.
Subsequence of A072587.
Cf. A001221, A001222, A065472, A162643, A162511.

Select[Range[10^2],OddQ[PrimeOmega[#]-PrimeNu[#]]&] (* Enrique Pérez Herrero, Jul 07 2012 *)

A372379 The largest divisor of n whose number of divisors is a power of 2.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 8, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 8, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 24, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70

Offset: 1

Amiram Eldar, Apr 29 2024

First differs from A350390 at n = 32.
The largest term of A036537 dividing n.
The largest divisor of n whose exponents in its prime factorization are all of the form 2^k-1 (A000225).

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

Cf. A000225, A000523, A036537, A138302, A162643, A282940, A350390, A353897, A372380, A372381.

f[p_, e_] := p^(2^Floor[Log2[e + 1]] - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2^exponent(f[i, 2]+1)-1));}

from math import prod
from sympy import factorint
def A372379(n): return prod(p**((1<<(e+1).bit_length()-1)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Apr 30 2024

Multiplicative with a(p^e) = p^(2^floor(log_2(e+1)) - 1).
a(n) = n if and only if n is in A036537.
a(A162643(n)) = A282940(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 0.7907361848... = Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = 2^floor(log_2(k))-1 for k >= 1, and f(0) = 0.

A337534 Nontrivial squares together with nonsquares whose square part's square root is in the sequence.

Original entry on oeis.org

4, 9, 16, 25, 32, 36, 48, 49, 64, 80, 81, 96, 100, 112, 121, 144, 160, 162, 169, 176, 196, 208, 224, 225, 240, 243, 256, 272, 289, 304, 324, 336, 352, 361, 368, 400, 405, 416, 441, 464, 480, 484, 486, 496, 512, 528, 529, 544, 560, 567, 576, 592, 608, 624, 625

Offset: 1

Peter Munn, Aug 31 2020

The appearance of a number is determined by its prime signature.
No terms are squarefree, as the square root of the square part of a squarefree number is 1.
If the square part of k is a 4th power, other than 1, k appears.
Every positive integer k is the product of a unique subset S_k of the terms of A050376, which are arranged in array form in A329050 (primes in column 0, squares of primes in column 1, 4th powers of primes in column 2 and so on). k is in this sequence if and only if there is m >= 1 such that column m of A329050 contains a member of S_k, but column m - 1 does not.

			4 is square and nontrivial (not 1), so 4 is in the sequence.
12 = 3 * 2^2 is nonsquare, but has square part 4, whose square root (2) is not in the sequence. So 12 is not in the sequence.
32 = 2 * 4^2 is nonsquare, and has square part 16, whose square root (4) is in the sequence. So 32 is in the sequence.

Eric Weisstein's World of Mathematics, Square part
Index to sequences related to prime signature

Complement of A337533.
Subsequences: A000290\{0,1}, A082294.
Subsequence of: A013929, A162643.
A209229, A267116 are used in a formula defining this sequence.
Cf. A008833, A050376, A329050.

A337534 := proc(n)
    option remember ;
    if n =1  then
        4;
    else
        for a from procname(n-1)+1 do
            if A209229(A267116(a)+1) = 0 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A337534(n),n=1..80) ; # R. J. Mathar, Feb 16 2021

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[625], ! pow2Q[1 + BitOr @@ (FactorInteger[#][[;; , 2]])] &] (* Amiram Eldar, Sep 18 2020 *)

Numbers k such that A209229(A267116(k) + 1) = 0.

A008833(a(n)) > 1.

cp's OEIS Frontend

A282900 Least non-infinitary divisor of A162643(n).

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A282940 Largest non-infinitary divisor of A162643(n) having no non-infinitary divisors.

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A036537 Numbers whose number of divisors is a power of 2.

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A072587 Numbers having at least one prime factor with an even exponent.

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A348271 a(n) is the sum of noninfinitary divisors of n.

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A162511 Multiplicative function with a(p^e) = (-1)^(e-1).

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A348341 a(n) is the number of noninfinitary divisors of n.

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