A386806 -id:A386806 - cp's OEIS frontend

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

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;

A386802 Numbers that have exactly three exponents in their prime factorization that are equal to 3.

Original entry on oeis.org

27000, 74088, 189000, 287496, 297000, 343000, 351000, 370440, 459000, 474552, 513000, 621000, 783000, 814968, 837000, 963144, 999000, 1029000, 1061208, 1107000, 1157625, 1161000, 1259496, 1269000, 1323000, 1331000, 1407672, 1431000, 1437480, 1481544, 1593000, 1647000

Offset: 1

Amiram Eldar, Aug 03 2025

Subsequence of A176359 and first differs from it at n = 173: A176359(173) = 9261000 = 2^3 * 3^3 * 5^3 * 7^3 is not a term of this sequence.
Numbers k such that A295883(k) = 3.
The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^3 + 1/p^4) * (s(1)^3 + 3*s(1)*s(2) + 2*s(3)) / 6 = 0.000018940548516752487509..., where s(m) = (-1)^(m-1) * Sum_{p prime} (1/(p^4/(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.

Subsequence of A176359.
Cf. A295883.
Numbers that have exactly three exponents in their prime factorization that are equal to k: A386798 (k=2), this sequence (k=3), A386806 (k=4), A386810 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 3: A386799 (m=0), A386800 (m=1), A386801 (m=2), this sequence (m=3).

f[p_, e_] := If[e == 3, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[2*10^6], s[#] == 3 &]

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

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;

A386804 Numbers that have exactly one exponent in their prime factorization that is equal to 4.

Original entry on oeis.org

16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 784, 810, 816, 848, 880, 891, 912, 944, 976, 1008, 1040, 1053, 1072, 1104, 1134, 1136, 1168, 1200, 1232, 1250

Offset: 1

Amiram Eldar, Aug 03 2025

Subsequence of A336595 and first differs from it at n = 21: A336595(21) = 512 = 2^9 is not a term of this sequence.

The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^4 + 1/p^5) * Sum_{p prime} (p-1)/(p^5 - p + 1) = 0.04058504714976055893... (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 A336595.
Numbers that have exactly one exponent in their prime factorization that is equal to k: A119251 (k=1), A386796 (k=2), A386800 (k=3), this sequence (k=4), A386808 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 4: A386803 (m=0), this sequence (m=1), A386805 (m=2), A386806 (m=3).

f[p_, e_] := If[e == 4, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[1300], s[#] == 1 &]

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

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;

A386810 Numbers that have exactly three exponents in their prime factorization that are equal to 5.

Original entry on oeis.org

24300000, 130691232, 170100000, 267300000, 315900000, 413100000, 461700000, 558900000, 653456160, 704700000, 753300000, 899100000, 996300000, 1044900000, 1142100000, 1190700000, 1252332576, 1287900000, 1433700000, 1437603552, 1482300000, 1628100000, 1680700000

Offset: 1

Amiram Eldar, Aug 03 2025

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^5 + 1/p^6) * (s(1)^3 + 3*s(1)*s(2) + 2*s(3)) / 6 = 1.38560245036673575581*10^(-8), where s(m) = (-1)^(m-1) * Sum_{p prime} (1/(p^6/(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.

Numbers that have exactly three exponents in their prime factorization that are equal to k: A386798 (k=2), A386802 (k=3), A386806 (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), A386809 (m=2), this sequence (m=3).

M:= 10^10: # for terms <= M
B:= select(t -> ifactors(t)[2][..,2]=[1,1,1],[$1..floor(M^(1/5))]):
R:= NULL:
for i from 1 to nops(B) do
  Q:= select(t -> igcd(t,B[i]) = 1 and not member(5, ifactors(t)[2][..,2]), [$1 .. M/B[i]^5]);
  R:= R, op(B[i]^5 * Q);
od:
sort([R]); # Robert Israel, Aug 03 2025

seq[lim_] := Module[{s = {}, sqfs = Select[Range[Surd[lim, 5]], SquareFreeQ[#] && PrimeNu[#] == 3 &]}, Do[s = Join[s, sqf^5 * Select[Range[lim/sqf^5], CoprimeQ[#, sqf] && !MemberQ[FactorInteger[#][[;; , 2]], 5] &]], {sqf, sqfs}]; Union[s]]; seq[2*10^9]

isok(k) = vecsum(apply(x -> if(x == 5, 1, 0), factor(k)[, 2])) == 3;

cp's OEIS Frontend

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

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A386802 Numbers that have exactly three exponents in their prime factorization that are equal to 3.

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A386803 Numbers without an exponent 4 in their prime factorization.

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A386804 Numbers that have exactly one exponent in their prime factorization that is equal to 4.

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

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A386810 Numbers that have exactly three exponents in their prime factorization that are equal to 5.

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