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 1-7 of 7 results.

A364151 Tetrahedral numbers that are products of smaller tetrahedral numbers.

Original entry on oeis.org

1, 560, 19600, 43680, 45760, 893200, 1521520, 7207200, 29269240, 2845642800, 22778408800, 26595476600, 59777945920, 199910480000, 239526427140, 249466897680, 283345302240, 3280499995500, 20894643369600, 115333903584900, 408688050971200, 706949015272500, 4613394351142500
Offset: 1

Views

Author

Pontus von Brömssen, Jul 15 2023

Keywords

Examples

			1 is a term because 1 is a tetrahedral number and equals the empty product.
560 is a term because 560 = C(16,3) = C(5,3) * C(8,3). (C(n,k) is the binomial coefficient.)
45760 is a term because 45760 = C(66,3) = C(4,3)^2 * C(5,3) * C(13,3).
3280499995500 is a term because 3280499995500 = C(27001,3) = C(4,3) * C(15,3) * C(31,3) * C(135,3).

Crossrefs

Cf. A000292, A068143 (analog for triangular numbers), A196568 (only two factors allowed), A363636, A364152.
Row n=3 of A374498.

Extensions

More terms from Jinyuan Wang, Jul 31 2023

A363634 Lucky numbers that are products of smaller lucky numbers.

Original entry on oeis.org

1, 9, 21, 49, 63, 75, 93, 99, 105, 111, 129, 135, 169, 189, 195, 201, 219, 231, 237, 259, 261, 273, 297, 357, 399, 429, 477, 483, 489, 495, 511, 553, 559, 579, 615, 621, 645, 651, 693, 723, 729, 735, 777, 801, 805, 819, 855, 867, 897, 903, 925, 931, 957, 961
Offset: 1

Views

Author

Pontus von Brömssen, Jun 19 2023

Keywords

Examples

			1 is a term because it is a lucky number and equals the empty product.
21 is a term because 21 = 3*7 and both 21 and the two factors 3 and 7 are lucky numbers.
729 is a term because 729 = 9*9*9 and both 729 and 9 are lucky numbers. (This is the first term that requires more than two factors.)

Crossrefs

Cf. A000959, A363635, A363636.

A363635 Ludic numbers that are products of smaller ludic numbers.

Original entry on oeis.org

1, 25, 77, 91, 115, 119, 121, 143, 161, 175, 221, 235, 265, 287, 301, 329, 377, 407, 415, 445, 481, 493, 497, 517, 535, 581, 595, 625, 667, 697, 749, 805, 841, 851, 865, 913, 943, 1015, 1043, 1045, 1105, 1177, 1207, 1225, 1247, 1351, 1363, 1375, 1391, 1403
Offset: 1

Views

Author

Pontus von Brömssen, Jun 19 2023

Keywords

Examples

			1 is a term because it is a ludic number and equals the empty product.
25 is a term because 25 = 5*5 and both 25 and 5 are ludic numbers.
1015 is a term because 1015 = 5*7*29 and both 1015 and the three factors 5, 7, and 29 are ludic numbers. (This is the first term that requires more than two factors.)

Crossrefs

Cf. A003309, A363634, A363636.

A363638 Primes p such that p+1 can be written as a product of smaller numbers that are also of the form prime+1.

Original entry on oeis.org

11, 17, 23, 31, 41, 47, 53, 59, 71, 79, 83, 89, 107, 113, 127, 131, 151, 167, 179, 191, 223, 227, 239, 251, 263, 269, 271, 293, 311, 359, 383, 419, 431, 439, 443, 449, 479, 491, 503, 521, 557, 587, 593, 599, 607, 631, 647, 659, 683, 701, 719, 727, 743, 773
Offset: 1

Views

Author

Pontus von Brömssen, Jun 19 2023

Keywords

Examples

			11 is a term because 11 is prime, 11+1 = 3*4 = (2+1)*(3+1), and 2 and 3 are prime.
223 is a term because 223 is prime, 223+1 = 4*4*14 = (3+1)^2*(13+1), and 3 and 13 are prime. (This is the first term that requires more than two factors, i.e., it is not a term of A066938.)

Crossrefs

Cf. A008864, A066938 (subsequence), A363636, A363750.

A363750 Primes p such that p-1 can be written as a product of smaller numbers that are also of the form prime-1.

Original entry on oeis.org

2, 5, 13, 17, 37, 41, 61, 73, 89, 97, 101, 109, 113, 157, 181, 193, 233, 241, 257, 277, 281, 313, 337, 349, 353, 397, 401, 409, 421, 433, 449, 457, 461, 521, 541, 577, 593, 601, 613, 617, 641, 661, 673, 701, 733, 757, 761, 769, 821, 829, 877, 881, 929, 937
Offset: 1

Views

Author

Pontus von Brömssen, Jun 19 2023

Keywords

Comments

Except for 2, this is a subsequence of primes of the form 4k+1 (A002144). - Davide Rotondo, Oct 07 2024

Examples

			2 is a term because 2 is prime and 2-1 = 1 equals the empty product.
5 is a term because 5 is prime, 5-1 = 2*2 = (3-1)*(3-1), and 3 is prime.
3329 is a term because 3329 is prime, 3329-1 = 4*16*52 = (5-1)*(17-1)*(53-1), and 5, 17, and 53 are prime. (This is the first term that requires more than two factors.)

Crossrefs

Cf. A006093, A363636, A363638.
Cf. A002144.

A363492 Numbers k such that the partition number p(k) = A000041(k) can be written as a product of smaller partition numbers.

Original entry on oeis.org

0, 1, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 24, 39
Offset: 1

Views

Author

Pontus von Brömssen, Jun 05 2023

Keywords

Comments

a(18) > 10000 (if it exists).

Examples

			0 and 1 are terms, because p(0) = p(1) = 1 is the empty product.
7 is a term, because p(7) = 15 = 3*5 = p(3)*p(4).
39 is a term, because p(39) = 31185 = 3^4*385 = p(3)^4*p(18).
33 is not a term, even though all prime factors of p(33) = 3^2 * 7^2 * 23 appear in smaller partition numbers. (In particular, 33 is a term of A194345.) This is because the only smaller partition number that is divisible by 23 is p(32) = 3 * 11^2 * 23, but p(33) is not divisible by 11.

Crossrefs

Cf. A000041, A034878, A363636.
Except for a(1) = 0, subsequence of A194345.

A363637 Indices of numbers of the form k^2-1, k >= 2, that can be written as a product of smaller numbers of that same form.

Original entry on oeis.org

5, 11, 19, 26, 29, 31, 41, 55, 65, 71, 89, 99, 109, 127, 131, 134, 151, 155, 161, 181, 191, 209, 239, 244, 251, 265, 271, 274, 287, 289, 305, 323, 341, 349, 351, 379, 419, 449, 461, 485, 491, 505, 511, 551, 575, 599, 647, 649, 701, 703, 755, 769, 811, 846, 869
Offset: 1

Views

Author

Pontus von Brömssen, Jun 19 2023

Keywords

Examples

			5 is a term because 5^2-1 = 24 = 3*8 = (2^2-1)*(3^2-1)
26 is a term because 26^2-1 = 675 = 3*15*15 = (2^2-1)*(4^2-1)^2. (This is the first term that requires more than two factors.)

Crossrefs

Cf. A005563, A363636.
Showing 1-7 of 7 results.

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