A114437 -id:A114437 - cp's OEIS frontend

A069904 Number of prime factors of n-th triangular number (with multiplicity).

0, 1, 2, 2, 2, 2, 3, 4, 3, 2, 3, 3, 2, 3, 5, 4, 3, 3, 3, 4, 3, 2, 4, 5, 3, 4, 5, 3, 3, 3, 5, 6, 3, 3, 5, 4, 2, 3, 5, 4, 3, 3, 3, 5, 4, 2, 5, 6, 4, 4, 4, 3, 4, 5, 5, 5, 3, 2, 4, 4, 2, 4, 8, 7, 4, 3, 3, 4, 4, 3, 5, 5, 2, 4, 5, 4, 4, 3, 5, 8, 5, 2, 4, 5, 3, 3, 5, 4, 4, 5

Offset: 1

Reinhard Zumkeller, Apr 10 2002

nonn

			A000217(8) = 8*(8+1)/2 = 36 = 2*2*3*3, therefore a(8) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A069901, A069902, A069903, A092405, A101745, A108815, A114435, A114436, A114437, A164977, A240527, A240528, A240529, A278253.

Array[Plus@@Last/@FactorInteger[ #*(#+1)/2]&,33] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2010 *)

A069904(n) = bigomega((n*(n+1))/2); \\ Antti Karttunen, Oct 07 2017

a(n) = A001222(A000217(n)).
From Antti Karttunen, Oct 07 2017: (Start)
a(n) = (A001222(n)+A001222(n+1))-1.
a(n) = A001222(A278253(n)). (End)
From Alois P. Heinz, Aug 05 2019: (Start)
a(n) = 2 <=> n in { A164977 }.
a(n) = 3 <=> n in { A108815 }.
a(n) = 4 <=> n in { A114435 }.
a(n) = 5 <=> n in { A114436 }.
a(n) = 6 <=> n in { A114437 }.
a(n) = 7 <=> n in { A240527 }.
a(n) = 8 <=> n in { A240528 }.
a(n) = 9 <=> n in { A240529 }.
a(n) = 10 <=> n im { A101745 }. (End)

A240527 Indices of 7-almost prime triangular numbers.

Original entry on oeis.org

64, 95, 127, 135, 143, 144, 159, 160, 175, 191, 192, 207, 215, 216, 242, 243, 272, 279, 296, 323, 335, 350, 360, 368, 375, 404, 405, 415, 416, 431, 432, 448, 455, 459, 464, 479, 485, 504, 527, 528, 543, 544, 559, 584, 594, 615, 620, 623, 647, 656, 672, 719

Offset: 1

Vincenzo Librandi, Apr 07 2014

nonn

			a(1) = 64 because A000217(64) = 64*(64+1)/2 = 2080 = 2^5 * 5 * 13 is a 7-almost prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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

Flatten[Position[Accumulate[Range[800]],_?(PrimeOmega[#]==7&)]]

{ m : A069904(m) = 7 }. - Alois P. Heinz, Aug 05 2019

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

Vincenzo Librandi, Apr 07 2014

nonn

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

Muniru A Asiru, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

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

Cf. A046310 (8-almost primes).

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

[n: n in [2..1200] | &+[d[2]: d in Factorization((n*(n+1)))] eq 9]; // Vincenzo Librandi, Dec 22 2018

Flatten[Position[Accumulate[Range[1500]], _?(PrimeOmega[#]== 8 &)]]

{ 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

Vincenzo Librandi, Apr 07 2014

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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

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

[n: n in [2..2600] | &+[d[2]: d in Factorization((n*(n+1)))] eq 10]; // Vincenzo Librandi, Dec 22 2018

Flatten[Position[Accumulate[Range[3500]], _?(PrimeOmega[#]== 9 &)]]
Select[Range[3000], PrimeOmega[(# (# + 1))/2] == 9 &] (* Harvey P. Dale, Jun 22 2017 *)

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

cp's OEIS Frontend

A069904 Number of prime factors of n-th triangular number (with multiplicity).

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A240527 Indices of 7-almost prime triangular numbers.

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A240528 Indices of 8-almost prime triangular numbers.

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A240529 Indices of 9-almost prime triangular numbers.

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