A386797 -id:A386797 - cp's OEIS frontend

A337050 Numbers without an exponent 2 in their prime factorization.

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

A386796 Numbers that have exactly one exponent in their prime factorization that is equal to 2.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268

Offset: 1

Amiram Eldar, Aug 02 2025

First differs from its subsequence A060687 at n = 16: a(16) = 72 is not a term of A060687.
Differs from A286228 by having the terms 60, 72, 84, 90, ..., and not having the term 1.
Numbers k such that A369427(k) = 1.
The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^2 + 1/p^3) * Sum_{p prime} (p-1)/(p^3 - p + 1) = 0.22661832022705616779... (the product is A330596) (Elma and Martin, 2024).

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

A060687 is a subsequence.
Cf. A286228, A330596, A369427.
Numbers that have exactly one exponent in their prime factorization that is equal to k: A119251 (k=1), this sequence (k=2), A386800 (k=3), A386804 (k=4), A386808 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 2: A337050 (m=0), this sequence (m=1), A386797 (m=2), A386798 (m=3).

f[p_, e_] := If[e == 2, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[300], s[#] == 1 &]

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

A386798 Numbers that have exactly three exponents in their prime factorization that are equal to 2.

Original entry on oeis.org

900, 1764, 4356, 4900, 6084, 6300, 8820, 9900, 10404, 11025, 11700, 12100, 12996, 14700, 15300, 16900, 17100, 19044, 19404, 20700, 21780, 22050, 22932, 23716, 26100, 27225, 27900, 28900, 29988, 30276, 30420, 30492, 33124, 33300, 33516, 34596, 36100, 36300, 36900, 38025, 38700

Offset: 1

Amiram Eldar, Aug 02 2025

Numbers k such that A369427(k) = 2.

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^2 + 1/p^3) * (s(1)^3 + 3*s(1)*s(2) + 2*s(3)) / 6 = 0.0011175284878980531468... (the product is A330596), where s(m) = (-1)^(m-1) * Sum_{p prime} (1/(p^3/(p-1)-1))^m (Elma and Martin, 2024).

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

Cf. A330596, A369427.
Numbers that have exactly three exponents in their prime factorization that are equal to k: this sequence (k=2), A386802 (k=3), A386806 (k=4), A386810 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 2: A337050 (m=0), A386796 (m=1), A386797 (m=2), this sequence (m=3).

f[p_, e_] := If[e == 2, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[40000], s[#] == 3 &]

isok(k) = vecsum(apply(x -> if(x == 2, 1, 0), factor(k)[, 2])) == 3;

A386801 Numbers that have exactly two exponents in their prime factorization that are equal to 3.

Original entry on oeis.org

216, 1000, 1080, 1512, 2376, 2744, 2808, 3000, 3375, 3672, 4104, 4968, 5400, 6264, 6696, 6750, 7000, 7560, 7992, 8232, 8856, 9000, 9261, 9288, 10152, 10584, 10648, 11000, 11448, 11880, 12744, 13000, 13176, 13500, 13720, 14040, 14472, 15336, 15768, 16632, 17000

Offset: 1

Amiram Eldar, Aug 03 2025

Subsequence of A109399 and first differs from it at n = 64: A109399(64) = 27000 = 2^3 * 3^3 * 5^3 is not a term of this sequence.
Numbers k such that A295883(k) = 2.
The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^3 + 1/p^4) * ((Sum_{p prime} (p-1)/(p^4 - p + 1))^2 - Sum_{p prime} ((p-1)^2/(p^4 - p + 1)^2)) / 2 = 0.0024403883082851652103... (Elma and Martin, 2024).

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

Subsequence of A109399.
Cf. A295883.
Numbers that have exactly two exponents in their prime factorization that are equal to k: A386797 (k=2), this sequence (k=3), A386805 (k=4), A386809 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 3: A386799 (m=0), A386800 (m=1), this sequence (m=2), A386802 (m=3).

f[p_, e_] := If[e == 3, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[17000], s[#] == 2 &]

isok(k) = vecsum(apply(x -> if(x == 3, 1, 0), factor(k)[, 2])) == 2;

A386805 Numbers that have exactly two exponents in their prime factorization that are equal to 4.

Original entry on oeis.org

1296, 6480, 9072, 10000, 14256, 16848, 22032, 24624, 29808, 30000, 32400, 37584, 38416, 40176, 45360, 47952, 50625, 53136, 55728, 60912, 63504, 68688, 70000, 71280, 76464, 79056, 84240, 86832, 90000, 92016, 94608, 99792, 101250, 102384, 107568, 110000, 110160, 115248

Offset: 1

Amiram Eldar, Aug 03 2025

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^4 + 1/p^5) * ((Sum_{p prime} (p-1)/(p^5 - p + 1))^2 - Sum_{p prime} ((p-1)^2/(p^5 - p + 1)^2)) / 2 = 0.00032582100547959312658... (Elma and Martin, 2024).

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

Numbers that have exactly two exponents in their prime factorization that are equal to k: A386797 (k=2), A386801 (k=3), this sequence (k=4), A386809 (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 4: A386803 (m=0), A386804 (m=1), this sequence (m=2), A386806 (m=3).

f[p_, e_] := If[e == 4, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[120000], s[#] == 2 &]

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

A386809 Numbers that have exactly two exponents in their prime factorization that are equal to 5.

Original entry on oeis.org

7776, 38880, 54432, 85536, 100000, 101088, 132192, 147744, 178848, 194400, 225504, 241056, 272160, 287712, 300000, 318816, 334368, 365472, 381024, 412128, 427680, 458784, 474336, 505440, 520992, 537824, 552096, 567648, 598752, 614304, 645408, 660960, 692064, 700000

Offset: 1

Amiram Eldar, Aug 03 2025

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^5 + 1/p^6) * ((Sum_{p prime} (p-1)/(p^6 - p + 1))^2 - Sum_{p prime} ((p-1)^2/(p^6 - p + 1)^2)) / 2 = 4.86539910559896710587...*10^(-5) (Elma and Martin, 2024).

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

Numbers that have exactly two exponents in their prime factorization that are equal to k: A386797 (k=2), A386801 (k=3), A386805 (k=4), this sequence (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 5: A386807 (m=0), A386808 (m=1), this sequence (m=2), A386810 (m=3).

f[p_, e_] := If[e == 5, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[700000], s[#] == 2 &]

isok(k) = vecsum(apply(x -> if(x == 5, 1, 0), factor(k)[, 2])) == 2;

cp's OEIS Frontend

A337050 Numbers without an exponent 2 in their prime factorization.

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A386796 Numbers that have exactly one exponent in their prime factorization that is equal to 2.

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A386798 Numbers that have exactly three exponents in their prime factorization that are equal to 2.

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A386801 Numbers that have exactly two exponents in their prime factorization that are equal to 3.

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A386805 Numbers that have exactly two exponents in their prime factorization that are equal to 4.

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A386809 Numbers that have exactly two exponents in their prime factorization that are equal to 5.

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