A057521 -id:A057521 - cp's OEIS frontend

A375145 Numbers whose prime factorization has exactly one exponent that is larger than 2.

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 120, 125, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 343, 344

Offset: 1

Amiram Eldar, Aug 01 2024

Subsequence of A046099 and first differs from it at n = 35: A046099(35) = 216 = 2^3 * 3^3 is not a term of this sequence.

The asymptotic density of this sequence is (1/zeta(3)) * Sum_{p prime} 1/(p^3-1) = A286229 / A002117 = 0.16148833663564192901... .

			8 = 2^3 is a term since its prime factorization has exactly one exponent, 3, that is larger than 2.

Subsequence of A046099, A356862.

Cf. A002117, A057521, A190641, A246549, A286229, A375146.

q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 2 &)] == 1; Select[Range[350], q]

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

A135544 Decimal expansion of (-1)^(I Pi).

Original entry on oeis.org

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

Offset: 0

Marvin Ray Burns, Feb 22 2008, Feb 23 2008

			(-1)^(I*Pi) = exp(-Pi)^(Pi) = 0.000051723186...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Doctor Bombelli, The Math Forum, Exp[ -I * Pi] = Cos[Pi] + i Sin[Pi] = -1.

Cf. A057521, A002388, A093580.

N[(-1)^(I Pi), 1000] FullSimplify[(-1)^(I Pi) == Exp[ -Pi]^Pi == (Exp[ -(1/2)*Pi])^(2*Pi) == Sqrt[Exp[ -Pi]^Pi/(Exp[Pi]^Pi)] == Exp[(-1/2*Pi)]^(Gamma[1/6]*Gamma[5/6]) == 1/(Sqrt[Exp[Pi]^(2*Pi)]) == (Exp[ -(1/2)*Pi])^(2*Pi) == Exp[ -Pi^2]]
Join[{0, 0, 0, 0}, RealDigits[(Exp[-Pi])^(Pi), 10, 96][[1]]] (* G. C. Greubel, Oct 18 2016 *)

exp(-Pi^2) \\ Charles R Greathouse IV, Jan 23 2025

real((-1)^(I*Pi)) \\ Charles R Greathouse IV, Jan 23 2025

a(n) = (-1)^(I Pi) = exp(-Pi)^Pi = (exp( -(1/2)*Pi))^(2*Pi) = sqrt(exp( -Pi)^Pi/(exp(Pi)^Pi)) = exp[(-1/2*Pi)]^(Gamma[1/6]*Gamma[5/6]) = 1/(sqrt[exp[Pi]^(2*Pi)]) = (exp[ -(1/2)*Pi])^(2*Pi) = exp[ -Pi^2].

Offset corrected R. J. Mathar, Jan 26 2009

A336222 Numbers k such that the square root of the largest square dividing k 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, 32, 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, 72, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95

Offset: 1

Amiram Eldar, Jul 12 2020

nonn

Numbers k such that A000188(k) is a term of A028260.
The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then A000188(k) = 1, 1 has 0 prime divisors, and 0 is even.
A number k is a term if and only if its powerful part, A057521(k), is a term.
The asymptotic density of this sequence is 7/10 (Cohen, 1964).
The corresponding sequence of numbers k such that the square root of the largest square dividing k has an even number of distinct prime divisors, i.e., numbers k such that A000188(k) is a term of A030231, is A333634.

			2 is a term since the largest square dividing 2 is 1, sqrt(1) = 1, 1 has 0 prime divisors, and 0 is even.
16 is a term since the largest square dividing 16 is 16, 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.

Cf. A000188, A001222, A005117, A028260, A030231, A057521, A333634.

f[p_, e_] := p^Floor[e/2]; Select[Range[100], EvenQ[PrimeOmega[Times @@ (f @@@ FactorInteger[#])]] &]

A336223 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of distinct prime divisors.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101, 102, 103

Offset: 1

Amiram Eldar, Jul 12 2020

nonn

First differs from A333634 at n = 47.
Terms k of A335275 such that A000188(k) is a term of A030231.
Numbers whose powerful part (A057521) is a square term of A030231.
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) = 1 has 0 prime divisors, and 0 is even.
The asymptotic density of this sequence is (Product_{p prime} (1 - 1/(p^2*(p+1))) + Product_{p prime} (1 - (2*p+1)/(p^2*(p+1))))/2 = (0.881513... + 0.394391...)/2 = 0.637952807730728551636349961980617856650450613867264... (Cohen, 1964; the first product is A065465).

			36 is a term since the largest square dividing 36 is 36, which is a unitary divisor, sqrt(36) = 6, 6 = 2 * 3 has 2 distinct 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 A333634 and A335275.

Cf. A000188, A001221, A005117, A030231, A057521, A065465.

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

A368104 The number of bi-unitary divisors of the powerful part of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 12 2023

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

Cf. A013661, A001694, A005117, A057521, A286324.

Similar sequences: A323308, A357669, A368106.

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

a(n) = vecprod(apply(x -> if(x == 1 || !(x%2), x, x+1), factor(n)[, 2]));

a(n) = A286324(A057521(n)).
Multiplicative with a(p^e) = e if e is even or e = 1, and e + 1 otherwise.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A286324(n), with equality if and only if n is powerful (A001694).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 2/p^3 - 1/p^4) = 2.12258268547914758409... .

A368106 The number of infinitary divisors of the powerful part of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 12 2023

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

Cf. A001694, A005117, A037445, A057521.

Similar sequences: A323308, A357669, A368104.

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

a(n) = vecprod(apply(x -> if(x == 1, 1, 2^hammingweight(x)), factor(n)[, 2]));

a(n) = A037445(A057521(n)).
Multiplicative with a(p) = 1 and a(p^e) = 2^A000120(e) for e >= 2.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A037445(n), with equality if and only if n is powerful (A001694).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.89684906463124350536..., where f(x) = (1-x) * (Product_{k>=0} (1 + 2*x^(2^k)) - x).

A370783 a(n) is the numerator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 02 2024

			Fractions begin with: 1, 1, 1, 3/2, 1, 1, 1, 3/2, 4/3, 1, 1, 3/2, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Cf. A005117, A057521, A157289, A295295, A332880, A370784 (denominators).

a[n_] := Numerator[Times @@ (1 + 1/Select[FactorInteger[n], Last[#] > 1 &][[;; , 1]])]; Array[a, 100]

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

a(n) = A332880(A057521(n)).
Let f(n) = a(n)/A370784(n):
f(n) is multiplicative with f(p) = 1 and f(p^e) = 1 + 1/p for e >= 2.
f(n) = 1 if and only if n is squarefree (A005117).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = zeta(3)/zeta(6) = 1.181564... (A157289) (Jakimczuk, 2024).

A370784 a(n) is the denominator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Mar 02 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Cf. A005117, A057521, A295295, A332881, A370783 (numerators).

a[n_] := Denominator[Times @@ (1 + 1/Select[FactorInteger[n], Last[#] > 1 &][[;; , 1]])]; Array[a, 100]

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

a(n) = A332881(A057521(n)).

a(n) = 1 if n is squarefree (A005117).

A375074 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no higher exponents.

Original entry on oeis.org

72, 108, 200, 360, 392, 500, 504, 540, 600, 675, 756, 792, 936, 968, 1125, 1176, 1188, 1224, 1323, 1350, 1352, 1368, 1372, 1400, 1404, 1500, 1656, 1800, 1836, 1960, 2052, 2088, 2200, 2232, 2250, 2312, 2484, 2520, 2600, 2646, 2664, 2700, 2888, 2904, 2952, 3087

Offset: 1

Amiram Eldar, Jul 29 2024

Numbers whose powerful part (A057521) is a term of A375073.

The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(3) + 1/zeta(2) - zeta(6)/(zeta(2) * zeta(3)) * c = A215267 - A088453 + A059956 - A068468 * c = 0.0156712080080470088619..., where c = Product_{p prime} (1 + 2/p^3 - 1/p^4 + 1/p^5).

Equals A046100 \ (A004709 UNION A336591).
Disjoint union of A375073 and A375075.
Cf. A051903, A057521.
Cf. A002117, A013661, A013662, A013664, A059956, A068468, A088453, A215267.

Select[Range[3000], Union[Select[FactorInteger[#][[;; , 2]], # > 1 &]] == {2, 3} &]

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

A051903(a(n)) = 3.

A375144 Numbers whose prime factorization has exactly two exponents that equal 2 and has no higher exponents.

Original entry on oeis.org

36, 100, 180, 196, 225, 252, 300, 396, 441, 450, 468, 484, 588, 612, 676, 684, 700, 828, 882, 980, 1044, 1089, 1100, 1116, 1156, 1225, 1260, 1300, 1332, 1444, 1452, 1476, 1521, 1548, 1575, 1692, 1700, 1900, 1908, 1980, 2028, 2100, 2116, 2124, 2156, 2178, 2196

Offset: 1

Amiram Eldar, Aug 01 2024

Numbers of the form m * p^2 * q^2, where p < q are primes, and m is a squarefree number such that gcd(m, p*q) = 1.
Numbers whose powerful part (A057521) is a square of a squarefree semiprime (A085986).
The asymptotic density of this sequence is ((Sum_{p prime} 1/(p*(p+1)))^2 - Sum_{p prime} 1/(p*(p+1))^2)/(2*zeta(2)) = 0.022124574473271163980012... .

			36 = 2^2 * 3^2 is a term since its prime factorization has exactly two exponents and both are equal to 2.

Cf. A057521, A060687, A085986, A179119.

Subsequence: A179643.

q[n_] := Module[{e = Sort[FactorInteger[n][[;; , 2]], Greater]}, Length[e] > 1 && e[[1;;2]] == {2, 2} && If[Length[e] > 2, e[[3]] == 1, True]]; Select[Range[2200], q]

is(k) = {my(e = vecsort(factor(k)[,2], , 4)~); #e > 1 && e[1..2] == [2,2] && if(#e > 2, e[3] == 1, 1);}

cp's OEIS Frontend

A375145 Numbers whose prime factorization has exactly one exponent that is larger than 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A135544 Decimal expansion of (-1)^(I Pi).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A336222 Numbers k such that the square root of the largest square dividing k has an even number of prime divisors (counted with multiplicity).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A336223 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of distinct prime divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A368104 The number of bi-unitary divisors of the powerful part of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A368106 The number of infinitary divisors of the powerful part of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A370783 a(n) is the numerator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A370784 a(n) is the denominator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

Views

Author

Keywords