A175496 -id:A175496 - cp's OEIS frontend

A372691 Numbers k such that k and k+1 are both nonsquarefree numbers whose number of divisors is a power of 2 (A175496).

135, 296, 343, 375, 999, 1160, 1431, 1592, 1624, 2295, 2375, 2456, 2727, 2943, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4887, 4913, 5048, 5144, 5319, 5480, 6183, 6344, 6375, 6858, 7479, 7624, 7640, 7749, 7911, 8072, 8375, 8936, 9207, 9368, 9624, 9855, 10071, 10232

Offset: 1

Amiram Eldar, May 10 2024

First differs from A176313 at n = 14.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 0, 0, 5, 43, 404, 4086, 40839, 408366, 4083039, 40830831, ... . Apparently, the asymptotic density of this sequence exists and equals 0.004083... .

			135 is a term since 135 = 3^3 * 5 and 136 = 2^3 * 17 are both nonsquarefree numbers, and the number of divisors of 135 and 136 are both 8 = 2^3.
343 is a term since 343 = 7^3 and 344 = 2^3 * 43 are both nonsquarefree numbers, the number of divisors of 343 is 4 = 2^2, and the number of divisors of 344 is 8 = 2^3.

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

Subsequence of A013929, A068781, A175496 and A372690.

Cf. A176313.

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; q[n_] := q[n] = Module[{e = FactorInteger[n][[;;, 2]]}, Max[e] > 1 && pow2Q[Times @@ (e+1)]]; Select[Range[500], q[#] && q[# + 1] &]

is(n) = {my(f = factor(n), d = numdiv(f)); n > 1 && vecmax(f[, 2]) > 1 && d >> valuation(d, 2) == 1;}
lista(kmax) = {my(is1 = is(1), is2); for(k = 2, kmax, is2 = is(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2);}

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

A337050 Numbers without an exponent 2 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87

Offset: 1

Amiram Eldar, Aug 12 2020

Numbers k such that the powerful part (A057521) of k is a cubefull number (A036966).
Numbers k such that A003557(k) = k/A007947(k) is a powerful number (A001694).
The asymptotic density of this sequence is Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596).
A304364 is apparently a subsequence.
These numbers were named semi-2-free integers by Suryanarayana (1971). - Amiram Eldar, Dec 29 2020

			6 = 2^1 * 3^1 is a term since none of the exponents in its prime factorization is equal to 2.
9 = 3^2 is not a term since it has an exponent 2 in its prime factorization.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
D. Suryanarayana, Semi-k-free integers, Elemente der Mathematik, Vol. 26 (1971), pp. 39-40.
D. Suryanarayana and R. Sitaramachandra Rao, Distribution of semi-k-free integers, Proceedings of the American Mathematical Society, Vol. 37, No. 2 (1973), pp. 340-346.

Complement of A038109.
Cf. A003557, A007947, A057521, A304364, A330596.
A005117, A036537, A036966, A048109, A175496, A268335 and A336590 are subsequences.
Numbers without an exponent k in their prime factorization: A001694 (k=1), this sequence (k=2), A386799 (k=3), A386803 (k=4), A386807 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 2: this sequence (m=0), A386796 (m=1), A386797 (m=2), A386798 (m=3).

q:= n-> andmap(i-> i[2]<>2, ifactors(n)[2]):
select(q, [$1..100])[];  # Alois P. Heinz, Aug 12 2020

Select[Range[100], !MemberQ[FactorInteger[#][[;;, 2]], 2] &]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 2, return(0))); 1; } \\ Amiram Eldar, Oct 21 2023

Sum_{n>=1} 1/a(n)^s = zeta(s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)), for s > 1. - Amiram Eldar, Oct 21 2023

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A372691 Numbers k such that k and k+1 are both nonsquarefree numbers whose number of divisors is a power of 2 (A175496).

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

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A337050 Numbers without an exponent 2 in their prime factorization.

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