A380732 -id:A380732 - cp's OEIS frontend

A380730 Numbers k such that the greatest prime dividing k is smaller than the minimum exponent in the prime factorization of k.

8, 16, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 1296, 2048, 2187, 2592, 3888, 4096, 5184, 6561, 7776, 8192, 10368, 11664, 15552, 15625, 16384, 19683, 20736, 23328, 31104, 32768, 34992, 41472, 46656, 59049, 62208, 65536, 69984, 78125, 82944, 93312, 104976, 124416

Offset: 1

Amiram Eldar, Jan 31 2025

nonn

Numbers k such that A006530(k) < A051904(k).

Disjoint union of the sequences S_k, k >= 1, where S_k is the sequence of p-smooth numbers (numbers whose prime factors are all less than or equal to p), with p = prime(k), that are (prime(k)+1)-full but not (prime(k+1)+1)-full numbers (k-full numbers are numbers whose prime factorization exponents are all larger than or equal to k). S_1 contains only the term 8, and S_k is infinite for k >= 2. The sum of the reciprocals of the terms of S_k is rational for all k: 1/8, 583/5184, 19757609/777600000, ... (see the Formula section).

			8 = 2^3 is a term since 2 < 3.
9 = 3^2 is not a term since 3 > 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number.
Wikipedia, Powerful number: Generalization (k-full number).
Wikipedia, Smooth number.
Index entries for sequences related to powerful numbers.

Cf. A006530, A051904, A380692.

Subsequence of A036966, A380731, A380732 and A380733.

Select[Range[2, 125000], Module[{f = FactorInteger[#]}, f[[-1, 1]] < Min[f[[;;, 2]]]] &]

isok(k) = if(k == 1, 0, my(f = factor(k), e = f[,2]); f[#f~, 1] < vecmin(e));

Sum_{n>=1} 1/a(n) = Sum_{k>=1} f(k) = 0.27091620709274155136..., where f(k) = Sum_{i>=1} 1 / S_k(i) = g(prime(k), k) - g(prime(k+1), k), g(p, k) = Product_{j=1..k} (1 + Sum_{i >= p+1} 1/prime(j)^i), and S_k is defined in the Comments section.

A380731 Numbers k such that the largest prime dividing k is smaller than or equal to the minimum exponent in the prime factorization of k.

Original entry on oeis.org

4, 8, 16, 27, 32, 64, 81, 128, 216, 243, 256, 432, 512, 648, 729, 864, 1024, 1296, 1728, 1944, 2048, 2187, 2592, 3125, 3456, 3888, 4096, 5184, 5832, 6561, 6912, 7776, 8192, 10368, 11664, 13824, 15552, 15625, 16384, 17496, 19683, 20736, 23328, 27648, 31104, 32768

Offset: 1

Amiram Eldar, Jan 31 2025

nonn

Numbers k such that A006530(k) <= A051904(k).

Disjoint union of the sequences S_k, k >= 1, where S_k is the sequence of p-smooth numbers (numbers whose prime factors are all less than or equal to p), with p = prime(k), that are prime(k)-full but not prime(k+1)-full numbers (k-full numbers are numbers whose prime factorization exponents are all larger than or equal to k). S_1 contains only the term 4, and S_k is infinite for k >= 2. The sum of the reciprocals of the terms of S_k is rational for all k: 1/4, 649/2592, 61992313/1166400000, ... (see the Formula section).

			4 = 2^2 is a term since A006530(4) = A051904(4) = 2.
9 = 3^2 is not a term since 3 > 2.

Amiram Eldar, Table of n, a(n) for n = 1..5416 (terms below 10^18)
Eric Weisstein's World of Mathematics, Smooth Number.
Wikipedia, Powerful number: Generalization (k-full number).
Wikipedia, Smooth number.
Index entries for sequences related to powerful numbers.

Cf. A006530, A051904, A380693.
Subsequence of A001694, A380732 and A380733.
A380730 is a subsequence.

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  max(F[..,1]) <= min(F[..,2])
end proc:
select(filter, [$2..50000]); # Robert Israel, Jan 31 2025

Select[Range[2, 33000], Module[{f = FactorInteger[#]}, f[[-1, 1]] <= Min[f[[;;, 2]]]] &]

isok(k) = if(k == 1, 0, my(f = factor(k), e = f[,2]); f[#f~, 1] <= vecmin(e));

Sum_{n>=1} 1/a(n) = Sum_{k>=1} f(k) = 0.56987350769329353172..., where f(k) = Sum_{i>=1} 1 / S_k(i) = g(prime(k), k) - g(prime(k+1), k), g(p, k) = Product_{j=1..k} (1 + Sum_{i >= p} 1/prime(j)^i), and S_k is defined in the Comments section.

A380733 Numbers k such that the prime index of the largest prime dividing k is smaller than or equal to the minimum exponent in the prime factorization of k; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 27, 32, 36, 64, 72, 81, 108, 125, 128, 144, 216, 243, 256, 288, 324, 432, 512, 576, 625, 648, 729, 864, 972, 1000, 1024, 1152, 1296, 1728, 1944, 2000, 2048, 2187, 2304, 2401, 2592, 2916, 3125, 3375, 3456, 3888, 4000, 4096, 4608, 5000, 5184, 5832

Offset: 1

Amiram Eldar, Jan 31 2025

nonn

Numbers k such that A061395(k) <= A051904(k).
Except for 2, all the terms are powerful (A001694).
Disjoint union of the sequences S_k, k >= 1, where S_k is the sequence of p-smooth numbers (numbers whose prime factors are all less than or equal to p), with p = prime(k), that are k-full but not (k+1)-full numbers (k-full numbers are numbers whose prime factorization exponents are all larger than or equal to k). S_1 contains only the term 2, and S_k is infinite for k >= 2. The sum of the reciprocals of the terms of S_k is rational for all k: 1/2, 31/72, 2657/14400, ... (see the Formula section).

			4 = 2^2 is a term since PrimePi(2) = 1 <= 2.
25 = 5^2 is not a term since PrimePi(5) = 3 is larger than the exponent 2.

Amiram Eldar, Table of n, a(n) for n = 1..10226 (terms below 3*10^15)
Eric Weisstein's World of Mathematics, Smooth Number.
Wikipedia, Powerful number: Generalization (k-full number).
Wikipedia, Smooth number.
Index entries for sequences related to powerful numbers.

Cf. A001694, A006530, A051904, A061395, A380695.

Subsequences: A380730, A380731, A380732.

Select[Range[6000], Module[{f = FactorInteger[#]}, f[[-1, 1]] <= Prime[Min[f[[;;, 2]]]]] &]

isok(k) = if(k == 1, 1, my(f = factor(k), e = f[,2]); f[#f~, 1] <= prime(vecmin(e)));

Sum_{n>=1} 1/a(n) = Sum_{k>=1} f(k) = 1.263759868201220952806..., where f(k) = Sum_{i>=1} 1 / S_k(i) = g(k, k) - g(k+1, k), g(e, k) = Product_{j=1..k} (1 + Sum_{i >= e} 1/prime(j)^i), and S_k is defined in the Comments section.

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A380730 Numbers k such that the greatest prime dividing k is smaller than the minimum exponent in the prime factorization of k.

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A380731 Numbers k such that the largest prime dividing k is smaller than or equal to the minimum exponent in the prime factorization of k.

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A380733 Numbers k such that the prime index of the largest prime dividing k is smaller than or equal to the minimum exponent in the prime factorization of k; a(1) = 1 by convention.

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