A354179 -id:A354179 - cp's OEIS frontend

A354178 Numbers whose number of divisors is coprime to 30.

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...

A358250 Numbers whose square has a number of divisors coprime to 210.

Original entry on oeis.org

1, 32, 64, 243, 256, 512, 729, 2048, 3125, 6561, 7776, 15552, 15625, 16384, 16807, 19683, 23328, 32768, 46656, 62208, 100000, 117649, 124416, 161051, 177147, 186624, 200000, 209952, 262144, 371293, 373248, 390625, 419904, 497664, 500000, 537824, 629856, 759375

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^2), 210) = 1, where d(k) is the number of divisors of k (A000005).
Also numbers with no exponents = 1 mod 3, 2 mod 5, or 3 mod 7; also numbers whose square has a number of divisors coprime to 105. - Charles R Greathouse IV, Dec 08 2022

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

Subsequence of A069492 and hence of A036967, A036966, and A001694.
Subsequence of other sequences of numbers k such that gcd(d(k^2), m) = 1: A350014 (m=6), A354179 (m=30).
Cf. A000005, A000290, A008364.

With[{nn = 2^20}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], CoprimeQ[DivisorSigma[0, #^2], 210] &]]

is(n,f=factor(n))=if(n<32, return(n==1)); my(t=f[,2]%105, N=19200959813818273241621521446046); for(i=1,#t, if(bittest(N,t[i]), return(0))); 1 \\ Charles R Greathouse IV, Dec 08 2022

Sum_{n>=1} 1/a(n) = Product_{p prime} (Sum_{k=2..210, gcd(k-1,210)=1} p^(k/2))/(p^105-1) = 1.05981355805... . - Amiram Eldar, Dec 06 2022

cp's OEIS Frontend

A354178 Numbers whose number of divisors is coprime to 30.

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