A327939 -id:A327939 - cp's OEIS frontend

A327936 Multiplicative with a(p^e) = p if e >= p, otherwise 1.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Antti Karttunen, Oct 01 2019

			For n = 12 = 2^2 * 3^1, only prime factor p = 2 satisfies p^p | 12, thus a(12) = 2.
For n = 108 = 2^2 * 3^3, both prime factors p = 2 and p = 3 satisfy p^p | 108, thus a(108) = 2*3 = 6.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001221, A129251, A327937, A327938, A327939.

Differs from A129252 for the first time at n=108.

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; IntegerQ@ p :> If[e >= p, p, 1]] &, 120] (* Michael De Vlieger, Oct 01 2019 *)

A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };

Multiplicative with a(p^e) = p if e >= p, otherwise 1.
A001221(a(n)) = A129251(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (p-1)/p^p) = 1.3443209052633459342... . - Amiram Eldar, Nov 07 2022

A327938 Multiplicative with a(p^e) = p^(e mod p).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 2, 9, 10, 11, 3, 13, 14, 15, 1, 17, 18, 19, 5, 21, 22, 23, 6, 25, 26, 1, 7, 29, 30, 31, 2, 33, 34, 35, 9, 37, 38, 39, 10, 41, 42, 43, 11, 45, 46, 47, 3, 49, 50, 51, 13, 53, 2, 55, 14, 57, 58, 59, 15, 61, 62, 63, 1, 65, 66, 67, 17, 69, 70, 71, 18, 73, 74, 75, 19, 77, 78, 79, 5, 3, 82, 83, 21, 85, 86, 87, 22

Offset: 1

Antti Karttunen, Oct 01 2019

All terms are in A048103.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A048103, A129251, A327936, A327937, A327939, A327965.

Differs from A065883 for the first time at n=27.

f[p_, e_] := p^Mod[e, p]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 07 2022 *)

A327938(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]%f[k,1])); factorback(f); };

Multiplicative with a(p^e) = p^(e mod p).
a(n) = n / A327939(n).
For all n, A129251(a(n)) = 0, A327936(a(n)) = 1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1/(1+1/p^p)) = 0.38559042841678887219... . - Amiram Eldar, Nov 07 2022

A342007 Multiplicative with a(p^e) = p^floor(e/p).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Mar 04 2021

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A327936, A327938, A327939, A341997, A342017 [= a(A276086(n))].

Array[Times @@ Map[#1^Floor[#2/#1] & @@ # &, FactorInteger[#]] &, 105] (* Michael De Vlieger, Mar 12 2021 *)

A342007(n) = { my(f = factor(n)); for(k=1, #f~, f[k, 2] = floor(f[k, 2]/f[k, 1])); factorback(f); };

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (p-1)/(p^p-p)) = 1.6270951877598772517... . - Amiram Eldar, Nov 07 2022

A365632 The number of divisors of n that are terms of A072873.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 14 2023

The sum of these divisors is A365633(n) and the largest of them is A327939(n).

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

Cf. A072873, A327939, A365633, A365634.

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

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

Multiplicative with a(p^e) = 1 + floor(e/p).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} p^p/(p^p-1) = 1.3850602852... .

Data corrected by Amiram Eldar, Sep 20 2023

A368329 The largest term of A054743 that divide n.

Original entry on oeis.org

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))).

A368333 The largest term of A054744 that divide n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 1, 64, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1

Offset: 1

Amiram Eldar, Dec 21 2023

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. A034444, A048103, A054744, A327939, A368329, A368332, A368334, A368335.

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)) = A368334(n).
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= n, with equality if and only if n is in A054744.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^(p*s) + 1/p^(p*(s-1)) + 1/p^((p+1)*s-1) - 1/p^((p+1)*(s-1)+1)).

A368334 The number of terms of A054744 that are unitary divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 21 2023

First differs from A081117 at n = 28.

Also, the number of terms of A072873 that are unitary divisors of n.

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

Cf. A034444, A048103, A054744, A072873, A327939, A368330, A368332, A368333, A368335.

Cf. A081117.

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(A368333(n)).
a(n) = A034444(A327939(n)).
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= A034444(n), with equality if and only if n is in A054744.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(p*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^p) = 1.29671268566745796443... .

A368336 The number of divisors of the largest term of A072873 that divides of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 21 2023

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

Cf. A000005, A048103, A072873, A327939, A365632, A368331, A368335.

f[p_, e_] := (e - Mod[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~, f[i,2] - f[i,2]%f[i,1] + 1);}

a(n) = A000005(A327939(n)).
Multiplicative with a(p^e) = e - (e mod p) + 1.
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= A000005(n), with equality if and only if n is in A072873.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + p/(p^p-1)) = 1.86196549645040699446... .

A359594 Multiplicative with a(p^e) = p^e if p divides e, 1 otherwise.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 27, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 27, 1, 1, 1, 1, 1, 4, 1, 1, 1, 64, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 108

Offset: 1

Antti Karttunen, Jan 09 2023

Each term a(n) divides both A085731(n) and A327939(n).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A359593.

Cf. also A085731, A327939.

f[p_, e_] := If[Divisible[e, p], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 09 2023 *)

A359594(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^(f[k,2]*!(f[k, 2]%f[k, 1]))); };

from math import prod
from sympy import factorint
def A359594(n): return prod(p**e for p, e in factorint(n).items() if not e%p) # Chai Wah Wu, Jan 10 2023

a(n) = n / A359593(n).

A365633 The sum of divisors of n that are terms of A072873.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 14 2023

The number of these divisors is A365632(n) and the largest of them is A327939(n).

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

Cf. A072873, A327939, A332653, A365632.

f[p_, e_] := (p^(Floor[e/p] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = (p^(floor(e/p)+1) - 1)/(p - 1).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (A332653(p)/(p^(p-1)-1) - 1/(p*(p-1))) = 2.253624924813... .

cp's OEIS Frontend

A327936 Multiplicative with a(p^e) = p if e >= p, otherwise 1.

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A327938 Multiplicative with a(p^e) = p^(e mod p).

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A342007 Multiplicative with a(p^e) = p^floor(e/p).

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A365632 The number of divisors of n that are terms of A072873.

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A368329 The largest term of A054743 that divide n.

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A368333 The largest term of A054744 that divide n.

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A368334 The number of terms of A054744 that are unitary divisors of n.

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A368336 The number of divisors of the largest term of A072873 that divides of n.

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