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.

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A240528 Indices of 8-almost prime triangular numbers.

Original entry on oeis.org

63, 80, 128, 256, 287, 288, 319, 320, 324, 383, 399, 440, 447, 480, 495, 539, 560, 567, 576, 608, 640, 648, 671, 675, 703, 720, 729, 799, 831, 863, 927, 935, 972, 975, 1000, 1007, 1055, 1056, 1071, 1079, 1080, 1104, 1119, 1120, 1160, 1175, 1183, 1184
Offset: 1

Views

Author

Vincenzo Librandi, Apr 07 2014

Keywords

Examples

			a(1) = 63 because A000217(63) = 63*(63+1)/2 = 2016 = 2^5 * 3^2 * 7 is an 8-almost prime.

Links

Crossrefs

Cf. A000217, A069904, A108815, A114435, A114436, A114437, A164977.
Cf. A046310 (8-almost primes).

Programs

  • GAP
    F:=List([1..1200],n->Length(Factors(n*(n+1)/2)));; a:=Filtered([1..Length(F)],i->F[i]=8); # Muniru A Asiru, Dec 22 2018
  • Magma
    [n: n in [2..1200] | &+[d[2]: d in Factorization((n*(n+1)))] eq 9]; // Vincenzo Librandi, Dec 22 2018
  • Mathematica
    Flatten[Position[Accumulate[Range[1500]], _?(PrimeOmega[#]== 8 &)]]

Formula

{ m : A069904(m) = 8 }. - Alois P. Heinz, Aug 05 2019

A240529 Indices of 9-almost prime triangular numbers.

Original entry on oeis.org

224, 351, 624, 704, 735, 768, 783, 800, 832, 864, 895, 944, 959, 960, 999, 1151, 1152, 1224, 1279, 1343, 1344, 1375, 1440, 1520, 1539, 1824, 1855, 1935, 1943, 1944, 1952, 2000, 2144, 2159, 2176, 2187, 2295, 2367, 2430, 2432, 2464, 2495, 2496, 2499, 2511
Offset: 1

Views

Author

Vincenzo Librandi, Apr 07 2014

Keywords

Examples

			a(1) = 224 because A000217(224) = 224*(224+1)/2 = 25200 = 2^4 * 3^2 * 5^2 * 7 is a 9-almost prime.

Links

Crossrefs

Cf. A000217, A069904, A108815, A114435, A114436, A114437, A164977.

Programs

  • GAP
    F:=List([1..2600],n->Length(Factors(n*(n+1)/2)));; a:=Filtered([1..Length(F)],i->F[i]=9); # Muniru A Asiru, Dec 22 2018
  • Magma
    [n: n in [2..2600] | &+[d[2]: d in Factorization((n*(n+1)))] eq 10]; // Vincenzo Librandi, Dec 22 2018
  • Mathematica
    Flatten[Position[Accumulate[Range[3500]], _?(PrimeOmega[#]== 9 &)]]
Select[Range[3000], PrimeOmega[(# (# + 1))/2] == 9 &] (* Harvey P. Dale, Jun 22 2017 *)

Formula

{ m : A069904(m) = 9 }. - Alois P. Heinz, Aug 05 2019

A375657 The smallest triangular number that begins a run of at least n consecutive triangular numbers with the same number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, 6, 6, 6, 724206, 32365035, 32365035, 9127288495, 497232340606, 54524401634046, 192541553136345, 3119282531272578, 1584264619108935753, 34399764958387086910, 34399764958387086910
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 23 2024

Keywords

Examples

			a(4) = 6 because 6 is the smallest triangular number that begins a run of 4 consecutive triangular numbers (6, 10, 15, 21) with the same number of prime factors (counted with multiplicity), i.e. 2.
a(5) = 724206 because 724206 is the smallest triangular number that begins a run of 5 consecutive triangular numbers (724206, 725410, 726615, 727821, 729028) with the same number of prime factors (counted with multiplicity), i.e. 5.

Crossrefs

Cf. A000217, A069904, A075088.
Cf. A375658.
Previous Showing 11-13 of 13 results.

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

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