A368714 -id:A368714 - cp's OEIS frontend

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

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]));

A369939 Numbers whose maximal exponent in their prime factorization is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Amiram Eldar, Feb 06 2024

First differs from its subsequence A115063 at n = 2448. a(2448) = 2592 = 2^5 * 3^4 is not a term of A115063.
First differs from A209061 at n = 62.
Numbers k such that A051903(k) is a Fibonacci number.
The asymptotic density of this sequence is 1/zeta(4) + Sum_{k>=5} (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k))) = 0.94462177878047854647... .

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

Cf. A000045, A013662, A051903, A209061.
Subsequences: A005117, A062503, A062838, A113850, A115063.
Similar sequences: A368714, A369937, A369938.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}];
Select[Range[100], fibQ[Max[FactorInteger[#][[;; , 2]]]] &]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
is(n) = n == 1 || isfib(vecmax(factor(n)[, 2]));

A369937 Numbers whose maximal exponent in their prime factorization is square.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Amiram Eldar, Feb 06 2024

First differs from A366762 at n = 84, and from A197680, A361177 and A369210 at n = 95.
Numbers k such that A051903(k) is square.
The asymptotic density of this sequence is 1/zeta(2) + Sum_{k>=2} (1/zeta(k^2+1) - 1/zeta(k^2)) = 0.64939447949574562687... .

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

Cf. A000290, A013661, A051903.
Subsequences: A002416, A005117, A030514, A030635, A113849, A178739, A179665, A179692, A197680.
Similar sequences: A368714, A369938, A369939.
Cf. A361177, A366762, A369210.

Select[Range[100], IntegerQ@ Sqrt[Max[FactorInteger[#][[;; , 2]]]] &]

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

A374590 Numbers whose maximum exponent in their prime factorization is an evil number (A001969).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 64, 72, 88, 96, 104, 108, 120, 125, 135, 136, 152, 160, 168, 184, 189, 192, 200, 216, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 320, 328, 343, 344, 351, 352, 360, 375, 376, 378, 392, 408, 416, 424, 440, 448, 456, 459

Offset: 1

Amiram Eldar, Jul 12 2024

The asymptotic density of this sequence is Sum_{k in A001969} (1/zeta(k+1) - 1/zeta(k)) = 0.12101890210392912747... .

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

Cf. A001969, A051903.
Subsequence of A013929 and A262675 \ {1}.
Similar sequences: A368714, A369937, A369938, A369939, A374588, A374589.

evilQ[n_] := EvenQ[DigitCount[n, 2, 1]]; q[n_] := evilQ[Max[FactorInteger[n][[;; , 2]]]]; Select[Range[500], q]

is(n) = n > 1 && !(hammingweight(vecmax(factor(n)[, 2])) % 2);

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, ", ")));

A374324 The maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is even.

Original entry on oeis.org

0, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 6, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 6, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 8, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2

Offset: 1

Amiram Eldar, Jul 04 2024

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

Cf. A051903, A368714.

Similar sequences: A374325, A374326, A374327, A374328.

f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[EvenQ[e], e, Nothing]]; Array[f, 350]

lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(!(e % 2), print1(e, ", ")));}

a(n) = A051903(A368714(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (2*k * (1/zeta(2*k+1) - 1/zeta(2*k))) / Sum_{k>=2} (-1)^k * (1 - 1/zeta(k)) = 2.48584683692026915946... .

A376142 Nonsquarefree numbers whose prime factorization has a maximum exponent that is odd.

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, 440, 456, 459, 472, 480, 486, 488, 500

Offset: 1

Amiram Eldar, Sep 11 2024

Subsequence of A060476 and differs from it by not having the terms 1, 256, 768, 1280, 1792, 2304, ... .
Subsequence of A295661 and first differs from it at n = 51: A295661(51) = 432 is not a term of this sequence.
First differs from A325990 at n = 30: A325990(30) = 256 is not a term of this sequence.
Nonsquarefree numbers k such that A051903(k) is odd, or equivalently, numbers k such that A051903(k) is an odd number that is larger than 1.
The asymptotic density of this sequence is Sum_{k>=3} (-1)^(k+1) * (1 - 1/zeta(k)) = 0.11615617754774636364... .

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

Complement of A368714 within A013929.

Cf. A051903, A060476, A295661, A325990.

q[n_] := n > 1 && OddQ[n]; Select[Range[500], q[Max[FactorInteger[#][[;; , 2]]]] &]

is(k) = k > 1 && apply(x -> (x > 1 && x % 2), vecmax(factor(k)[, 2]));

A374588 Numbers whose maximum exponent in their prime factorization is a composite number.

Original entry on oeis.org

16, 48, 64, 80, 81, 112, 144, 162, 176, 192, 208, 240, 256, 272, 304, 320, 324, 336, 368, 400, 405, 432, 448, 464, 496, 512, 528, 560, 567, 576, 592, 624, 625, 648, 656, 688, 704, 720, 729, 752, 768, 784, 810, 816, 832, 848, 880, 891, 912, 944, 960, 976, 1008

Offset: 1

Amiram Eldar, Jul 12 2024

Subsequence of A322448 and first differs from it at n = 138: A322448(138) = 2592 = 2^5 * 3^4 is not a term of this sequence.

The asymptotic density of this sequence is d = Sum_{k composite} (1/zeta(k+1) - 1/zeta(k)) = 0.05296279266796920306... . The asymptotic density of this sequence within the nonsquarefree numbers (A013929) is d / (1 - 1/zeta(2)) = 0.13508404411123191108... .

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

Cf. A002808, A013661, A051903, A229099.
Complement of A074661 within A013929.
Subsequence of A322448 and A322449 \ {1}.
Similar sequences: A368714, A369937, A369938, A369939, A374589, A374590.

filter:= proc(n) local m;
  m:= max(ifactors(n)[2][..,2]);
  m > 1 and not isprime(m)
end proc:
select(filter, [$1..10000]); # Robert Israel, Jul 14 2024

Select[Range[1200], CompositeQ[Max[FactorInteger[#][[;; , 2]]]] &]

iscomposite(n) = n > 1 && !isprime(n);
is(n) = n > 1 && iscomposite(vecmax(factor(n)[, 2]));

A374589 Numbers whose maximum exponent in their prime factorization is a powerful number larger than 1.

Original entry on oeis.org

16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 256, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 512, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 768, 784, 810, 816, 848, 880, 891, 912, 944, 976, 1008, 1040, 1053, 1072, 1104, 1134, 1136, 1168, 1200

Offset: 1

Amiram Eldar, Jul 12 2024

Subsequence of A130897 and first differs from it at n = 115: A130897(115) = 2592 = 2^5 * 3^4 is not a term of this sequence.

The asymptotic density of this sequence is d = Sum_{k > 1 and in A001694} (1/zeta(k+1) - 1/zeta(k)) = 0.043523813088759413253... . The asymptotic density of this sequence within A130897 is d/(1 - A262276) = 0.98744988886705430331... .

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

Cf. A001694, A051903, A262276, A361177.
Subsequence of A013929, A130897 and A372405.
Similar sequences: A368714, A369937, A369938, A369939, A374588, A374590.

powQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 1; q[n_] := powQ[Max[ FactorInteger[n][[;; , 2]] ]]; Select[Range[1200], q]

ispow(n) = n > 1 && ispowerful(n);
is(n) = n > 1 && ispow(vecmax(factor(n)[, 2]))

A381950 Odd numbers whose prime factorization has an even maximum exponent.

Original entry on oeis.org

1, 9, 25, 45, 49, 63, 75, 81, 99, 117, 121, 147, 153, 169, 171, 175, 207, 225, 245, 261, 275, 279, 289, 315, 325, 333, 361, 363, 369, 387, 405, 423, 425, 441, 475, 477, 495, 507, 525, 529, 531, 539, 549, 567, 575, 585, 603, 605, 625, 637, 639, 657, 693, 711, 725

Offset: 1

Amiram Eldar, Mar 11 2025

Odd numbers k such that A051903(k) is even.

The asymptotic density of this sequence is (1/2) * Sum_{k>=2} (-1)^k * (1 - 2^k/((2^k-1)*zeta(k))) = 0.075617194130991839249... .

			9 = 3^2 is a term since it is odd and 2 is even.
45 = 3^2 * 5 is a term since it is odd and 2 is even.
125 = 5^3 is not a term since 3 is odd.

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

Intersection of A005408 and A368714.
Subsequence of A381956.
A381823 is a subsequence.
Cf. A051903, A375039, A381951.

Select[Range[1, 1000, 2], # == 1 || EvenQ[Max[FactorInteger[#][[;; , 2]]]] &]

isok(k) = if(k == 1, 1, k % 2 && !(vecmax(factor(k)[, 2]) % 2));

cp's OEIS Frontend

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

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A369939 Numbers whose maximal exponent in their prime factorization is a Fibonacci number.

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A369937 Numbers whose maximal exponent in their prime factorization is square.

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A374590 Numbers whose maximum exponent in their prime factorization is an evil number (A001969).

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

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A374324 The maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is even.

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A376142 Nonsquarefree numbers whose prime factorization has a maximum exponent that is odd.

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A374588 Numbers whose maximum exponent in their prime factorization is a composite number.

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