A057521 -id:A057521 - cp's OEIS frontend

A375146 Numbers whose prime factorization has exactly one exponent that is larger than 3.

16, 32, 48, 64, 80, 81, 96, 112, 128, 144, 160, 162, 176, 192, 208, 224, 240, 243, 256, 272, 288, 304, 320, 324, 336, 352, 368, 384, 400, 405, 416, 432, 448, 464, 480, 486, 496, 512, 528, 544, 560, 567, 576, 592, 608, 624, 625, 640, 648, 656, 672, 688, 704, 720

Offset: 1

Amiram Eldar, Aug 01 2024

Subsequence of A046101 and first differs from it at n = 98: A046101(98) = 1296 = 2^4 * 3^4 is not a term of this sequence.
Numbers k such that the powerful part of k, A057521(k), is a prime power whose exponent is larger than 3 (A246550).
The asymptotic density of this sequence is (1/zeta(4)) * Sum_{p prime} 1/(p^4-1) = 0.075131780079404733755... .

			16 = 2^4 is a term since its prime factorization has exactly one exponent, 4, that is larger than 3.

Subsequence of A046101.

Cf. A013662, A057521, A190641, A246550, A375145.

q[n_] := Count[FactorInteger[n][[;;, 2]], _?(# > 3 &)] == 1; Select[Range[1000], q]

is(k) = #select(x -> x > 3, factor(k)[, 2]) == 1;

A380160 a(n) is the value of the Euler totient function when applied to the powerful part of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 6, 1, 1, 2, 1, 1, 1, 8, 1, 6, 1, 2, 1, 1, 1, 4, 20, 1, 18, 2, 1, 1, 1, 16, 1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 2, 6, 1, 1, 8, 42, 20, 1, 2, 1, 18, 1, 4, 1, 1, 1, 2, 1, 1, 6, 32, 1, 1, 1, 2, 1, 1, 1, 24, 1, 1, 20, 2, 1, 1, 1, 8, 54, 1, 1, 2

Offset: 1

Amiram Eldar, Jan 13 2025

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

Cf. A000010, A001694, A005117, A057521, A295294, A295295, A323333, A349379, A357669, A380161.

f[p_, e_] := If[e == 1, 1, (p-1)*p^(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] == 1, 1, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1)));}

a(n) = A000010(A057521(n)).
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A000010(n), with equality if and only if n is powerful (A001694).
Multiplicative with a(p) = 1, and a(p^e) = (p-1)*p^(e-1) if e >= 2.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(s-1) + 1/p^(2*s-2) - 2/p^(2*s-1)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 2/p^(3/2) - 1/p^2 - 2/p^(5/2)) = 1.96428740396979919886... .

A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69

Offset: 1

Amiram Eldar, Jun 15 2025

First differs from A053165 at n = 32 = 2^5: a(32) = 1 while A053165(32) = 2.
First differs from A383764 at n = 32 = 2^5: a(32) = 1 while A383764(32) = 32.
Equivalently, a(n) is the least divisor d of n such that n/d is a 4-full number (A036967).

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

Cf. A013661, A036967, A046100, A053165, A058035, A077610, A383764.

The largest unitary divisor of n that is: A000265 (odd), A006519 (power of 2), A055231 (squarefree), A057521 (powerful), A065330 (5-rough), A065331 (3-smooth), A350388 (square), A350389 (exponentially odd), A360539 (cubefree), A360540 (cubefull), A366126 (cube), A367168 (exponentially 2^n), this sequence (biquadratefree).

f[p_, e_] := If[e < 4, p^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] < 4, f[i, 1]^f[i, 2], 1)); }

a(n) = 1 if and only if n is a 4-full number (A036967).
a(n) = n if and only if n is a biquadratefree number (A046100).
Multiplicative with a(p^e) = p^e if e <= 3, and 1 otherwise.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(-s) + p^(2-2*s) - p^(1-2*s) - p^(2-3*s) + p^(3-3*s) - p^(3-4*s) + p^(-4*s)).
Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^6 + 1/p^8 - 1/p^9) = 0.56331392082909224894... .

A328015 Decimal expansion of the growth constant for the number of terms of A328014 (numbers whose powerful part is larger than their powerfree part).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 01 2019

Cloutier et al. showed that the number of terms of A328014 below x is D0 * x^(3/4) + O(x^(2/3)*log(x)), where D0 is this constant.

			1.115436631111013698931934302941096327033286649113053...

Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.

Cf. A057521, A055231, A328014.

$MaxExtraPrecision = 500; m = 500; f[x_] := (1 - x)*(1 + (1 - x)*x/(1 + x^(3/2))); c = LinearRecurrence[{2, -3, 2, 1, -3, 3, -1}, {0, 0, 0, -8, -5, 6, 14}, m]; RealDigits[(4/3)*(Zeta[3/2]/Zeta[3])*f[1/2]*f[1/3]*Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - 1/2^(n/2) - 1/3^(n/2))/n, {n, 3, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals (4/3)*(zeta(3/2)/zeta(3)) * Product_{p prime} (1 - 1/p)*(1 + (1-1/p)/(p*(1 + 1/p^(3/2)))).

A336224 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100

Offset: 1

Amiram Eldar, Jul 12 2020

Terms k of A335275 such that A000188(k) is a term of A028260.
Numbers whose powerful part (A057521) is the square of a term of A028260.
The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then the largest square dividing k is 1 which is a unitary divisor, sqrt(1) has 0 prime divisors, and 0 is even.
The asymptotic density of this sequence is (5 * Product_{p prime} (1 - 1/(p^2*(p+1))) + 2 * Product_{p prime} (1 + 1/(p^2*(p+1))))/10 = (5 * 0.881513... + 2 * 1.125606...)/10 = 0.665878294481337275662425136416469977597382409701642... (Cohen, 1964; the first product is A065465).

			16 is a term since the largest square dividing 16 is 16, which is a unitary divisor, sqrt(16) = 4, 4 = 2 * 2 has 2 prime divisors, and 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.

Intersection of A335275 and A336222.

Cf. A000188, A001222, A005117, A028260, A057521, A065465.

seqQ[n_] := AllTrue[(e = FactorInteger[n][[;; , 2]]), # == 1 || EvenQ[#] &] && EvenQ @ Total[Select[e, # > 1 &]/2]; Select[Range[100], seqQ]

A349574 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344696(i) = A344696(j) and A344697(i) = A344697(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 2, 1, 1, 1, 5, 1, 4, 1, 2, 1, 1, 1, 3, 6, 1, 7, 2, 1, 1, 1, 8, 1, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 5, 10, 6, 1, 2, 1, 7, 1, 3, 1, 1, 1, 2, 1, 1, 4, 11, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 6, 2, 1, 1, 1, 5, 13, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 1, 8, 1, 10, 4, 14, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Nov 22 2021

nonn

Restricted growth sequence transform of the ordered pair [A344696(n), A344697(n)].
For all i, j, A003557(i) = A003557(j) => a(i) = a(j); in other words, this sequence is a function of A003557. This follows because A344696(n) = A344696(A057521(n)), A344697(n) = A344696(A057521(n)), and A057521(n) = A064549(A003557(n)).
Apparently, A081770 gives the positions of 2's, which occur on those n where A344696(n) = 7 and A344697(n) = 6.

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

Cf. A000203, A001615, A003557, A005117 (positions of 1's), A057521, A064549, A081770, A280292, A344695, A344696, A344697.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
Aux349574(n) = { my(s=sigma(n),u=A001615(n),g=gcd(u,s)); [s/g, u/g]; };
v349574 = rgs_transform(vector(up_to, n, Aux349574(n)));
A349574(n) = v349574[n];

For all n >= 1, a(n) = a(A057521(n)). [See comments]

A371601 Nonsquarefree numbers whose largest nonunitary prime divisor is smaller than their smallest unitary prime divisor, if it exists.

Original entry on oeis.org

4, 8, 9, 12, 16, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 52, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 169, 171, 172, 175, 176

Offset: 1

Amiram Eldar, Mar 29 2024

Subsequence of A283050 and first differs from it at n = 100: A283050(100) = 300 = 2^2 * 3 * 5^2 is not a term of this sequence.
Powerful numbers and nonpowerful numbers k such that 1 < A249740(k) < A277698(k), or equivalently, 1 < A006530(A057521(k)) < A020639(A055231(k)).
The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} f(p)/(p^2-p+1) = 0.32131800923..., where f(p) = Product_{primes q <= p} (q^2-q+1)/(q^2-1).

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

Subsequence of A013929 and A283050.

Cf. A001694, A006530, A020639, A052485, A055231, A057521, A059956, A249740, A277698.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[e] > 1 && (Min[e] > 1 || Max[e[[FirstPosition[e, 1][[1]] ;; -1]]] == 1)]; Select[Range[200], q]

is(n) = {my(e = apply(x->if(x > 1, 2, 1), factor(n)[,2])); n > 1 && vecmax(e) > 1 && vecsort(e, , 4) == e;}

A375342 The maximum exponent in the prime factorization of the numbers whose powerful part is a power of a squarefree number that is larger than 1.

Original entry on oeis.org

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 2, 3, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 2, 3, 2, 3, 6, 2, 2, 2, 2, 4, 2, 3, 2, 5, 2, 2, 3, 2, 2, 4, 2, 5, 2, 2, 3, 3, 2, 8, 2, 2, 3, 2, 3, 4, 2, 2, 2, 3

Offset: 1

Amiram Eldar, Aug 12 2024

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

Cf. A051903, A057521, A067259, A375142, A375341.

s[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] > 0 && SameQ @@ e, e[[1]], Nothing]]; Array[s, 300]

lista(kmax) = {my(e); for(k = 1, kmax, e = select(x -> x > 1, factor(k)[,2]); if(#e > 0 && vecmin(e) == vecmax(e), print1(e[1], ", ")));}

a(n) = A051903(A375142(n)).
a(n) = 2 if and only if A375142(n) is in A067259.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} k*d(k) / Sum_{k>=2} d(k) = 2.70113273169250927084..., where d(k) = (f(k)-1)/zeta(2) is the asymptotic density of terms m of A375142 with A051903(m) = k, f(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i), if k is even, and f(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i) if k is odd > 1.

A383795 Dirichlet g.f.: zeta(2s-2) zeta(s)^2.

Original entry on oeis.org

1, 2, 2, 7, 2, 4, 2, 12, 12, 4, 2, 14, 2, 4, 4, 33, 2, 24, 2, 14, 4, 4, 2, 24, 28, 4, 22, 14, 2, 8, 2, 54, 4, 4, 4, 84, 2, 4, 4, 24, 2, 8, 2, 14, 24, 4, 2, 66, 52, 56, 4, 14, 2, 44, 4, 24, 4, 4, 2, 28, 2, 4, 24, 139, 4, 8, 2, 14, 4, 8, 2, 144, 2, 4, 56, 14, 4, 8, 2, 66, 113

Offset: 1

Vaclav Kotesovec, May 10 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A035316, A057521.

f[p_, e_] := If[OddQ[e], Sum[(e+1-2*k) * p^(2*k), {k, 0, (e-1)/2}], Sum[(e+1-2*k) * p^(2*k), {k, 0, e/2}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 24 2025 *)

for(n=1, 100, print1(direuler(p=2, n, 1/((1-p^2*X^2)*(1-X)^2))[n], ", "))

Sum_{k=1..n} a(k) ~ zeta(3/2)^2 * n^(3/2)/3 - n*(log(n) + 2*log(2*Pi) + 2*gamma - 1)/2, where gamma is the Euler-Mascheroni constant A001620.

Multiplicative with a(p^e) = Sum_{k=0..(e-1)/2} (e+1-2*k) * p^(2*k) if e is odd, and Sum_{k=0..e/2} (e+1-2*k) * p^(2*k) if e is even. - Amiram Eldar, May 24 2025

A173518 Solutions z of the Diophantine equation x^3 + y^3 = 6z^3.

Original entry on oeis.org

21, 960540, 16418498901144294337512360, 436066841882071117095002459324085167366543342937477344818646196279385305441506861017701946929489111120

Offset: 1

Michel Lagneau, Feb 20 2010

nonn

A. Nitaj proved Erdős's conjecture (1975) and claimed that there exist infinitely many triples of 3-powerful numbers a,b,c with (a,b) = 1, such that a+b=c, because the equation x^3 + y^3 = 6z^3 admits an infinite number of solutions, and given by the recurrence equations (see formula). It is proved that a=x(k)^3, b=y(k)^3, and c=6c^3, and are 3-powerful numbers for each k >= 1.

			37^3 + 17^3 = 6*21^3.

J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses, 2008, p. 348.
Mordell, L. J. (1969). Diophantine equations. Academic Press. ISBN 0-12-506250-8

P. Erdős, C. Pomerance, and A. Sárközy, On locally repeated values of certain arithmetic functions, II, Acta Math. Hungarica 49 (1987), pp. 251-259. [alternate link]
A. Nitaj, On a conjecture of Erdős on 3-powerful numbers, Bull. London Math. Soc. 27 (1995), no. 4, 317-318.
Wikipedia, Diophantine equation

Cf. A050240, A050241, A057521, A060859, A113839, A115645, A115651, A115676, A115686, A115687, A115689, A115691, A115693, A115695, A115697, A116064, A140172.

x0:=37:y0:=17:z0:=21: for p from 1 to 5 do: x1:=x0*(x0^3+ 2*y0^3):y1:=-y0*(2*x0^3+ y0^3):z1:=z0*(x0^3- y0^3): print(z1) : x0 :=x1 :y0 :=y1 :z0 :=z1 :od :

We generate the solutions (x(k),y(k),z(k)) from the initial solution x(0) = 37, y(0)=17, z(0)=21 x(k+1) = x(k)*(x(k)^3 + 2*y(k)^3) y(k+1) = -y(k)*(2*x(k)^3 + y(k)^3) z(k+1) = z(k)*(x(k)^3 - y(k)^3).

cp's OEIS Frontend

A375146 Numbers whose prime factorization has exactly one exponent that is larger than 3.

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A380160 a(n) is the value of the Euler totient function when applied to the powerful part of n.

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A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

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A328015 Decimal expansion of the growth constant for the number of terms of A328014 (numbers whose powerful part is larger than their powerfree part).

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A336224 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of prime divisors (counted with multiplicity).

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A349574 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344696(i) = A344696(j) and A344697(i) = A344697(j), for all i, j >= 1.

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A371601 Nonsquarefree numbers whose largest nonunitary prime divisor is smaller than their smallest unitary prime divisor, if it exists.

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A375342 The maximum exponent in the prime factorization of the numbers whose powerful part is a power of a squarefree number that is larger than 1.

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A383795 Dirichlet g.f.: zeta(2s-2) zeta(s)^2.