A330596 -id:A330596 - cp's OEIS frontend

A330594 Decimal expansion of Product_{primes p} (1 + 1/p^2 - 2/p^3).

1, 1, 0, 6, 9, 6, 0, 1, 1, 1, 9, 5, 3, 2, 1, 7, 6, 7, 6, 6, 5, 1, 1, 7, 9, 1, 3, 0, 0, 0, 7, 4, 3, 9, 5, 9, 2, 9, 4, 9, 5, 4, 8, 8, 3, 3, 6, 5, 8, 1, 2, 2, 4, 1, 9, 0, 4, 3, 1, 3, 4, 0, 4, 4, 9, 7, 8, 7, 7, 7, 3, 3, 2, 4, 1, 2, 3, 7, 3, 7, 0, 7, 8, 0, 4, 4, 4, 9, 8, 5, 6, 5, 9, 5, 9, 1, 2, 5, 3, 7, 2, 4, 9, 1, 6

Offset: 1

Vaclav Kotesovec, Dec 19 2019

			1.106960111953217676651179130007439592949548833658122419043134044978777...

Cf. A065464, A065473, A065476, A208133, A328017, A330595, A330596.

Do[Print[N[Exp[-Sum[q = Expand[(-p^2 + 2*p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]

prodeulerrat(1 + 1/p^2 - 2/p^3) \\ Amiram Eldar, Mar 16 2021

A056551 Smallest cube divisible by n divided by largest cube which divides n.

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 1, 27, 1000, 1331, 216, 2197, 2744, 3375, 8, 4913, 216, 6859, 1000, 9261, 10648, 12167, 27, 125, 17576, 1, 2744, 24389, 27000, 29791, 8, 35937, 39304, 42875, 216, 50653, 54872, 59319, 125, 68921, 74088, 79507, 10648

Offset: 1

Henry Bottomley, Jun 25 2000

			a(16) = 8 since smallest cube divisible by 16 is 64 and smallest cube which divides 16 is 8 and 64/8 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.

Cf. A000189, A000578, A008834, A019555, A048798, A050985, A053149, A053150, A056552.

Cf. A002117, A013670, A330596.

f[p_, e_] := p^If[Divisible[e, 3], 0, 1]; a[n_] := (Times @@ (f @@@ FactorInteger[ n]))^3; Array[a, 100] (* Amiram Eldar, Aug 29 2019*)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%3, f[i,1], 1))^3; } \\ Amiram Eldar, Oct 28 2022

a(n) = A053149(n)/A008834(n) = A048798(n)*A050985(n) = A056552(n)^3.
From Amiram Eldar, Oct 28 2022: (Start)
Multiplicative with a(p^e) = 1 if e is divisible by 3, and a(p^e) = p^3 otherwise.
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(12)/(4*zeta(3))) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A013670 * A330596 / (4*A002117) = 0.1557163105... . (End)
Dirichlet g.f.: zeta(3*s) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(2*s-3)). - Amiram Eldar, Sep 16 2023

A134193 a(1) = 1; for n>1, a(n) = the smallest positive integer not occurring among the exponents in the prime-factorization of n.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Jan 13 2008

nonn

From Amiram Eldar, Jun 30 2025: (Start)
The first position of k = 1, 2, 3, ... is A006939(k-1).
Let d(k) be the asymptotic density of the occurrences of k = 1, 2, ... in this sequence.
d(1) = 0 = the density of the powerful numbers (A001694).
d(2) = Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596) = the density of A337050.
d(3) = Product_{primes p} (1 - 1/p^3 + 1/p^4) - Product_{primes p} (1 - 1/p^2 + 1/p^4) = 0.23548870893364493209...
d(4) = Product_{primes p} (1 - 1/p^4 + 1/p^5) - Product_{primes p} (1 - 1/p^3 + 1/p^5) - Product_{primes p} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5) + Product_{primes p} (1 - 1/p^2 + 1/p^5) = 0.01580134256336122613... .
d(5) = 0.000174471282..., d(6) = 0.000000217516..., etc.
In general, d(k) = Sum_{s subset of {2, 3, ..., k-1}} (-1)^card(s) * Product_{p prime} (1 -Sum_{i=1..card(s)} 1/p^s(i) + 1/p^(s(i)+1) - 1/p^k + 1/p^(k+1)).
The asymptotic mean of this sequence is Sum_{k>=1} k*d(k) = 2.26761567808299143335... . (End)

			The prime factorization of 24 is 2^3 * 3^1. The exponents are 3 and 1. Therefore a(24) = 2 is the smallest positive integer not occurring among (3,1).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001694, A006939, A136567, A181819, A257993, A330596, A337050.

Join[{1}, Table[Complement[Range[n], Table[FactorInteger[n][[i, 2]], {i, 1, Length[FactorInteger[n]]}]][[1]], {n, 2, 120}]] (* Stefan Steinerberger, Jan 21 2008 *)

a(n) = if (n==1, 1, my(f=factor(n)); ve = vecsort(f[,2],,8); k = 1; while(vecsearch(ve, k), k++); k;); \\ Michel Marcus, Jul 28 2017

(define (A134193 n) (A257993 (A181819 n))) ;; Antti Karttunen, Jul 28 2017

a(n) = A257993(A181819(n)). - Antti Karttunen, Jul 28 2017

More terms from Stefan Steinerberger, Jan 21 2008

A336652 Sum of positive divisors of odd part of n that are divisible by every (odd) prime dividing it: a(n) = A057723(A000265(n)).

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 12, 5, 11, 3, 13, 7, 15, 1, 17, 12, 19, 5, 21, 11, 23, 3, 30, 13, 39, 7, 29, 15, 31, 1, 33, 17, 35, 12, 37, 19, 39, 5, 41, 21, 43, 11, 60, 23, 47, 3, 56, 30, 51, 13, 53, 39, 55, 7, 57, 29, 59, 15, 61, 31, 84, 1, 65, 33, 67, 17, 69, 35, 71, 12, 73, 37, 90, 19, 77, 39, 79, 5, 120, 41, 83, 21, 85, 43

Offset: 1

Antti Karttunen, Jul 30 2020

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A000593, A057723, A204455, A330596, A336649, A336651.

f[2, e_] := 1; f[p_, e_] := (p^(e+1) - 1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

A336652(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],1,-1+(((f[i,1]^(1+f[i,2]))-1) / (f[i,1]-1)))));

Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (p + p^2 + ... + p^e) = sigma(p^e)-1.
a(n) = A057723(A000265(n)).
a(n) = A204455(n) * A336649(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/21) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = (Pi^2/21) * A330596 = 0.3517974711... . - Amiram Eldar, Nov 12 2022

A372636 a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(j*k) / phi(k).

Original entry on oeis.org

1, 5, 14, 31, 58, 93, 148, 219, 306, 407, 550, 695, 898, 1103, 1323, 1610, 1963, 2293, 2738, 3152, 3597, 4116, 4773, 5362, 6073, 6808, 7611, 8437, 9492, 10348, 11557, 12728, 13868, 15143, 16425, 17753, 19482, 21083, 22687, 24350, 26481, 28186, 30535, 32641

Offset: 1

Seiichi Manyama, May 08 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^3 for n = 1..10000
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A330596, A372619, A372633.

Table[Sum[Sum[EulerPhi[j*k], {j, 1, n}] / EulerPhi[k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, May 08 2024 *)
s = 1; Join[{1}, Table[s += Sum[EulerPhi[j*n] / EulerPhi[j], {j, 1, n}] + Sum[EulerPhi[j*n], {j, 1, n-1}] / EulerPhi[n], {n, 2, 50}]] (* Vaclav Kotesovec, May 08 2024 *)

a(n) = sum(j=1, n, sum(k=1, n, eulerphi(j*k)/eulerphi(k)));

a(n) ~ c * n^3, where c = A330596 / 2 = 0.374267629841... . - Amiram Eldar, May 09 2024

A366763 The number of divisors of n that have no exponent 2 in their prime factorization.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 5, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 6, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 21 2023

The number of terms of A337050 that divide n.

The sum of these divisors is A366764(n), and the largest of them is A366765(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A004709, A005117, A034444, A337050, A366764, A366765.

Cf. A330596.

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

a(n) = vecprod(apply(x -> max(x, 2), factor(n)[, 2]));

Multiplicative with a(p^e) = max(e, 2);
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
a(n) >= A034444(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)).
From Vaclav Kotesovec, Apr 20 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 + 1/p^3) = A330596 = 0.74853525968236356464421504863791060164164034300532440451585279392592558689...,
f'(1) = f(1) * Sum_{p prime} (2*p-3)*log(p)/(p^3-p+1) = f(1) * 0.560697508735949606541137451100554649565120075155278833396722097786365686597...
and gamma is the Euler-Mascheroni constant A001620. (End)

A380325 The sum of the square roots of the squares that divide the n-th exponentially odd number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 20 2025

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

Cf. A013661, A065463, A069290, A268335, A330596, A380323, A380324.

f[p_, e_] := If[OddQ[e], (p^((e+1)/2) - 1)/(p - 1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 0 &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]^((f[i, 2]+1)/2) - 1)/(f[i, 1] - 1), 0));}
list(lim) = select(x -> x > 0, vector(lim, i, s(i)));

a(n) = A069290(A268335(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * A330596 / d = 1.74789521005721521109..., where d = A065463 is the asymptotic density of the exponentially odd numbers.

A159253 a(n) is the smallest positive integer not yet in the sequence such that n * a(n) is a cube.

Original entry on oeis.org

1, 4, 9, 2, 25, 36, 49, 8, 3, 100, 121, 18, 169, 196, 225, 32, 289, 12, 361, 50, 441, 484, 529, 72, 5, 676, 27, 98, 841, 900, 961, 16, 1089, 1156, 1225, 6, 1369, 1444, 1521, 200, 1681, 1764, 1849, 242, 75, 2116, 2209, 288, 7, 20, 2601, 338, 2809, 108, 3025, 392

Offset: 1

Franklin T. Adams-Watters, Apr 07 2009

This is a self-inverse permutation of the positive integers.

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

Cf. A064429 (function on exponents)

Cf. A330596, A347328.

f[p_, e_] := If[(r = Mod[e, 3]) == 0, p^e, p^(e - (-1)^r)]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

a(n) = {my(f = factor(n), r); prod(i = 1, #f~, r=f[i,2]%3; f[i,1]^if(r == 0, f[i,2], f[i,2]-(-1)^r));} \\ Amiram Eldar, Dec 01 2022

Multiplicative with a(p^(3*n)) = p^(3*n), a(p^(3*n+1)) = p^(3*n+2), and a(p^(3*n+2)) = p^(3*n+1).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(6)/(3*zeta(3))) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A347328 * A330596 / 3 = 0.2111705... . - Amiram Eldar, Dec 01 2022

A375847 The maximum exponent in the prime factorization of the largest unitary cubefree divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Aug 31 2024

Cf. A004709, A036966, A038109, A051903, A330596, A337050, A360539.

Cf. A007424 (analogous with the largest cubefree divisor, for n >= 2).

a[n_] := Max[Join[{0}, Select[FactorInteger[n][[;; , 2]], # <= 2 &]]]; a[1] = 0; Array[a, 100]

a(n) = {my(e = select(x -> x <= 2, factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

a(n) = A051903(A360539(n)).
a(n) = 0 if and only if n is cubefull (A036966).
a(n) = 1 if and only if n is in A337050 \ A036966.
a(n) = 2 if and only if n is in A038109.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 - A330596 = 1.25146474031763643535... .

cp's OEIS Frontend

A330594 Decimal expansion of Product_{primes p} (1 + 1/p^2 - 2/p^3).

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A056551 Smallest cube divisible by n divided by largest cube which divides n.

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A134193 a(1) = 1; for n>1, a(n) = the smallest positive integer not occurring among the exponents in the prime-factorization of n.

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A336652 Sum of positive divisors of odd part of n that are divisible by every (odd) prime dividing it: a(n) = A057723(A000265(n)).

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A372636 a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(j*k) / phi(k).

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A366763 The number of divisors of n that have no exponent 2 in their prime factorization.

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A380325 The sum of the square roots of the squares that divide the n-th exponentially odd number.

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A159253 a(n) is the smallest positive integer not yet in the sequence such that n * a(n) is a cube.

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