A335275 -id:A335275 - cp's OEIS frontend

A357684 The squarefree part (A007913) of numbers whose squarefree part is a unitary divisor (A335275).

1, 2, 3, 1, 5, 6, 7, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 1, 26, 7, 29, 30, 31, 33, 34, 35, 1, 37, 38, 39, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 55, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 73, 74, 3, 19, 77, 78, 79

Offset: 1

Amiram Eldar, Oct 09 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts, International Mathematical Forum, Vol. 12, No. 7 (2017), pp. 331-338.

Cf. A000290, A005117, A007913, A008833, A065465, A069891, A335275.

s[n_] := If[AllTrue[(f = FactorInteger[n])[[;; , 2]], # == 1 || EvenQ[#] &], i = Position[f[[;; , 2]], 1] // Flatten; Times @@ f[[i, 1]], Nothing]; Array[s, 100]

s(n) = {my(f = factor(n), ans = 1); for(k = 1, #f~, if(f[k,2] > 1 && f[k,2]%2, ans = 0)); if(ans, ans = prod(k = 1, #f~, if(f[k,2] == 1, f[k,1], 1))) };
for(n = 1, 100, if(s(n) > 0, print1(s(n), ", ")))

a(n) = A007913(A335275(n)).
a(n) = 1 iff A335275(n) is a square (A000290).
a(n) = A335275(n) iff A335275(n) is squarefree (A005117).
Sum_{k, a(k) <= x} ~ c*x^2 + o(x^2), where c = (3/Pi^2) * Sum_{k>=1} f(k)/k^4 = 0.32103327852028541131..., and f(k) = Product_{p prime | k} (p/(p+1)) (Jakimczuk, 2017).
Sum_{k=1..n} a(k) ~ c'*x^2 + o(x^2), where c' = c / (A065465)^2 = 0.41313480468422995583... .

A350388 a(n) is the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 1, 25, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4

Offset: 1

Amiram Eldar, Dec 28 2021

First differs from A056623 at n = 32.

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

Cf. A000290, A001222, A008833, A056623, A071974, A076479, A191414, A268335, A335275, A350386, A350389.

f[p_, e_] := If[EvenQ[e], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, 1, f[i,1]^f[i,2]));} \\ Amiram Eldar, Oct 01 2023

Multiplicative with a(p^e) = p^e if e is even and 1 otherwise.
a(n) = n/A350389(n).
a(n) = A071974(n)^2.
a(n) = A008833(n) if and only if n is in A335275.
A001222(a(n)) = A350386(n).
a(n) = 1 if and only if n is an exponentially odd number (A268335).
a(n) = n if and only if n is a positive square (A000290 \ {0}).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (1/3) * Product_{p prime} (1 + sqrt(p)/(1 + p + p^2)) = 0.59317173657411718128... [updated Oct 16 2022]
Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) - 1/p^(3*s-2)). - Amiram Eldar, Oct 01 2023
Sum_{d|n, gcd(d, n/d) == 1} A076479(d) * a(n/d) = A191414(sqrt(n)) if n is a square, and 0 otherwise. - Amiram Eldar, Jun 01 2025

A369938 Numbers whose maximal exponent in their prime factorization is a power of 2.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77

Offset: 1

Amiram Eldar, Feb 06 2024

First differs from its subsequence A138302 \ {1} at n = 378: a(378) = 432 = 2^4 * 3^3 is not a term of A138302.
First differs from A096432, A220218 \ {1}, A335275 \ {1} and A337052 \ {1} at n = 56, and from A270428 \ {1} at n = 113.
Numbers k such that A051903(k) is a power of 2.
The asymptotic density of this sequence is 1/zeta(3) + Sum_{k>=2} (1/zeta(2^k+1) - 1/zeta(2^k)) = 0.87442038669659566330... .

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

Cf. A000079, A002117, A051903.
Subsequences: A001146, A001248, A005117, A030514, A030635, A050376, A062503, A113849, A138302 \ {1}, A179645.
Similar sequences: A368714, A369937, A369939.
Cf. A096432, A220218, A270428, A335275, A337052.

pow2Q[n_] := n == 2^IntegerExponent[n, 2];
Select[Range[2, 100], pow2Q[Max[FactorInteger[#][[;; , 2]]]] &]
Select[Range[2,80],IntegerQ[Log2[Max[FactorInteger[#][[;;,2]]]]]&] (* Harvey P. Dale, Nov 06 2024 *)

ispow2(n) = n >> valuation(n, 2) == 1;
is(n) = n > 1 && ispow2(vecmax(factor(n)[, 2]));

A295661 Numbers with at least one odd exponent larger than one in their prime factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456, 459, 472, 480, 486, 488, 500, 504, 512, 513, 520, 536, 540

Offset: 1

Antti Karttunen, Nov 28 2017

nonn

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.1184861602... (= 1 - A065465). - Amiram Eldar, May 18 2022

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

Positions of nonzero terms in A295662 and A295663.
Subsequence of A046099 (64 = 2^6, although a cube, is not in this sequence).
Differs from A060476 (256 = 2^8 is not a member of this sequence).
Complement of A335275.
Cf. A065465.

Select[Range[540], Count[FactorInteger[#][[All, -1]], ?(And[OddQ@ #, # > 1] &)] > 0 &] (* _Michael De Vlieger, Nov 28 2017 *)

from sympy import factorint
def ok(n):
    return max((e for e in factorint(n).values() if e%2), default=-1) > 1
print(list(filter(ok, range(541)))) # Michael S. Branicky, Aug 24 2021

A368167 The largest unitary divisor of n that is a cubefull exponentially odd number (A335988).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 14 2023

First differs from A056191 and A366126 at n = 32, and from A367513 at n = 64.
Also, the largest exponentially odd unitary divisor of the powerful part on n.
Also, the powerful part of the largest exponentially odd unitary divisor of n.

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

Cf. A057521, A077610, A335275, A335988, A350389, A368168, A368169.

Cf. A056191, A366126, A367513.

f[p_, e_] := If[e == 1 || EvenQ[e], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, f[i, 1]^f[i, 2]));}

Multiplicative with a(p^e) = p^e if e is odd that is larger than 1, and 1 otherwise.
a(n) = A350389(A057521(n)).
a(n) = A057521(A350389(n)).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= n, with equality if and only if n is in A335988.

A368168 The number of unitary divisors of n that are cubefull exponentially odd numbers (A335988).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 14 2023

First differs from A359411 and A367516 at n = 64.

Also, the number of unitary divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

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

Cf. A013661, A034444, A055076, A325837, A335275, A335988, A368167, A368169.

Cf. A359411, A367516.

f[p_, e_] := If[e == 1 || EvenQ[e], 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, 2));}

a(n) = A034444(A368167(n)).
Multiplicative with a(p^e) = 2 if e is odd that is larger than 1, and 1 otherwise.
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= n, with equality if and only if n is in A335988.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 1.12560687309375943599... .

A375032 The maximum odd exponent in the prime factorization of n, or 0 if no such exponent exists.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 3, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 0, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 0, 1, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Jul 28 2024

The asymptotic density of the occurrences of 0's is 0 (the asymptotic density of squares).
The asymptotic density of the occurrences of 1's is d(0) = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465, asymptotic density of A335275).
The asymptotic density of the occurrences of 2*k+1, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+2)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k)*(p+1))).

Cf. A000290, A051903, A065465, A162642, A335275, A350389, A375033.

a[n_] := Max[0, Max[Select[FactorInteger[n][[;; , 2]], OddQ]]]; a[1] = 0; Array[a, 100]

a(n) = {my(e = select(x -> (x % 2), factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

max(a(n), A375033(n)) = A051903(n).
a(n) = 0 if and only if n is a square (A000290).
a(n) = 1 if and only if n is in A335275 \ A000290.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} (2*k+1) * d(k) = 1.30000522546018852138..., where d(k) is defined in the Comments section above.
a(n) = A051903(A350389(n)). - Amiram Eldar, Aug 17 2024

A377020 Numbers whose prime factorization has exponents that are all numbers of the form m*k!, where 1 <= m <= k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset: 1

Amiram Eldar, Oct 13 2024

First differs from A138302 and A270428 at n = 57: a(57) = 64 is not a term of A138302 and A270428.
First differs from A337052 at n = 193: A337052(193) = 216 is not a term of this sequence.
First differs from A335275 at n = 227: A335275(227) = 256 is not a term of this sequence.
First differs from A220218 at n = 903: A220218(903) = 1024 is not a term of this sequence.
Numbers k such that A376886(k) = A001221(k).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + (1 - 1/p) * (Sum_{k>=3} 1/p^A051683(k))) = 0.87902453718626485582... .
a(n) = A096432(n-1) for 2<=n<380, but then the sequences start to differ: A096432 contains 432, 648, 1024, 1728, 2000, 2160,... which are not in this sequence. - R. J. Mathar, Oct 15 2024

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

Cf. A001221, A051683, A376886, A377021, A377022.
Subsequences: A005117, A004709, A377019.
Cf. A138302, A220218, A270428, A335275, A337052.

expQ[n_] := expQ[n] = Module[{m = n, k = 2}, While[Divisible[m, k], m /= k; k++]; m < k]; q[n_] := AllTrue[FactorInteger[n][[;;, 2]], expQ]; Select[Range[100], q]

isf(n) = {my(k = 2); while(!(n % k), n /= k; k++); n < k;}
is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isf(e[i]), return(0))); 1;}

A336615 Numbers of the form p * m^2, where p is prime and m > 0 is not divisible by p.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 112, 113, 116, 117, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153, 157, 162

Offset: 1

Amiram Eldar, Jul 27 2020

nonn

Numbers k such that A008833(k) is a unitary divisor of k and A007913(k) = k / A008833(k) is a prime number.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes, IX. On the set of integers representable as a product of a prime and square, Acta Arithmetica, Vol. 7 (1962), pp. 417-420.

Intersection of A229125 and A335275.
Cf. A007913, A008833, A013661.
Subsequences: A000040, A054753, A179643.

Select[Range[2, 200], Select[FactorInteger[#][[;;, 2]], OddQ] == {1} &]

from math import isqrt
from sympy import primepi, primefactors
def A336615(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(m:=x//y**2)-sum(1 for p in primefactors(y) if p<=m) for y in range(1,isqrt(x)+1))
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

The number of terms not exceeding x is (Pi^2/6) * x/log(x) + O(x/(log(x))^2) (Cohen, 1962).

A337052 Numbers k such that the powerful part of k has an even number of prime divisors counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset: 1

Amiram Eldar, Aug 12 2020

nonn

Numbers k such that A001222(A057521(k)) == 0 (mod 2).
Numbers k such that A057521(k) is in A028260.
Differs from A096432 by having the additional terms 1 and 216, 256, 768, 864, ... and not having the terms 432, 648, ...
First differs from both A220218 and A335275 at n = 193: a(193) = 216 is not a term of these two sequences.
Cohen (1964) proved that this sequence has an asymptotic density, and gave the value 1/2 + (1/5) * Product_{p prime} (1 + (p^2 + p + 1)/(p^3 * (p + 1))) = 0.8172707179... But the numbers of terms not exceeding 10^k for k = 1, 2, ... are 9, 90, 885, 8849, 88499, 884993, 8849889, 88498711, 884987643, 8849876178, ... indicating that the asymptotic density is about 0.88498...

			2 is a term since the powerful part of 2 is 1, which has 0 prime divisors, and 0 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112, No. 2 (1964), pp. 214-227. See corollary 4.2.2, pp. 226-227.

Cf. A001222, A028260, A057521, A096432.

Subsequences: A000290, A004709, A005117, A138302, A220218, A335275 and A336615.

Select[Range[100], EvenQ @ Total @ Select[FactorInteger[#][[;; , 2]], #1 > 1 &] &]

cp's OEIS Frontend

A357684 The squarefree part (A007913) of numbers whose squarefree part is a unitary divisor (A335275).

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A350388 a(n) is the largest unitary divisor of n that is a square.

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A369938 Numbers whose maximal exponent in their prime factorization is a power of 2.

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A295661 Numbers with at least one odd exponent larger than one in their prime factorization.

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A368167 The largest unitary divisor of n that is a cubefull exponentially odd number (A335988).

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A368168 The number of unitary divisors of n that are cubefull exponentially odd numbers (A335988).

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A375032 The maximum odd exponent in the prime factorization of n, or 0 if no such exponent exists.

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A377020 Numbers whose prime factorization has exponents that are all numbers of the form m*k!, where 1 <= m <= k.

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