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.

A386796 Numbers that have exactly one exponent in their prime factorization that is equal to 2.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268
Offset: 1

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Author

Amiram Eldar, Aug 02 2025

Keywords

Comments

First differs from its subsequence A060687 at n = 16: a(16) = 72 is not a term of A060687.
Differs from A286228 by having the terms 60, 72, 84, 90, ..., and not having the term 1.
Numbers k such that A369427(k) = 1.
The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^2 + 1/p^3) * Sum_{p prime} (p-1)/(p^3 - p + 1) = 0.22661832022705616779... (the product is A330596) (Elma and Martin, 2024).

Links

Crossrefs

A060687 is a subsequence.
Cf. A286228, A330596, A369427.
Numbers that have exactly one exponent in their prime factorization that is equal to k: A119251 (k=1), this sequence (k=2), A386800 (k=3), A386804 (k=4), A386808 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 2: A337050 (m=0), this sequence (m=1), A386797 (m=2), A386798 (m=3).

Programs

  • Mathematica
    f[p_, e_] := If[e == 2, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[300], s[#] == 1 &]
  • PARI
    isok(k) = vecsum(apply(x -> if(x == 2, 1, 0), factor(k)[, 2])) == 1;

A386800 Numbers that have exactly one exponent in their prime factorization that is equal to 3.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 432, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584
Offset: 1

Views

Author

Amiram Eldar, Aug 03 2025

Keywords

Comments

First differs from its subsequence A381315 at n = 40: a(40) = 432 = 2^4 * 3^3 is not a term of A381315.
Numbers k such that A295883(k) = 1.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + 1/p^4) * Sum_{p prime} (p-1)/(p^4 - p + 1) = 0.092831691827595439609... (Elma and Martin, 2024).

Links

Crossrefs

A381315 is subsequence.
Cf. A295883.
Numbers that have exactly one exponent in their prime factorization that is equal to k: A119251 (k=1), A386796 (k=2), this sequence (k=3), A386804 (k=4), A386808 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 3: A386799 (m=0), this sequence (m=1), A386801 (m=2), A386802 (m=3).

Programs

  • Mathematica
    f[p_, e_] := If[e == 3, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[300], s[#] == 1 &]
  • PARI
    isok(k) = vecsum(apply(x -> if(x == 3, 1, 0), factor(k)[, 2])) == 1;

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;

A386808 Numbers that have exactly one exponent in their prime factorization that is equal to 5.

Original entry on oeis.org

32, 96, 160, 224, 243, 288, 352, 416, 480, 486, 544, 608, 672, 736, 800, 864, 928, 972, 992, 1056, 1120, 1184, 1215, 1248, 1312, 1376, 1440, 1504, 1568, 1632, 1696, 1701, 1760, 1824, 1888, 1944, 1952, 2016, 2080, 2144, 2208, 2272, 2336, 2400, 2430, 2464, 2528
Offset: 1

Views

Author

Amiram Eldar, Aug 03 2025

Keywords

Comments

Subsequence of A362841 and first differs from it at n = 145: A362841(145) = 7776 = 2^5 * 3 ^ 5 is not a term of this sequence.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^5 + 1/p^6) * Sum_{p prime} (p-1)/(p^6 - p + 1) = 0.0185875810803524107305... (Elma and Martin, 2024).

Links

Crossrefs

Cf. A362841.
Numbers that have exactly one exponent in their prime factorization that is equal to k: A119251 (k=1), A386796 (k=2), A386800 (k=3), A386804 (k=4), this sequence (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 5: A386807 (m=0), this sequence (m=1), A386809 (m=2), A386810 (m=3).

Programs

  • Mathematica
    f[p_, e_] := If[e == 5, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[3000], s[#] == 1 &]
  • PARI
    isok(k) = vecsum(apply(x -> if(x == 5, 1, 0), factor(k)[, 2])) == 1;
Showing 1-4 of 4 results.

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