A096432 -id:A096432 - cp's OEIS frontend

A368714 Numbers whose maximal exponent in their prime factorization is even.

1, 4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 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, 204, 207, 208

Offset: 1

Amiram Eldar, Jan 04 2024

First differs from A240112 at n = 30.
Numbers k such that A051903(k) is even.
The asymptotic density of this sequence is Sum_{k>=2} (-1)^k * (1 - 1/zeta(k)) = 0.27591672059822700769... .

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

Cf. A051903, A240112.
Subsequences: A000290, A000302, A000583, A001014, A001016, A001025, A008454, A062503, A067259.
Similar sequences: A060476, A074661, A096432, A336064, A368715.

Select[Range[210], # == 1 || EvenQ[Max[FactorInteger[#][[;;, 2]]]] &]

lista(kmax) = for(k = 1, kmax, if(k == 1 || !(vecmax(factor(k)[,2])%2), print1(k, ", ")));

A335275 Numbers k such that the largest square dividing k is a unitary divisor of 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, Jul 06 2020

nonn

Numbers k such that gcd(A008833(k), k/A008833(k)) = 1.
Numbers whose prime factorization contains exponents that are either 1 or even.
Numbers whose powerful part (A057521) is a square.
First differs from A220218 at n = 227: a(227) = 256 is not a term of A220218.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465).
Complement of A295661. - Vaclav Kotesovec, Jul 07 2020
Differs from A096432 in having or not having 1, 256, 432, 648, 768, 1280, 1728, 1792, 2000, 2160, 2304,... - R. J. Mathar, Jul 22 2020
Equivalently, numbers k whose squarefree part (A007913) is a unitary divisor, or gcd(A007913(k), A008833(k)) = 1. - Amiram Eldar, Oct 09 2022

			12 is a term since the largest square dividing 12 is 4, and 4 and 12/4 = 3 are coprime.

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 3.1.2, p. 222.

A000290, A138302 and A220218 are subsequences.

Cf. A007913, A008833, A057521, A065465, A295661.

seqQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # == 1 || EvenQ[#] &];  Select[Range[100], seqQ]

isok(k) = my(d=k/core(k)); gcd(d, k/d) == 1; \\ Michel Marcus, Jul 07 2020

A060476 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is nonprime.

Original entry on oeis.org

1, 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, 256, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 440, 456, 459, 472, 480

Offset: 1

N. J. A. Sloane, Sep 18 2008

nonn

The old entry with this sequence number was a duplicate of A005171.

The asymptotic density of this sequence is Sum_{c composite} (1/zeta(c) - 1/zeta(c-1)) = 0.1182437806... - Amiram Eldar, Oct 18 2020

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

Cf. A096432, A074661.

Cf. A010051, A051903, A055229.

a060476 n = a060476_list !! (n-1)
a060476_list = filter ((== 0) . a010051' . (+ 1) . a051903) [1..]
-- Reinhard Zumkeller, Nov 30 2015

Join[{1}, Select[Range[500], !PrimeQ[1+Max[FactorInteger[#][[All, 2]]]]&]] (* Jean-François Alcover, Aug 02 2018 *)

isA060476(n) = if(n<2,1,!isprime(vecmax(factor(n)[,2])+1))

From Reinhard Zumkeller, Nov 30 2015: (Start)
A010051(A051903(a(n)+1)) = 1.
a(A055229(n)) > 1 for n > 1. (End)

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]));

A074661 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that max{e_2, e_3, ...} is prime.

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 164, 168, 169, 171, 172, 175, 180, 184

Offset: 1

N. J. A. Sloane, Sep 18 2008

nonn

The old entry with this sequence number was a duplicate of A056594.
The largest exponent of the prime factors of n is prime. - Harvey P. Dale, Mar 09 2012
The asymptotic density of this sequence is Sum_{p prime} (1/zeta(p+1) - 1/zeta(p)) = 0.3391101054... - Amiram Eldar, Oct 18 2020

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

Cf. A096432, A060476.

Select[Range[200],PrimeQ[Max[Transpose[FactorInteger[#]][[2]]]]&] (* Harvey P. Dale, Mar 09 2012 *)

isA074661(n) = if(n<4,0,isprime(vecmax(factor(n)[,2])))

A220218 Numbers where all exponents in its prime factorization are one less than a prime.

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, 77, 78, 79, 80

Offset: 1

Charles R Greathouse IV, Dec 07 2012

nonn

Sequence has positive density, between 0.83 and 0.89; probably about 0.87951.
The numbers of terms not exceeding 10^k, for k=1,2,..., are 9, 90, 880, 8796, 87956, 879518, 8795126, 87951173, 879511794, ... The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + Sum_{q prime >= 5} (p-1)/p^q) = 0.87951176583716527413... - Amiram Eldar, Mar 20 2021
Numbers whose sets of unitary divisors (A077610) and modified exponential divisors (A379027) coincide. - Amiram Eldar, Dec 14 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Paul Erdős and Leon Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc., Vol. s3-2, No. 1 (1952), pp. 257-271; alternative link.

Apart from the first term, a subsequence of A096432.

Cf. A010051, A124010, A077610, A379027.

a220218 n = a220218_list !! (n-1)
a220218_list = 1 : filter
               (all (== 1) . map (a010051' . (+ 1)) . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 30 2015

Select[Range[100],AllTrue[Transpose[FactorInteger[#]][[2]]+1,PrimeQ]&] (* Harvey P. Dale, Sep 29 2014 *)

is(n)=vecmin(apply(n->isprime(n+1),factor(max(n,2))[,2])) \\ Charles R Greathouse IV, Dec 07 2012

A368715 Numbers that are not coprime to the maximal exponent in their prime factorization.

Original entry on oeis.org

4, 12, 16, 18, 20, 24, 27, 28, 36, 44, 48, 50, 52, 54, 60, 64, 68, 72, 76, 80, 84, 90, 92, 98, 100, 108, 112, 116, 120, 124, 126, 132, 135, 140, 144, 148, 150, 156, 160, 162, 164, 168, 172, 176, 180, 188, 189, 192, 196, 198, 204, 208, 212, 216, 220, 228, 234, 236, 240, 242, 244

Offset: 1

Amiram Eldar, Jan 04 2024

Subsequence of A137257 and first differs from it at n = 51.
Numbers k such that gcd(k, A051903(k)) > 1.
Includes all the nonsquarefree terms of A336064.
The asymptotic density of this sequence is 1 - 1/zeta(2) - Sum_{k>=2} (1/(f(k+1, k) * zeta(k+1)) - 1/(f(k, k) * zeta(k))) = 0.24998449199080279703..., where f(e, m) = Product_{primes p|m} ((1-1/p^e)/(1-1/p)).

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

Cf. A051903.
Subsequence of A013929 and A137257.
Similar sequences: A060476, A074661, A096432, A336064, A368714.

Select[Range[210], !CoprimeQ[#, Max[FactorInteger[#][[;;, 2]]]] &]

lista(kmax) = for(k = 2, kmax, if(gcd(k, vecmax(factor(k)[,2])) > 1, print1(k, ", ")));

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;}

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 &] &]

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A368714 Numbers whose maximal exponent in their prime factorization is even.

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A335275 Numbers k such that the largest square dividing k is a unitary divisor of k.

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A060476 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is nonprime.

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

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A074661 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that max{e_2, e_3, ...} is prime.

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A220218 Numbers where all exponents in its prime factorization are one less than a prime.

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A368715 Numbers that are not coprime to the maximal exponent in their prime factorization.

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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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