A361177 -id:A361177 - cp's OEIS frontend

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

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, ", ")));

A366762 Numbers whose canonical prime factorization contains only exponents which are congruent to 1 modulo 3.

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, Oct 21 2023

First differs from A274034 at n = 42, and from A197680 and A361177 at n = 84.

The asymptotic density of this sequence is zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A002117 * A330523 = A253905 * A065465 = 0.644177671086029533405... .

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

Similar sequences with exponents of a given form: A000290 (2*k), A268335 (2*k+1), A000578 (3*k), A182120 (3*k+2).
Cf. A002117, A065465, A253905, A330523.
Cf. A197680, A274034, A361177, A366761.

q[n_] := AllTrue[FactorInteger[n][[;; , 2]], Mod[#, 3] == 1 &]; Select[Range[120], q]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2]%3 != 1, return(0))); 1;}

Sum_{n>=1} 1/a(n)^s = zeta(3*s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)), for s > 1.

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]))

A369210 Numbers k such that the number of divisors of k^2 is a power of 3.

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, Jan 16 2024

First differs from A197680 at n = 331, from A274034 at n = 42, from A361177 at n = 167, and from A366762 at n = 84.
Equivalently, square roots of the numbers whose number of divisors is a power of 3.
The asymptotic density of this sequence is Product_{p prime} ((1 - 1/p) * Sum_{k>=0} 1/p^((3^k-1)/2)) = 0.64033435998103973346... .

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

Cf. A000005, A000244, A000290.

Cf. A197680, A274034, A361177, A366762.

pow3q[n_] := n == 3^IntegerExponent[n, 3]; Select[Range[100], pow3q[DivisorSigma[0, #^2]] &]

ispow3(n) = n == 3^valuation(n, 3);
is(n) = ispow3(numdiv(n^2));

Sum_{n>=1} 1/a(n)^2 = Product_{p prime} Sum_{k>=0} 1/p^(3^k-1) = 1.52478035628964060288... .

A372405 Exponentially powerful numbers whose prime factorization exponents are all powerful numbers > 1.

Original entry on oeis.org

1, 16, 81, 256, 512, 625, 1296, 2401, 6561, 10000, 14641, 19683, 20736, 28561, 38416, 41472, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 314928, 320000, 390625, 456976, 614656, 707281, 810000, 923521, 1185921, 1229312, 1336336, 1500625, 1679616

Offset: 1

David James Sycamore, Apr 29 2024

nonn

In other words, numbers m such that if p^k is the greatest power of any prime p which divides m, then k is a term > 1 in A001694.
Subsequence of A001694 (since all prime exponents are > 1).
Compare with A361177, of which this is a subsequence (see Formula).
Distinct from A277562; A277652(26) = 331776 = 2^12 * 3^4 is not in this sequence. - Michael De Vlieger, Apr 30 2024
1 and 41472 are two terms here that are not in A277562. - David A. Corneth, Apr 30 2024

			16 = 2^4 and 4 = A001694(2) is a powerful number.
a(7) = 1296 = 2^4*3^4.
a(12) = 19683 = 3^9 (9 = A001694(4) is a powerful number).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1580 terms from Michael De Vlieger)

Intersection of A001694 and A361177.

Subsequence of A036967.

nn = 2^21; f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]]; Select[Union@ Flatten@ Table[a^7*b^6*c^5*d^4, {d, Surd[nn, 4]}, {c, Surd[nn/d^4, 5]}, {b, Surd[nn/(c^5*d^4), 6]}, {a, Surd[nn/(b^6*c^5*d^4), 7]}], AllTrue[FactorInteger[#][[All, -1]], Divisible[#, f[#]^2] &] &] (* Michael De Vlieger, Apr 29 2024 *)

isok(k) = if (ispowerful(k), my(f=factor(k)[,2]); #select(ispowerful, f) == #f); \\ Michel Marcus, Apr 30 2024

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=2} 1/p^A001694(k)) = 1.08410926642148594327... . - Amiram Eldar, May 12 2024

More terms from Michael De Vlieger, Apr 29 2024

A382065 Exponentially refactorable numbers: numbers whose exponents in their canonical prime factorization are all refactorable numbers (A033950).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 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, 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, 78, 79

Offset: 1

Amiram Eldar, Mar 14 2025

First differs from A377019 at n = 55: A377019(55) = 64 is not a term of this sequence.
First differs from A344742 at n = 62: A344742(62) = 72 is not a term of this sequence.
All the cubefree numbers (A004709) are terms. The least term that is not cubefree is a(215) = 256 = 2^8.
Subsequence of A382063 and first differs from it at n = 362: A382063(362) = 432 = 2^4 * 3^3 is not a term of this sequence.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + (1 - 1/p) * (Sum_{k>=3} 1/p^A033950(k))) = 0.83493143539605138255... .
The relative density of this sequence within A382063 is the ratio between the densities of the two sequences: 0.997553... .

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

Cf. A033950, A344742.
Subsequence of A382063.
Subsequence: A004709.
Similar sequences: A197680, A209061, A138302, A268335, A361177, A377019.

refQ[k_] := Divisible[k, DivisorSigma[0, k]]; q[k_] := AllTrue[FactorInteger[k][[;; , 2]], refQ]; Select[Range[100], q]

isref(n) = !(n % numdiv(n));
isok(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isref(e[i]), return(0))); 1; }

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

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A366762 Numbers whose canonical prime factorization contains only exponents which are congruent to 1 modulo 3.

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

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A369210 Numbers k such that the number of divisors of k^2 is a power of 3.

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A372405 Exponentially powerful numbers whose prime factorization exponents are all powerful numbers > 1.

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A382065 Exponentially refactorable numbers: numbers whose exponents in their canonical prime factorization are all refactorable numbers (A033950).

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