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.

A386803 Numbers without an exponent 4 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70
Offset: 1

Views

Author

Amiram Eldar, Aug 03 2025

Keywords

Comments

First differs from its subsequence A209061 at n = 246: a(246) = 256 = 2^8 is not a term of A209061.
First differs from its subsequences A115063 and A369939 at n = 62: a(62) = 64 = 2^6 is not a term of A115063.
The complement of this sequence is a subsequence of A336595.
These numbers were named semi-4-free integers by Suryanarayana (1971).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^4 + 1/p^5) = 0.95908865419555719109... (Suryanarayana, 1971).

Links

Crossrefs

Subsequences: A115063, A209061, A369939.
Numbers without an exponent k in their prime factorization: A001694 (k=1), A337050 (k=2), A386799 (k=3), this sequence (k=4), A386807 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 4: this sequence (m=0), A386804 (m=1), A386805 (m=2), A386806 (m=3).

Programs

  • Mathematica
    Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 4] &]
  • PARI
    isok(k) = vecsum(apply(x -> if(x == 4, 1, 0), factor(k)[, 2])) == 0;

A386804 Numbers that have exactly one exponent in their prime factorization that is equal to 4.

Original entry on oeis.org

16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 784, 810, 816, 848, 880, 891, 912, 944, 976, 1008, 1040, 1053, 1072, 1104, 1134, 1136, 1168, 1200, 1232, 1250
Offset: 1

Views

Author

Amiram Eldar, Aug 03 2025

Keywords

Comments

Subsequence of A336595 and first differs from it at n = 21: A336595(21) = 512 = 2^9 is not a term of this sequence.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^4 + 1/p^5) * Sum_{p prime} (p-1)/(p^5 - p + 1) = 0.04058504714976055893... (Elma and Martin, 2024).

Links

Crossrefs

Subsequence of A336595.
Numbers that have exactly one exponent in their prime factorization that is equal to k: A119251 (k=1), A386796 (k=2), A386800 (k=3), this sequence (k=4), A386808 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 4: A386803 (m=0), this sequence (m=1), A386805 (m=2), A386806 (m=3).

Programs

  • Mathematica
    f[p_, e_] := If[e == 4, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[1300], s[#] == 1 &]
  • PARI
    isok(k) = vecsum(apply(x -> if(x == 4, 1, 0), factor(k)[, 2])) == 1;

A336596 Numbers whose number of divisors is divisible by 7.

Original entry on oeis.org

64, 192, 320, 448, 576, 704, 729, 832, 960, 1088, 1216, 1344, 1458, 1472, 1600, 1728, 1856, 1984, 2112, 2240, 2368, 2496, 2624, 2752, 2880, 2916, 3008, 3136, 3264, 3392, 3520, 3645, 3648, 3776, 3904, 4032, 4160, 4288, 4416, 4544, 4672, 4800, 4928, 5056, 5103
Offset: 1

Views

Author

Amiram Eldar, Jul 26 2020

Keywords

Comments

The asymptotic density of this sequence is 1 - zeta(7)/zeta(6) = 0.0088404638... (Sathe, 1945).

Examples

			64 is a term since A000005(64) = 7 is divisible by 7.

Links

Crossrefs

Cf. A000005, A008589, A059269, A336595.
Cf. A030516, A113851 and A138031 are subsequences.

Programs

  • Maple
    q:= n-> is(irem(numtheory[tau](n), 7)=0):
select(q, [$1..5500])[];  # Alois P. Heinz, Jul 26 2020
  • Mathematica
    Select[Range[5000], Divisible[DivisorSigma[0, #], 7] &]

Formula

A030516 UNION A030632 UNION A137484 UNION A137491 UNION A175745 UNION A175750 UNION ... - R. J. Mathar, May 05 2023

A369209 Numbers whose number of divisors has the largest prime factor 3.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228
Offset: 1

Views

Author

Amiram Eldar, Jan 16 2024

Keywords

Comments

Subsequence of A059269 and first differs from it at n = 36: A059269(136) = 44 has 15 = 3 * 5 divisors and thus is not a term of this sequence.
Numbers k such that A000005(k) is in A065119.
Numbers k such that A071188(k) = 3.
Equals the complement of A354181, without the terms of A036537 (i.e., complement(A354181) \ A036537).
The asymptotic density of this sequence is Product_{p prime} (1-1/p) * (Sum_{k>=1} 1/p^(A003586(k)-1)) - A327839 = 0.26087647470200496716... .

Links

Crossrefs

Cf. A000005, A003586, A006530, A036537, A065119, A336595, A071188, A211337, A211338, A327839, A354181.
Subsequence of A013929 and A059269.
Subsequences: A001248, A030627, A050997, A054753, A062503, A067259, A079395, A085986, A085987, A086975, A095990, A096156, A138032, A162143, A179643, A179645.

Programs

  • Mathematica
    gpf[n_] := FactorInteger[n][[-1, 1]]; Select[Range[300], gpf[DivisorSigma[0, #]] == 3 &]
  • PARI
    gpf(n) = if(n == 1, 1, vecmax(factor(n)[, 1]));
is(n) = gpf(numdiv(n)) == 3;
Showing 1-4 of 4 results.

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