cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-37 of 37 results.

A377819 Powerful numbers that have no more than one even exponent in their prime factorization.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 72, 81, 108, 121, 125, 128, 169, 200, 216, 243, 256, 288, 289, 343, 361, 392, 432, 500, 512, 529, 625, 648, 675, 729, 800, 841, 864, 961, 968, 972, 1000, 1024, 1125, 1152, 1323, 1331, 1352, 1369, 1372, 1568, 1681, 1728, 1849, 1944, 2000
Offset: 1

Views

Author

Amiram Eldar, Nov 09 2024

Keywords

Comments

Powerful numbers k such that A350388(k) is either 1 or a prime power with an even positive exponent (A056798 \ {1}).

Links

Crossrefs

Disjoint union of A335988 and A377818.
Intersection of A001694 and the complement of A377817.
Cf. A056798, A065487, A350388.

Programs

  • Mathematica
    With[{max = 2000}, Select[Union@ Flatten@Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], _?EvenQ] <= 1 &]]
  • PARI
    is(k) = if(k == 1, 1, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> !(x%2), e) <= 1);

Formula

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) * (1 + Sum_{p prime} (p/(p^3-p+1))) = 1.84528389659572754387... .

A377991 Numbers k such that A351568(k) and A351569(k) are not coprime, where A351568 and A351569 are the sum of divisors of the largest unitary divisor of n that is a square, and of the largest unitary divisor of n that is an exponentially odd number, respectively.

Original entry on oeis.org

52, 98, 156, 164, 245, 260, 294, 332, 338, 364, 388, 392, 468, 490, 492, 539, 556, 572, 668, 722, 724, 735, 780, 820, 833, 845, 882, 884, 892, 927, 972, 976, 980, 988, 996, 1004, 1014, 1078, 1092, 1125, 1127, 1148, 1164, 1172, 1176, 1196, 1228, 1274, 1300, 1352, 1396, 1404, 1421, 1470, 1476, 1508, 1525, 1568, 1573
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2024

Keywords

Examples

			A351568(52) = 7 and A351569(52) = 14, so they share a factor (7), and therefore 52 is included as a term.

Links

Crossrefs

Positions k where A377990(k) is larger than A051027(k).
Cf. A350388, A350389, A351568, A351569.
Subsequence of A336548.

Programs

  • PARI
    A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351568(n) = sigma(A350388(n));
isA377991(n) = (1A351568(n), sigma(n)/A351568(n)));

Formula

{k such that gcd(A351568(n),A351569(n)) > 1}.
{k such that A377990(k) > A051027(k)}.

A380164 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 6, 1, 1, 2, 1, 1, 1, 8, 1, 6, 1, 2, 1, 1, 1, 1, 20, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 8, 42, 20, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 6, 32, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 20, 2, 1, 1, 1, 8, 54, 1, 1, 2, 1, 1, 1, 1, 1, 6, 1, 2, 1, 1, 1, 1, 1, 42, 6, 40
Offset: 1

Views

Author

Amiram Eldar, Jan 14 2025

Keywords

Links

Crossrefs

Cf. A000010, A000290, A002117, A077591, A268335, A350388, A351568, A365401, A380165.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], 1, (p-1)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, 1, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1)));}

Formula

a(n) = A000010(A350388(n)).
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
a(n) <= A000010(n), with equality if and only if n is either a square (A000290) or twice an odd square (A077591 \ {1}).
Multiplicative with a(p^e) = (p-1)*p^(e-1) if e is even, and 1 otherwise.
Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s-1) - 1/p^(2*s) - 1/p^(3*s-2) + 1/p^(4*s-1)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^2 - 1/p^(5/2) - 1/p^3 + 1/p^5) = 1.16404670858123447768... .

A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 204, 207, 208, 212, 220, 228, 234, 236
Offset: 1

Views

Author

Amiram Eldar, Feb 19 2025

Keywords

Comments

Numbers k whose largest unitary divisor that is a square, A350388(k), is a prime power (A246655), or equivalently, A350388(k) is in A056798 \ {1}.
Numbers having exactly one non-unitary prime factor and its multiplicity is even.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m} with m >= 1, i.e., any number (including zero) of 1's and then a single even number.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} p/((p-1)*(p+1)^2) = 0.24200684327095676029... .

Links

Crossrefs

Complement of A381312 within A190641.
Cf. A057521, A118914, A246655, A350388.
Subsequence of: A377816.
Subsequences: A001248, A030514, A030516, A030629, A030631, A054753, A056798 \ {1}, A085987, A178739, A179644, A179645, A179668, A179672, A179693, A179747, A189982, A189983, A189985, A189987, A190110, A190292, A190388.

Programs

  • Mathematica
    q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;;,2]]]}, EvenQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000],q]
  • PARI
    isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); !(e[1] % 2) && (#e == 1 || e[2] == 1));

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

Views

Author

Amiram Eldar, Jun 15 2025

Keywords

Comments

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).

Links

Crossrefs

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).

Programs

  • Mathematica
    f[p_, e_] := If[e < 4, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] < 4, f[i, 1]^f[i, 2], 1)); }

Formula

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... .

A360158 a(n) is the number of unitary divisors of n that are odd squares minus the number of unitary divisors of n that are even squares.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Jan 29 2023

Keywords

Comments

The unitary analog of A344299.
The least term that is larger than 2 is a(225) = 4.

Links

Crossrefs

Cf. A002117, A013661, A016742, A016754, A056624, A344299, A350388, A360157.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], 1, 2]; f[2, e_] := If[OddQ[e], 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, 1, if(f[i, 1] == 2, 0, 2)));}

Formula

a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} (-1)^(d+1).
Multiplicative with a(2^e) = 1 if e is odd and 0 if e is even, and for p > 2, a(p^e) = 1 if e is odd and 2 if e is even.
Dirichlet g.f.: (zeta(s)*zeta(2*s)/zeta(3*s)) * (4^s + 2^s - 1)/(4^s + 2^s + 1).
Sum_{k=1..n} a(k) ~ c * n, where c = 5*zeta(2)/(7*zeta(3)) = 0.977451984014... .

A367989 The sum of square divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 1, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1, 1, 26, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 50, 1, 1, 1, 1, 1, 1, 1, 5, 10, 1, 1, 21, 50, 26, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 10, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 07 2023

Keywords

Links

Crossrefs

Cf. A002117, A035316, A268335, A350388, A367988.

Programs

  • Mathematica
    f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, 1, (f[i,1]^(f[i,2] + 2) - 1)/(f[i,1]^2 - 1)));}

Formula

a(n) = A035316(A350388(n)).
Multiplicative with a(p^e) = (p^(e+2)-1)/(p^2-1) if e is even and 1 otherwise.
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^(5/2)) = 0.69451968056653021193... .
Previous Showing 31-37 of 37 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.