A336590 -id:A336590 - 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

A352475 Numbers m such that gcd(d(m),6) = 1.

Original entry on oeis.org

1, 16, 64, 81, 625, 729, 1024, 1296, 2401, 4096, 5184, 10000, 11664, 14641, 15625, 28561, 38416, 40000, 46656, 50625, 59049, 65536, 82944, 83521, 117649, 130321, 153664, 194481, 234256, 250000, 262144, 279841, 331776, 455625, 456976, 531441, 640000, 707281, 746496

Offset: 1

Michael De Vlieger, Mar 26 2022

All terms are square since numbers coprime to 6 are odd.
The square roots of terms are in A001694.
Intersection of A000290 and A336590, i.e., numbers whose prime factorization has only exponents that are congruent to {0, 4} mod 6 (A047233). - Amiram Eldar, Mar 31 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Titus Hilberdink, How often is d(n) a power of a given integer?, Journal of Number Theory, Vol. 236 (2022), pp. 261-279.

Cf. A000005, A000290, A001694, A007310, A047233, A076400, A100044, A336590, A350014.

Select[Range[864]^2, GCD[DivisorSigma[0, #], 6] == 1 &] (* or, more efficiently, *)
With[{nn = 864}, Select[Union[Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}]]^2, Mod[DivisorSigma[0, #], 3] > 0 &]]

isok(m) = gcd(numdiv(m), 6) == 1; \\ Michel Marcus, Mar 29 2022

m = 100000; seq = direuler(p=2, m, (1 - X^8)/(1 - X^4)/(1 - X^6)); for(n=1, m, if(seq[n] != 0, print1(n, ", "))) \\ Vaclav Kotesovec, May 19 2022

a(n) = A350014(n)^2.
Sum_{n>=1} 1/a(n) = Pi^2/9 (A100044). - Amiram Eldar, Mar 31 2022
The number of terms <= x is (zeta(3/2)/zeta(2))*x^(1/4) + (zeta(2/3)/zeta(4/3))*x^(1/6) + O(x^(1/8 + eps)), for all eps > 0 (Hilberdink, 2022). - Amiram Eldar, May 18 2022

A354178 Numbers whose number of divisors is coprime to 30.

Original entry on oeis.org

1, 64, 729, 1024, 4096, 15625, 46656, 59049, 65536, 117649, 262144, 531441, 746496, 1000000, 1771561, 2985984, 3779136, 4194304, 4826809, 7529536, 9765625, 11390625, 16000000, 24137569, 34012224, 43046721, 47045881, 47775744, 60466176, 64000000, 85766121, 113379904

Offset: 1

Amiram Eldar, May 18 2022

nonn

Numbers k such that gcd(d(k), 30) = 1, where d(k) is the number of divisors of k (A000005).

All the terms are squares since their number of divisors is odd.

			64 is a term since A000005(64) = 7 and gcd(7, 30) = 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1709 terms from Amiram Eldar)
Titus Hilberdink, How often is d(n) a power of a given integer?, Journal of Number Theory, Vol. 236 (2022), pp. 261-279.
Index entries for sequences computed from exponents in factorization of n.

Subsequence of other sequences of numbers k such that gcd(d(k), m) = 1: A000290 (m=2), A336590 (m=3), A352475 (m=6).

Cf. A000005, A007775, A354179.

Select[Range[10^4]^2, CoprimeQ[DivisorSigma[0, #], 30] &]

isok(k) = gcd(numdiv(k), 30) == 1;
for(k=1, 10650, if(isok(k^2), print1(k^2,", ")))

a(n) = A354179(n)^2.
The number of terms <= x is (zeta(5)*zeta(5/3))/(zeta(4)*zeta(10/3))*x^(1/6) + (zeta(3)*zeta(3/5))/(zeta(2)*zeta(12/5))*x^(1/10) + O(x^(1/20 + eps)) for all eps > 0 (Hilberdink, 2022).
Sum_{n>=1} 1/a(n) = Product_{p prime} (p^2 + p^8 + p^12 + p^14 + p^18 + p^20 + p^24 + p^30)/(p^30 - 1) = 1.0183538548...

A358001 Numbers whose number of divisors is coprime to 210.

Original entry on oeis.org

1, 1024, 4096, 59049, 65536, 262144, 531441, 4194304, 9765625, 43046721, 60466176, 241864704, 244140625, 268435456, 282475249, 387420489, 544195584, 1073741824, 2176782336, 3869835264, 10000000000, 13841287201, 15479341056, 25937424601, 31381059609, 34828517376

Offset: 1

Michael De Vlieger, Dec 03 2022

nonn

210 is the product of the smallest 4 primes.
Numbers k such that gcd(d(k), 210) = 1, where d(k) is the number of divisors of k (A000005).
The square roots of terms are in A001694.

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Subsequence of other sequences of numbers k such that gcd(d(k), m) = 1: A000290 (m=2), A336590 (m=3), A352475 (m=6), A354178 (m=30).

Cf. A000005, A001694, A008364.

With[{nn = 200000}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], CoprimeQ[DivisorSigma[0, #^2], 210] &]^2]

a(n) = A358250(n)^2.

Sum_{n>=1} 1/a(n) = Product_{p prime} (Sum_{k=2..210, gcd(k-1,210)=1} p^k)/(p^210-1) = 1.001258995976... . - Amiram Eldar, Dec 06 2022

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A337050 Numbers without an exponent 2 in their prime factorization.

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A352475 Numbers m such that gcd(d(m),6) = 1.

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A354178 Numbers whose number of divisors is coprime to 30.

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A358001 Numbers whose number of divisors is coprime to 210.

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