A207481 -id:A207481 - cp's OEIS frontend

A368329 The largest term of A054743 that divide n.

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 81, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 21 2023

First differs from A360540 at n = 27.

The largest divisor d of n such that e > p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).

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

Cf. A054743, A034444, A207481, A327939, A360540, A368329, A368330, A368331, A368333.

f[p_, e_] := If[e <= p, 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= f[i,1], 1, f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = p^e if e > p.
A034444(a(n)) = A368330(n).
a(n) >= 1, with equality if and only if n is in A207481.
a(n) <= n, with equality if and only if n is in A054743.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^((p+2)*s-1) - 1/p^((p+2)*(s-1)+1) - 1/p^((p+1)*s) + 1/p^((p+1)*(s-1))).

A368331 The number of divisors of the largest term of A054743 that divides of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 21 2023

First differs from A366145 at n = 27.

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

Cf. A000005, A054743, A207481, A368328, A368329, A368330, A368335, A368336.

Cf. A366145.

f[p_, e_] := If[e <= p, 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= f[i,1], 1, f[i,2]+1));}

Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = e+1 if e > p.
a(n) = A000005(A368329(n)).
a(n) >= 1, with equality if and only if n is in A207481.
a(n) <= A000005(n), with equality if and only if n is in A054743.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s - 1/p^((p+2)*s-1) + 1/p^((p+1)*s) + 1/p^((p+1)*s-1)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^(p-1))) = 1.58396891058853238595... .

A368330 The number of terms of A054743 that are unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 21 2023

First differ from A043281 at n = 49.

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

Cf. A034444, A054743, A207481, A368328, A368329, A368331, A368334.

Cf. A043281.

f[p_, e_] := If[e <= p, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= f[i,1], 1, 2));}

Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = 2 if e > p.
a(n) = A034444(A368329(n)).
a(n) >= 1, with equality if and only if n is in A207481.
a(n) <= A034444(n), with equality if and only if n is in A054743.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^((p+1)*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^(p+1)) = 1.13896197534988330925... .

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

Amiram Eldar, Jan 30 2025

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.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Rough Number.
Wikipedia, Rough number.

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

q[k_] := k == 1 || Module[{f = FactorInteger[k]}, f[[1, 1]] >= Max[f[[;; , 2]]]]; Select[Range[100], q]

isok(k) = if(k == 1, 0, my(f = factor(k), e = f[, 2]); f[1, 1] >= vecmax(e));

A368328 The number of terms of A054743 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 21 2023

The number of divisors d of n such that e > p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).

The largest of these divisors is A368329(n).

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

Cf. A054743, A207481, A365632, A368329, A368330, A368331, A368332.

f[p_, e_] := If[e <= p, 1, e - p + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= f[i,1], 1, f[i,2] - f[i,1] + 1));}

Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = e - p + 1 if e > p.
a(n) >= 1, with equality if and only if n is in A207481.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^((p+1)*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^p)) = 1.27325025767774256043... .

cp's OEIS Frontend

A368329 The largest term of A054743 that divide n.

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A368331 The number of divisors of the largest term of A054743 that divides of n.

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A368330 The number of terms of A054743 that are unitary divisors of n.

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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.

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A368328 The number of terms of A054743 that divide n.

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