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.

Showing 1-4 of 4 results.

A377818 Powerful numbers that have a single even exponent in their prime factorization.

Original entry on oeis.org

4, 9, 16, 25, 49, 64, 72, 81, 108, 121, 169, 200, 256, 288, 289, 361, 392, 432, 500, 529, 625, 648, 675, 729, 800, 841, 961, 968, 972, 1024, 1125, 1152, 1323, 1352, 1369, 1372, 1568, 1681, 1728, 1849, 2000, 2209, 2312, 2401, 2592, 2809, 2888, 3087, 3200, 3267, 3481
Offset: 1

Views

Author

Amiram Eldar, Nov 09 2024

Keywords

Comments

Each term can be represented in a unique way as m * p^(2*k), k >= 1, where m is a cubefull exponentially odd number (A335988) and p is a prime that does not divide m.
Powerful numbers k such that A350388(k) is a prime power with an even positive exponent (A056798 \ {1}).

Links

Crossrefs

Intersection of A001694 and A377816.
Subsequence of A377819.
Cf. A056798, A065487, A335988, A350388.

Programs

  • Mathematica
    With[{max = 3500}, 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, 0, 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))) * Sum_{p prime} (p/(p^3-p+1)) = 0.61399274770712398109... .

A377817 Numbers that have more than one even exponent in their prime factorization.

Original entry on oeis.org

36, 100, 144, 180, 196, 225, 252, 300, 324, 396, 400, 441, 450, 468, 484, 576, 588, 612, 676, 684, 700, 720, 784, 828, 882, 900, 980, 1008, 1044, 1089, 1100, 1116, 1156, 1200, 1225, 1260, 1296, 1300, 1332, 1444, 1452, 1476, 1521, 1548, 1575, 1584, 1600, 1620, 1692, 1700, 1764, 1800
Offset: 1

Views

Author

Amiram Eldar, Nov 09 2024

Keywords

Comments

Subsequence of A072413 and differs from it by not having the terms 216, 1000, 1080, 1512, ... .
Each term can be represented in a unique way as m * k^2, where m is an exponentially odd number (A268335) and k is a composite number that is coprime to m.
Numbers k such that A350388(k) is a square of a composite number (A062312 \ {1}).
The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p*(p+1))) * (1 + Sum_{p prime} 1/(p^2+p-1)) = 0.032993560887093165933... .

Links

Crossrefs

Complement of the union of A268335 and A377816.
Subsequence of A072413.
Cf. A062312, A162645, A350388.

Programs

  • Mathematica
    Select[Range[1800], Count[FactorInteger[#][[;; , 2]], _?EvenQ] > 1 &]
  • PARI
    is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); #select(x -> !(x%2), e) > 1);

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));

A381314 Powerful numbers that have a single exponent in their prime factorization that equals 2.

Original entry on oeis.org

4, 9, 25, 49, 72, 108, 121, 144, 169, 200, 288, 289, 324, 361, 392, 400, 500, 529, 576, 675, 784, 800, 841, 961, 968, 972, 1125, 1152, 1323, 1352, 1369, 1372, 1568, 1600, 1681, 1849, 1936, 2025, 2209, 2304, 2312, 2500, 2704, 2809, 2888, 2916, 3087, 3136, 3200
Offset: 1

Views

Author

Amiram Eldar, Feb 19 2025

Keywords

Comments

Number of the form A036966(m)/p, m >= 2, where p is a prime divisor of A036966(m).

Links

Crossrefs

Subsequence of A001694, A377816 and A377818.
Subsequences: A001248, A143610, A179646, A179689, A179699, A189988, A189990, A190106, A190115, A190470, A190471.
Cf. A036966, A065483.

Programs

  • Mathematica
    With[{max = 3200}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], 2] == 1 &]]
  • PARI
    isok(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> (x==2), e) == 1);

Formula

Sum_{n>=1} 1/a(n) = Sum_{p prime}((p-1)/(p^3-p^2+1)) * Product_{p prime} (1 + 1/(p^2*(p-1))) = 0.53045141423939736076... .
Showing 1-4 of 4 results.

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

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