cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A380693 Numbers k such that the least prime dividing k is larger than or equal to the maximum exponent in the prime factorization of k; a(1) = 1 by convention.

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, 27, 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

Views

Author

Amiram Eldar, Jan 30 2025

Keywords

Comments

First differs from A047592, A187320, A207481 and A255805 at n = 48: A047592(48) = A187320(48) = A207481(48) = A255805(48) = 54 is not a term of this sequence.
Numbers k such that A020639(k) >= A051903(k).
Disjoint union of the sequences S_k, k >= 1, where S_k is the sequence of p-rough numbers (numbers whose prime factors are all greater than or equal to p), with p = nextprime(k) = A007918(k), whose maximum exponent in their prime factorization is k (i.e., numbers that are (k+1)-free but not k-free, where k-free numbers are numbers whose prime factorization exponents do not exceed k).
The asymptotic density of this sequence is Sum_{i>=1} d(i) = 0.84999238500582943243..., where d(i), the density of S_i, equals f(i+1) * Product_{primes p < i} ((1-1/p)/(1-1/p^(i+1))) - f(i) * Product_{primes p < i} ((1-1/p)/(1-1/p^i)), f(i) = 1/zeta(i) if i >= 2, and f(1) = 0.

Examples

			6 = 2^1 * 3^1 is a term since 2 >= 1.
8 = 2^3 is not a term since 2 < 3.

Links

Crossrefs

Cf. A020639, A051903, A007918.
Cf. A047592, A187320, A207481, A255805.
Subsequences: A380692, A380694, A380695.

Programs

  • Mathematica
    q[k_] := k == 1 || Module[{f = FactorInteger[k]}, f[[1, 1]] >= Max[f[[;; , 2]]]]; Select[Range[100], q]
  • PARI
    isok(k) = if(k == 1, 0, my(f = factor(k), e = f[, 2]); f[1, 1] >= vecmax(e));

A380694 Numbers k such that the prime index of the least prime dividing k is larger than the maximum exponent in the prime factorization of k.

Original entry on oeis.org

3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145, 149
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2025

Keywords

Comments

Differs from A352830 by having the terms 1225, 13475, 15925, ... and not having the terms 1, 147, 363, 507, 867, 1083, ... .
Differs from A320055 by having the terms 105, 165, 195, ... and not having the terms 1 and 2.
Differs from A320056 by having the terms 105, 165, 195, ... and not having the term 1.
Numbers k such that A055396(k) > A051903(k).
Disjoint union of the sequences S_k, k >= 1, where S_k is the sequence of p-rough numbers (numbers whose prime factors are all greater than or equal to p), with p = prime(k+1), whose maximum exponent in their prime factorization is k (i.e., numbers that are (k+1)-free but not k-free, where k-free numbers are numbers whose prime factorization exponents do not exceed k).
The asymptotic density of this sequence is Sum_{i>=1} d(i) = 0.43156823896267860476......, where d(i), the density of S_i, equals f(i+1) * Product_{primes p <= prime(i)} ((1-1/p)/(1-1/p^(i+1))) - f(i) * Product_{primes p <= prime(i)} ((1-1/p)/(1-1/p^i)), f(i) = 1/zeta(i) if i >= 2, and f(1) = 0.

Examples

			3 = 3^1 is a term since PrimePi(3) = 2 > 1.
9 = 3^2 is not a term since PrimePi(3) = 2 is not larger than the exponent 2.
25 = 5^2 is a term since PrimePi(5) = 3 > 2.

Links

Crossrefs

Cf. A000720, A020639, A051903, A055396.
Cf. A352830, A320055, A320056.
Subsequence of A380692, A380693 and A380695.

Programs

  • Mathematica
    q[k_] := k > 1 && Module[{f = FactorInteger[k]}, f[[1, 1]] > Prime[Max[f[[;; , 2]]]]]; Select[Range[150], q]
  • PARI
    isok(k) = if(k == 1, 0, my(f = factor(k), e = f[,2]); f[1,1] > prime(vecmax(e)));

A380695 Numbers k such that the prime index of the least prime dividing k is larger than or equal to the maximum exponent in the prime factorization of k; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 99
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2025

Keywords

Comments

First differs from A368110 at n = 68: A368110(68) = 98 is not a term of this sequence.
Subsequence of A380692 and first differs from it at n = 428: A380692(428) = 625 is not a term of this sequence.
Numbers k such that A055396(k) >= A051903(k).
Disjoint union of the sequences S_k, k >= 1, where S_k is the sequence of p-rough numbers (numbers whose prime factors are all greater than or equal to p), with p = prime(k), whose maximum exponent in their prime factorization is k (i.e., numbers that are (k+1)-free but not k-free, where k-free numbers are numbers whose prime factorization exponents do not exceed k).
The asymptotic density of this sequence is Sum_{i>=1} d(i) = 0.68165919742420048618..., where d(i), the density of S_i, equals f(i+1) * Product_{primes p < prime(i)} ((1-1/p)/(1-1/p^(i+1))) - f(i) * Product_{primes p < prime(i)} ((1-1/p)/(1-1/p^i)), f(i) = 1/zeta(i) if i >= 2, and f(1) = 0.

Examples

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

Links

Crossrefs

Cf. A000720, A020639, A051903, A055396, A368110.
Subsequence of A380692 and A380693.
A380694 is a subsequence.

Programs

  • Mathematica
    q[k_] := k == 1 || Module[{f = FactorInteger[k]}, f[[1, 1]] >= Prime[Max[f[[;; , 2]]]]]; Select[Range[100], q]
  • PARI
    isok(k) = if(k == 1, 1, my(f = factor(k), e = f[,2]); f[1,1] >= prime(vecmax(e)));

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

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jan 31 2025

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Cf. A006530, A051904, A380692.
Subsequence of A036966, A380731, A380732 and A380733.

Programs

  • Mathematica
    Select[Range[2, 125000], Module[{f = FactorInteger[#]}, f[[-1, 1]] < Min[f[[;;, 2]]]] &]
  • PARI
    isok(k) = if(k == 1, 0, my(f = factor(k), e = f[,2]); f[#f~, 1] < vecmin(e));

Formula

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.
Showing 1-4 of 4 results.

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