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.

A276941 Square array A(row,col): A(1,col) = A276937(col), and for row > 1, A(row,col) = A003961(A(row-1,col)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

2, 6, 3, 9, 15, 5, 10, 25, 35, 7, 14, 21, 49, 77, 11, 18, 33, 55, 121, 143, 13, 22, 75, 65, 91, 169, 221, 17, 26, 39, 245, 119, 187, 289, 323, 19, 30, 51, 85, 847, 209, 247, 361, 437, 23, 34, 105, 95, 133, 1859, 299, 391, 529, 667, 29, 38, 57, 385, 161, 253, 3757, 493, 551, 841, 899, 31, 42, 69, 115, 1001, 319, 377, 6137, 589, 713, 961, 1147, 37
Offset: 2

Views

Author

Antti Karttunen, Sep 25 2016

Keywords

Comments

The starting offset is 2 because 1 is not included in the array proper. With it the terms are a permutation of A276078.

Examples

			The top left corner of the array:
   2,   6,   9,   10,   14,    18,   22,   26,    30,   34,   38,    42
   3,  15,  25,   21,   33,    75,   39,   51,   105,   57,   69,   165
   5,  35,  49,   55,   65,   245,   85,   95,   385,  115,  145,   455
   7,  77, 121,   91,  119,   847,  133,  161,  1001,  203,  217,  1309
  11, 143, 169,  187,  209,  1859,  253,  319,  2431,  341,  407,  2717
  13, 221, 289,  247,  299,  3757,  377,  403,  4199,  481,  533,  5083
  17, 323, 361,  391,  493,  6137,  527,  629,  7429,  697,  731,  9367
  19, 437, 529,  551,  589, 10051,  703,  779, 12673,  817,  893, 13547
  23, 667, 841,  713,  851, 19343,  943,  989, 20677, 1081, 1219, 24679
  29, 899, 961, 1073, 1189, 27869, 1247, 1363, 33263, 1537, 1711, 36859

Links

Crossrefs

Transpose: A276942.
Topmost row: A276937, second row: A276938. Leftmost column: A000040.
Cf. A003961.
Cf. A276078 (sorted into ascending order).
Cf. also A276075, A276953.

Programs

Formula

A(1,col) = A276937(col), and for row > 1, A(row,col) = A003961(A(row-1,col)).

A276942 Square array A(row,col): A(row,1) = A276937(row), and for col > 1, A(row,col) = A003961(A(row,col-1)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

2, 3, 6, 5, 15, 9, 7, 35, 25, 10, 11, 77, 49, 21, 14, 13, 143, 121, 55, 33, 18, 17, 221, 169, 91, 65, 75, 22, 19, 323, 289, 187, 119, 245, 39, 26, 23, 437, 361, 247, 209, 847, 85, 51, 30, 29, 667, 529, 391, 299, 1859, 133, 95, 105, 34, 31, 899, 841, 551, 493, 3757, 253, 161, 385, 57, 38, 37, 1147, 961, 713, 589, 6137, 377, 319, 1001, 115, 69, 42
Offset: 2

Views

Author

Antti Karttunen, Sep 25 2016

Keywords

Comments

The starting offset is 2 because 1 is not included in the array proper. With it the terms are a permutation of A276078.
All terms on each row have the same prime signature.

Examples

			The top left corner of the array:
   2,  3,  5,   7,  11,  13,  17,  19,  23,   29,   31,   37,   41,   43
   6, 15, 35,  77, 143, 221, 323, 437, 667,  899, 1147, 1517, 1763, 2021
   9, 25, 49, 121, 169, 289, 361, 529, 841,  961, 1369, 1681, 1849, 2209
  10, 21, 55,  91, 187, 247, 391, 551, 713, 1073, 1271, 1591, 1927, 2279
  14, 33, 65, 119, 209, 299, 493, 589, 851, 1189, 1333, 1739, 2173, 2537

Links

Crossrefs

Transpose: A276941.
Leftmost column: A276937, second column: A276938.
Rows from the top: A000040, A006094, A001248 (from 9 onward), A090076, A090090.
Cf. A003961.
Cf. A276078 (sorted into ascending order).
Cf. also A276075, A276955.

Programs

Formula

A(row,1) = A276937(row); for col > 1, A(row,col) = A003961(A(row,col-1)).

A276938 Second row of A276941: a(n) = A003961(A276937(n)).

Original entry on oeis.org

3, 15, 25, 21, 33, 75, 39, 51, 105, 57, 69, 165, 175, 87, 147, 93, 111, 275, 195, 231, 123, 255, 129, 141, 525, 159, 363, 325, 285, 177, 273, 345, 425, 183, 201, 343, 825, 357, 213, 435, 219, 237, 735, 475, 429, 249, 267, 399, 575, 465, 291, 561, 555, 483, 303, 975, 309, 321, 725
Offset: 1

Views

Author

Antti Karttunen, Sep 24 2016

Keywords

Links

Crossrefs

Row 2 of A276941 (column 2 of A276942).
Cf. A003961, A276937.
Subsequence of A276078, but no common terms with A276936.

Programs

Formula

a(n) = A003961(A276937(n)).

A276936 Numbers m with at least one distinct prime factor prime(k) such that prime(k)^k divides, but prime(k)^(k+1) does not divide m.

Original entry on oeis.org

2, 6, 9, 10, 14, 18, 22, 26, 30, 34, 36, 38, 42, 45, 46, 50, 54, 58, 62, 63, 66, 70, 72, 74, 78, 82, 86, 90, 94, 98, 99, 102, 106, 110, 114, 117, 118, 122, 125, 126, 130, 134, 138, 142, 144, 146, 150, 153, 154, 158, 162, 166, 170, 171, 174, 178, 180, 182, 186, 190, 194, 198, 202, 206, 207, 210, 214, 218, 222, 225
Offset: 1

Views

Author

Antti Karttunen, Sep 24 2016

Keywords

Comments

Numbers m with at least one prime factor such that the exponent of its highest power in m is equal to the index of that prime.
The asymptotic density of this sequence is 1 - Product_{k>=1} (1 - 1/prime(k)^k + 1/prime(k)^(k+1)) = 0.31025035294364447031... - Amiram Eldar, Jan 09 2021

Examples

			2 is a member as 2 = prime(1) and as 2^1 divides but 2^2 does not divide 2.
3 is NOT a member as 3 = prime(2) but 3^2 does not divide 3.
4 is NOT a member as 2^2 divides 4.
6 is a member as 2 = prime(1) and 2^1 is a divisor of 6, but 2^2 is not.
9 is a member as 3 = prime(2) and 3^2 divides 9.

Links

Crossrefs

Cf. A109298, A276935.
Intersection with A276078 gives A276937.
Cf. A016825, A051063 (subsequences).
Complement of A325130.

Programs

  • Maple
    q:= n-> ormap(i-> numtheory[pi](i[1])=i[2], ifactors(n)[2]):
select(q, [$1..225])[];  # Alois P. Heinz, Nov 18 2024
  • Mathematica
    Select[Range[225], AnyTrue[FactorInteger[#], PrimePi[First[#1]] == Last[#1] &] &] (* Amiram Eldar, Jan 09 2021 *)
Showing 1-4 of 4 results.

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