A062503 -id:A062503 - cp's OEIS frontend

A368105 The number of bi-unitary divisors of n that are powerful (A001694).

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

Offset: 1

Amiram Eldar, Dec 12 2023

First differs from A095691 and A365552 at n = 32.

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

Cf. A013661, A001694, A005117, A062503, A286324,
Similar sequences: A005361, A323308, A360721.
Cf. A095691, A365552.

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

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

Multiplicative with a(p^e) = e if e = 2 or e is odd, 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 equals the square of a squarefree number (A062503).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 1/p^3 - 1/p^4 + 1/p^5) = 1.87133814920590891161... .

A368978 The number of bi-unitary divisors of n that are squares (A000290).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 11 2024

First differs from A007424, A278908, A307848, A323308, A358260 and A365549 at n = 32.

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

Cf. A000005, A000290, A005117, A013662, A062503, A222266, A286324, A368977.
Cf. A007424, A278908, A307848, A323308, A365549.
Similar sequences: A046951, A056624, A358260, A368980.

f[p_, e_] := If[OddQ[e], (e + 1)/2, 2*Floor[(e+2)/4]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2, (x+1)/2, 2*((x+2)\4)), factor(n)[, 2]));

Multiplicative with a(p^e) = (e + 1)/2 if e is odd, and 2*floor((e+2)/4) if e is even.
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 in A062503.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4 + 1/p^5) = 1.58922450321701775833... .

A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

Original entry on oeis.org

72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 1800, 2312, 2700, 2888, 3087, 3267, 3528, 4232, 4500, 4563, 5292, 5324, 5400, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10584, 10952, 11979, 12168, 12348, 13068, 13448, 13500, 14283, 14792

Offset: 1

Amiram Eldar, Jul 29 2024

Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {2, 3}.

Number k such that A051904(k) = 2 and A051903(k) = 3.

Equals A338325 \ (A062503 UNION A062838).
Subsequence of A001694 and A046100.
A143610 is a subsequence.
Cf. A051903, A051904, A136568.
Cf. A082020, A157289, A330595.

Select[Range[15000], Union[FactorInteger[#][[;; , 2]]] == {2, 3} &]

PARI
```
is(k) = Set(factor(k)[,2]) == [2, 3];
```

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) - 15/Pi^2 - zeta(3)/zeta(6) + 1 = A330595 - A082020 - A157289 + 1 = 0.047550294197921818806... .

A382889 The largest square dividing the n-th cubefree number.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 25, 1, 4, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 49, 25, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 1, 1, 25, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 49, 9, 100

Offset: 1

Amiram Eldar, Apr 07 2025

Also, the powerful part of the n-th cubefree number.

All the terms are squares of squarefree numbers (A062503).

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

Cf. A002117, A004709, A008833, A057521, A062503, A371188 (positions of 1's).

Similar sequences: A382888, A382890, A382891.

f[p_, e_] := p^If[e == 1, 0, 2]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 0, 2)), ", ")));}

a(n) = A008833(A004709(n)).
a(n) = A057521(A004709(n)).
a(n) = A382890(n)^2.
a(n) = A004709(n)/A382891(n).
a(n) = (A004709(n)/A382888(n))^2.
a(A371188(n)) = 1.
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3)^(3/2) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^2 - 1/p^(5/2)) = 1.48513488319516447978... .

A182120 Numbers for which the canonical prime factorization contains only exponents which are congruent to 2 modulo 3.

Original entry on oeis.org

1, 4, 9, 25, 32, 36, 49, 100, 121, 169, 196, 225, 243, 256, 288, 289, 361, 441, 484, 529, 676, 800, 841, 900, 961, 972, 1089, 1156, 1225, 1369, 1444, 1521, 1568, 1681, 1764, 1849, 2048, 2116, 2209, 2304, 2601, 2809, 3025, 3125, 3249, 3364, 3481, 3721, 3844

Offset: 1

Douglas Latimer, Apr 12 2012

By convention 1 is included as the first term.

			100 is included, as its canonical prime factorization (2^2)*(5^2) contains only exponents which are congruent to 2 modulo 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Douglas Latimer)

A062503 is a subsequence.
Subsequence of A001694.
Cf. A002117, A366762.

Join[{1},Select[Range[5000],Union[Mod[Transpose[FactorInteger[#]][[2]],3]] == {2}&]] (* Harvey P. Dale, Aug 18 2014 *)

{plnt=1; k=1; print1(k, ", "); plnt++;
mxind=76 ; mxind++ ; for(k=2, 2*10^6,
M=factor(k);passes=1;
sz = matsize(M)[1];
for(k=1,sz,  if( M[k,2] % 3 != 2, passes=0));
if( passes == 1 ,
print1(k, ", "); plnt++) ; if(mxind ==  plnt, break() ))}

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2]%3 != 2, return(0))); 1;} \\ Amiram Eldar, Oct 21 2023

Sum_{n>=1} 1/a(n) = zeta(3) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.56984817927051410948... . - Amiram Eldar, Oct 21 2023

A329376 Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 7, 3, 10, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Nov 16 2019

Product of those distinct prime factors that occur exactly twice in the prime factorization of n, that is, whose exponent is 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000196, A062503, A337050.

Row 3 of array A106177, and the square roots of its row 9.

f[p_, e_] := If[e == 2, p, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 11 2023 *)

A329376(n) = { my(f = factor(n)); prod(i=1,#f~,f[i, 1]^(2 == f[i, 2])); };

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X^2 + X + X^2/(-1 + 1/X)))[n], ", ")) \\ Vaclav Kotesovec, May 31 2024

Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.
a(n) <= A000196(n).
From Amiram Eldar, Feb 11 2023: (Start)
a(n) <= sqrt(n), with equality if and only if n is in A062503.
a(n) = 1 if and only if n is in A337050. (End)
From Vaclav Kotesovec, May 31 2024: (Start)
Dirichlet g.f.: zeta(2*s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(5*s-1) + 1/p^(5*s-2) + 1/p^(4*s-1) - 1/p^(4*s-2) - 1/p^(3*s-1) + 1/p^(3*s) - 1/p^(2*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(5*s-1) + 1/p^(5*s-2) + 1/p^(4*s-1) - 1/p^(4*s-2) - 1/p^(3*s-1) + 1/p^(3*s) - 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = Product_{p prime} (1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.33718787379158997196169281615215824494915412775816393888028828465611936...,
f'(1) = f(1) * Sum_{p prime} (9*p^2 - 12*p + 5) * log(p) / (p^4 - 3*p^2 + 3*p - 1) = f(1) * 3.78385641685861932254178374972226733621783278751462026270346293...
and gamma is the Euler-Mascheroni constant A001620. (End)

A368977 The number of bi-unitary divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 11 2024

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

Cf. A000005, A005117, A062503, A222266, A268335, A286324, A368978.

Similar sequences: A055076, A322483, A363825, A368979.

f[p_, e_] := If[OddQ[e], (e+3)/2, 2*Floor[e/4]+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2, (x+3)/2, 2*(x\4)+1), factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2 + 2*X^3 - X^4)/(1 - X - X^4 + X^5))[n], ", ")) \\ Vaclav Kotesovec, Jan 11 2024

Multiplicative with a(p^e) = (e+3)/2 if e is odd, and 2*floor(e/4)+1 if e is even.
a(n) >= 1, with equality if and only if n is in A062503.
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
From Vaclav Kotesovec, Jan 11 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - (1 - p^s + 2*p^(2*s)) / (p^s*(1 + p^s)*(1 + p^(2*s)))).
Let f(s) = Product_{p prime} (1 - (1 - p^s + 2*p^(2*s)) / (p^s*(1 + p^s)*(1 + p^(2*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*p^2) / (p*(1 + p)*(1 + p^2))) = 0.5715031234451924252215041182933420817059774181158824297150124265420835...,
f'(1) = f(1) * Sum_{p prime} (4*p^5 - p^4 + 2*p^3 + 2*p + 1) * log(p) / (p^7 + 2*p^6 + p^5 + 3*p^4 + p^3 + p - 1) = f(1) * 1.1422556395248477875508983912036578244050011522937179465478688905880430...
and gamma is the Euler-Mascheroni constant A001620. (End)

A081084 Nonsquarefree numbers m such that rad(m+1)=rad(m)+1, where rad(m)=A007947(m) is the squarefree kernel of m.

Original entry on oeis.org

8, 48, 224, 960, 65024, 261120, 1046528, 4190208, 268402688, 1073676288, 4294836224, 17179607040, 70368727400448, 4503599493152768, 18014398241046528, 72057593501057024, 288230375077969920

Offset: 1

Reinhard Zumkeller, Mar 04 2003

nonn

For k >= 3, 2^k*(2^(k-2)-1) is in the sequence if and only if 2^(k-1)-1 and 2^(k-2)-1 are squarefree. So if m is a term, m+1=2^(k-1)-1 is a squarefree number squared. - Lambert Herrgesell (zero815(AT)googlemail.com), Feb 18 2007

			48 = 2^4*3 is in the sequence because it is not squarefree, its squarefree kernel is 6 and the squarefree kernel of 49 = 7^2 is 7.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 48, p. 18, Ellipses, Paris 2008.

Cf. A081083, A062503.

with(numtheory): rad:=proc(n) local fs, c: fs:=convert(factorset(n),list): c:=nops(fs): product(fs[j],j=1..c) end: b:=proc(n) if mobius(n)=0 and rad(n+1)=rad(n)+1 then n else fi end:seq(b(n),n=1..1000); # Emeric Deutsch

rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
is(n)=!issquarefree(n) && rad(n+1)==rad(n)+1 \\ Charles R Greathouse IV, Aug 08 2013

a(5)-a(8) from Emeric Deutsch, Mar 29 2005

Edited and a(9) onwards supplied by Lambert Herrgesell (zero815(AT)googlemail.com), Feb 18 2007

A321070 Squares divisible by more than one cube > 1.

Original entry on oeis.org

64, 256, 576, 729, 1024, 1296, 1600, 2304, 2916, 3136, 4096, 5184, 6400, 6561, 7744, 9216, 10000, 10816, 11664, 12544, 14400, 15625, 16384, 18225, 18496, 20736, 23104, 25600, 26244, 28224, 30976, 32400, 33856, 35721, 36864, 38416, 40000, 43264, 46656, 50176

Offset: 1

Hugo Pfoertner, Oct 27 2018

nonn

			a(1) = 64 because 16^2 is divisible by 2^3 and by 4^3.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000290, A000578, A062503, A062320, A216427, A320965.

Select[Range[225]^2, Max[(e = FactorInteger[#][[;; , 2]])] > 4 || (Length[e] > 1 && Sort[e, Greater][[2]] > 2) &] (* Amiram Eldar, Jun 25 2022 *)

iscubes(n) = {my(nb = 0); fordiv(n, d, if ((d>1) && ispower(d, 3), nb++; if (nb > 1, return(1))););}
isok(n) = issquare(n) && iscubes(n); \\ Michel Marcus, Oct 27 2018

From Amiram Eldar, Jun 25 2022: (Start)
Equals A000290 \ (union of A062503 and A320965).
Sum_{n>=1} 1/a(n) = Pi^2/6 - (15/Pi^2) * (1 + Sum_{k>=2} (-1)^k * P(2*k)) = 0.029082273527998239268... . (End)

A327274 Dirichlet g.f.: 1 / (zeta(s)^2 * (1 - 2^(1 - s))).

Original entry on oeis.org

1, 0, -2, 1, -2, 0, -2, 2, 1, 0, -2, -2, -2, 0, 4, 4, -2, 0, -2, -2, 4, 0, -2, -4, 1, 0, 0, -2, -2, 0, -2, 8, 4, 0, 4, 1, -2, 0, 4, -4, -2, 0, -2, -2, -2, 0, -2, -8, 1, 0, 4, -2, -2, 0, 4, -4, 4, 0, -2, 4, -2, 0, -2, 16, 4, 0, -2, -2, 4, 0, -2, 2, -2, 0, -2, -2, 4, 0, -2, -8

Offset: 1

Ilya Gutkovskiy, Oct 22 2019

Dirichlet inverse of A048272.

Moebius transform of A067856.

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

Cf. A007427, A008683, A048272, A062503 (positions of 1's), A067856, A327268.

a[1] = 1; a[n_] := Sum[Sum[(-1)^j, {j, Divisors[n/d]}] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 80}]
f[p_, e_] := Switch[e, 1, -2, 2, 1, , 0]; f[2, e] := 2^(e-2); f[2, 1] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

A067856(n) = { my(k); if(n<1, 0, k=valuation(n, 2); moebius(n/2^k)*2^max(0, k-1)); }; \\ From A067856
A327274(n) = sumdiv(n,d,moebius(n/d)*A067856(d));

a(1) = 1; a(n) = -Sum_{d|n, dA048272(n/d) * a(d).
a(n) = Sum_{d|n} mu(n/d) * A067856(d).
a(n) = 0 if n == 2 (mod 4). - Bernard Schott, Dec 07 2021
Multiplicative with a(2) = 0, a(2^e) = 2^(e-2) for e >= 2, and for an odd prime p, a(p) = -2, a(p^2) = 1, and a(p^e) = 0 for e >= 3. - Amiram Eldar, Sep 15 2023

cp's OEIS Frontend

A368105 The number of bi-unitary divisors of n that are powerful (A001694).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A368978 The number of bi-unitary divisors of n that are squares (A000290).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A382889 The largest square dividing the n-th cubefree number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A182120 Numbers for which the canonical prime factorization contains only exponents which are congruent to 2 modulo 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A329376 Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A368977 The number of bi-unitary divisors of n that are exponentially odd numbers (A268335).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI