A369939 -id:A369939 - 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]));

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, ", ")));

A386803 Numbers without an exponent 4 in their prime factorization.

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, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Amiram Eldar, Aug 03 2025

First differs from its subsequence A209061 at n = 246: a(246) = 256 = 2^8 is not a term of A209061.
First differs from its subsequences A115063 and A369939 at n = 62: a(62) = 64 = 2^6 is not a term of A115063.
The complement of this sequence is a subsequence of A336595.
These numbers were named semi-4-free integers by Suryanarayana (1971).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^4 + 1/p^5) = 0.95908865419555719109... (Suryanarayana, 1971).

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.
Index entries for sequences computed from indices in prime factorization.

Subsequences: A115063, A209061, A369939.
Numbers without an exponent k in their prime factorization: A001694 (k=1), A337050 (k=2), A386799 (k=3), this sequence (k=4), A386807 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 4: this sequence (m=0), A386804 (m=1), A386805 (m=2), A386806 (m=3).

Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 4] &]

isok(k) = vecsum(apply(x -> if(x == 4, 1, 0), factor(k)[, 2])) == 0;

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);

A375768 The maximum exponent in the prime factorization of the numbers whose maximum exponent in their prime factorization is a Fibonacci number.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1

Offset: 1

Amiram Eldar, Aug 27 2024

First differs from A375766 at n = 2448.

All the terms are Fibonacci numbers by definition.

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

Cf. A000045, A051903, A369939, A375766, A375769.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]; s[n_] := Module[{e = Max[FactorInteger[n][[;; , 2]]]}, If[fibQ[e], e, Nothing]]; s[1] = 0; Array[s, 100]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(isfib(e), print1(e, ", ")));}

a(n) = A051903(A369939(n)).
a(n) = A000045(A375769(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/zeta(2) + Sum_{k>=3} Fibonacci(k) * (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k)))) / d = 1.52660290991620063268..., where d = 1/zeta(4) + Sum_{k>=5} (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k))) = 0.94462177878047854647... is the density of A369939.

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]))

A375769 The indices of the terms of A375768 in the Fibonacci sequence.

Original entry on oeis.org

0, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 4, 3, 2, 4, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 5, 2, 3, 3, 3, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2

Offset: 1

Amiram Eldar, Aug 27 2024

First differs from A375767 at n = 2448.

Since 1 appears twice in the Fibonacci sequence (1 = Fibonacci(1) = Fibonacci(2)), its index here is chosen to be 2.

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

Cf. A000045, A130233, A369939, A375767, A375768.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]; A130233[n_] := Module[{k = 2}, While[Fibonacci[k] <= n, k++]; k-1]; s[n_] := Module[{e = Max[FactorInteger[n][[;; , 2]]]}, If[fibQ[e], A130233[e], Nothing]]; s[1] = 0; Array[s, 100]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
A130233(n) = {my(k = 2); while(fibonacci(k) <= n, k++); k-1;}
lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(isfib(e), print1(A130233(e), ", ")));}

a(n) = A130233(A375768(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (2/zeta(2) + Sum_{k>=3} k * (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k)))) / d = 2.4999593748274972257073..., where d = 1/zeta(4) + Sum_{k>=5} (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k))) = 0.94462177878047854647... is the density of A369939.
If the chosen index for 1 is 1 instead of 2, then the asymptotic mean is (1/zeta(2) + Sum_{k>=3} k * (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k)))) / d = 1.85639269500896710302009... .

cp's OEIS Frontend

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

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

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A386803 Numbers without an exponent 4 in their prime factorization.

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

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A375768 The maximum exponent in the prime factorization of the numbers whose maximum exponent in their prime factorization is a Fibonacci number.

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

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

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A375769 The indices of the terms of A375768 in the Fibonacci sequence.

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