A069904 -id:A069904 - cp's OEIS frontend

A164977 Numbers m such that the set {1..m} has only one nontrivial decomposition into subsets with equal element sum.

3, 4, 5, 6, 10, 13, 22, 37, 46, 58, 61, 73, 82, 106, 157, 166, 178, 193, 226, 262, 277, 313, 346, 358, 382, 397, 421, 457, 466, 478, 502, 541, 562, 586, 613, 661, 673, 718, 733, 757, 838, 862, 877, 886, 982, 997, 1018, 1093, 1153, 1186, 1201, 1213, 1237, 1282

Offset: 1

Alois P. Heinz, Sep 03 2009

Numbers m such that the triangular number T(m) = m*(m+1)/2 has exactly two divisors >= m.
Also numbers m such that m*(m+1)/2 is the product of two primes.
Contains all numbers in A005383. - Harry Richman, Jan 09 2025
Contains all numbers in A077065. - Alois P. Heinz, Jan 19 2025

			10 is in the sequence, because there is only one nontrivial decomposition of {1..10} into subsets with equal element sum: {1,10}, {2,9}, {3,8}, {4,7}, {5,6}; 11|55.
13 is in the sequence with decomposition of {1..13}: {1,12}, {2,11}, {3,10}, {4,9}, {5,8}, {6,7}, {13}; 13|91.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000217, A001222, A164978, A035470, A068443, A069904, A001358.

Cf. A005383, A077065 (distinct subsequences).

a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, 2, a(n-1))
      while not (andmap(isprime, [k, (k+1)/2]) or
                 andmap(isprime, [k+1, k/2]))
      do od; k
    end:
seq(a(n), n=1..100);

Select[Range@1304, PrimeOmega[#] + PrimeOmega[# + 1] == 3 &] (* Robert G. Wilson v, Jun 28 2010 and updated Sep 21 2018 *)

is(n)=if(isprime(n),bigomega(n+1)==2, isprime(n+1) && bigomega(n)==2) \\ Charles R Greathouse IV, Sep 08 2015

is(n)=if(n%2, isprime((n+1)/2) && isprime(n), isprime(n/2) && isprime(n+1)) \\ Charles R Greathouse IV, Mar 16 2022

list(lim)=my(v=List()); forprime(p=3,lim, if(isprime((p+1)/2), listput(v,p))); forprime(p=5,lim+1, if(isprime(p\2), listput(v,p-1))); Set(v) \\ Charles R Greathouse IV, Mar 16 2022

{ m :  A035470(m) = 2 }.
{ m :  A164978(m) = 2 }.
{ m : |{d|m*(m+1)/2 : d>=m}| = 2 }.
{ m :  m*(m+1)/2 in {A068443} }.
{ m :  m*(m+1)/2 in {A001358} }.
{ m :  A069904(m) = 2 }.
{ m :  A001222(n) + A001222(n+1) = 3 }. - Alois P. Heinz, Jan 08 2022
{ A005383 } union { A077065 }. - Alois P. Heinz, Jan 19 2025

A069902 Largest prime factor of n(n+1)/2, the n-th triangular number.

Original entry on oeis.org

1, 3, 3, 5, 5, 7, 7, 3, 5, 11, 11, 13, 13, 7, 5, 17, 17, 19, 19, 7, 11, 23, 23, 5, 13, 13, 7, 29, 29, 31, 31, 11, 17, 17, 7, 37, 37, 19, 13, 41, 41, 43, 43, 11, 23, 47, 47, 7, 7, 17, 17, 53, 53, 11, 11, 19, 29, 59, 59, 61, 61, 31, 7, 13, 13, 67

Offset: 1

Reinhard Zumkeller, Apr 10 2002

Essentially the same as A074399, which has many comments, references and links.

			A000217(9) = 9*(9+1)/2 = 45 = 3*3*5, therefore a(9) = 5.

Related properties of triangular numbers: A069901, A069903, A069904.

Cf. A000217, A006530, A074399.

PrimeFactors[n_]:=Flatten[Table[ #[[1]],{1}]&/@FactorInteger[n]]; Table[PrimeFactors[n*(n-1)/2][[ -1]],{n,2,6!}] (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
(* Second program: *)
Array[FactorInteger[PolygonalNumber[#]][[-1, 1]] &, 66] (* Michael De Vlieger, Sep 14 2023 *)

\\ written for a(n), n >= 2
a(n)=vecmax(factor(n*(n+1)/2)[,1]) \\ M. F. Hasler, May 02 2015

a(n) = A006530(A000217(n)).

Edited by Peter Munn, Sep 14 2023

A069901 Smallest prime factor of n-th triangular number.

Original entry on oeis.org

1, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 2, 7, 3, 2, 2, 3, 3, 2, 2, 3, 11, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 2, 19, 3, 2, 2, 3, 3, 2, 2, 3, 23, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 3, 29, 2, 2, 31, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 2, 37, 3, 2, 2, 3, 3, 2, 2, 3, 41, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 2, 7, 3, 2

Offset: 1

Reinhard Zumkeller, Apr 10 2002

Or, a(1) = 1, then the smallest nontrivial k (>1) which divides the sum of (next n) numbers from k+1 to k+n or smallest k > 1 that divides nk + n(n+1)/2. - Amarnath Murthy, Sep 22 2002. For example, a(7) = 4, which is the smallest nontrivial number that divides the sum 5+6+...+11, of 7 numbers.

			A000217(10) = 10*(10+1)/2 = 55 = 5*11, therefore a(10) = 5.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000217, A020639, A069902, A069903, A069904.

FactorInteger[#][[1,1]]&/@Accumulate[Range[100]] (* Harvey P. Dale, Apr 05 2014 *)

a(n) = if (n==1, 1, vecmin(factor(n*(n+1)/2)[,1]));

a(n) = A020639(A000217(n)).
a(4*k-1) = a(4*k) = 2.
From Zak Seidov, Jun 06 2013: (Start)
a(n) = 3 for n = {2, 5, 6, 9} + 12*k;
a(n) = 5 for n = {10, 25, 34, 49} + 60*k;
a(n) = 7 for n = {13, 97, 118, 133, 181, 202, 217, 238, 286, 301, 322, 406} + 420*k, etc. (End)

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of Franklin T. Adams-Watters

A069903 Number of distinct prime factors of n-th triangular number.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 4, 3, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 4, 3, 2, 3, 4, 3, 3, 3, 3, 4, 3, 2, 3, 3, 2, 3, 4, 3, 2, 3, 4, 4, 3, 2, 4, 4, 2, 3, 3, 3, 4, 3, 3, 4, 4, 3, 3, 3, 2, 3, 4, 4, 4, 3, 3, 3, 2, 2, 4, 5, 3, 3, 4, 3, 3, 4

Offset: 1

Reinhard Zumkeller, Apr 10 2002

			A000217(11) = 11*(11+1)/2 = 66 = 2*3*11, therefore a(11) = 3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000217, A001221, A059957, A069901, A069902, A069904, A077761.

PrimeNu[#]&/@Accumulate[Range[90]] (* Harvey P. Dale, Oct 06 2016 *)

a(n) = omega(n*(n+1)/2); \\ Michel Marcus, Feb 05 2021

a(n)=onega(n/gcd(n,2))+omega((n+1)/gcd(n+1)) \\ Charles R Greathouse IV, Sep 21 2024

a(n) = A001221(A000217(n)).

Sum_{k=1..n} a(k) = 2 * n * (log(log(n)) + B - 1/4) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 21 2024

A108815 Indices of triangular numbers which are products of 3 primes.

Original entry on oeis.org

7, 9, 11, 12, 14, 17, 18, 19, 21, 25, 28, 29, 30, 33, 34, 38, 41, 42, 43, 52, 57, 66, 67, 70, 78, 85, 86, 93, 94, 97, 101, 102, 109, 113, 118, 121, 122, 130, 133, 137, 138, 141, 142, 145, 148, 158, 163, 172, 173, 177, 181, 190, 201, 202, 205, 211, 213, 214, 217, 218

Offset: 1

Jonathan Vos Post, Jul 10 2005

Indices of 3-almost prime triangular numbers.

			a(1) = 7 because T(7) = TriangularNumber(7) = 7*(7+1)/2 = 28 = 2^2 * 7 is a 3-almost prime.
a(2) = 9 because T(9) = 9*(9+1)/2 = 45 = 3^2 * 5 is a 3-almost prime.
a(3) = 11 because T(11) = 11*(11+1)/2 = 66 = 2 * 3 * 11.
a(31) = 101 because T(101) = 101*(101+1)/2 = 5151 = 3 * 17 * 101.
a(49) = 173 because T(173) = 173*(173+1)/2 = 15051 = 3 * 29 * 173.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Triangular Number.
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A000217, A001222, A014612, A069904.

Select[Range[225], Plus @@ Last /@ FactorInteger[ #*(# + 1)/2] == 3 &] (* Ray Chandler, Jul 16 2005 *)

issemi(n)=bigomega(n)==2
is(n)=if(isprime(n/gcd(n,2)), issemi((n+1)/gcd(n+1,2)), isprime((n+1)/gcd(n+1,2)) && issemi(n/gcd(n,2))) \\ Charles R Greathouse IV, Feb 05 2017

{a(n)} = {k such that A001222(A000217(k)) = 3}. {a(n)} = {k such that k*(k+1)/2 has exactly 3 prime factors, with multiplicity}. {a(n)} = {k such that A000217(k) is an element of A014612}.
n such that n*(n+1)/2 is an element of A014612. n such that A000217(n) is an element of A014612. n such that C(n+1, 2) is an element of A014612.
{ m :  A069904(m) = 3 }. - Alois P. Heinz, Aug 05 2019

Extended by Ray Chandler, Jul 16 2005

Edited by N. J. A. Sloane, May 07 2007

A114435 Indices of 4-almost prime triangular numbers.

Original entry on oeis.org

8, 16, 20, 23, 26, 36, 40, 45, 49, 50, 51, 53, 59, 60, 62, 65, 68, 69, 74, 76, 77, 83, 88, 89, 91, 92, 100, 103, 105, 110, 114, 115, 117, 123, 126, 129, 131, 136, 139, 146, 149, 150, 151, 154, 156, 165, 169, 182, 185, 186, 187, 194, 196, 197, 198, 206, 210

Offset: 1

Jonathan Vos Post, Feb 13 2006

			a(1) = 8 because T(8) = TriangularNumber(8) = 8*(8+1)/2 = 36 = 2^2 * 3^2 is a 4-almost prime.
a(2) = 16 because T(16) = 16*(16+1)/2 = 136 = 2^3 * 17 is a 4-almost prime.
a(3) = 20 because T(20) = 20*(20+1)/2 = 210 = 2 * 3 * 5 * 7 (210 = primorial 4#).
a(4) = 23 because T(23) = 23*(23+1)/2 = 276 = 2^2 * 3 * 23.
a(5) = 26 because T(26) = 26*(26+1)/2 = 351 = 3^3 * 13.
a(6) = 36 because T(36) = 36*(36+1)/2 = 666 = 2 * 3^2 * 37.
a(27) = 100 because T(100) = 100*(100+1)/2 = 5050 = 2 * 5^2 * 101.
a(57) = 210 because T(210) = 210*(210+1)/2 = 22155 = 3 * 5 * 7 * 211 (again, 210 = primorial 4#).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Triangular Number.
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A000217, A001222, A014613, A069904.

Flatten[Position[Accumulate[Range[800]], ?(PrimeOmega[#]== 4 &)]] (* _Vincenzo Librandi, Apr 09 2014 *)

is(n)=my(t=bigomega(n/gcd(n,2))); if(t<3, bigomega((n+1)/gcd(n+1,2))+t==4, t==3 && isprime((n+1)/gcd(n+1,2))) \\ Charles R Greathouse IV, Jun 14 2017

{a(n)} = {k such that A001222(A000217(k)) = 4}. {a(n)} = {k such that k*(k+1)/2 has exactly 4 prime factors, with multiplicity}. {a(n)} = {k such that A000217(k) is an element of A014613}.

{ m : A069904(m) = 4 }. - Alois P. Heinz, Aug 05 2019

A114436 Indices of 5-almost prime triangular numbers.

Original entry on oeis.org

15, 24, 27, 31, 35, 39, 44, 47, 54, 55, 56, 71, 72, 75, 79, 81, 84, 87, 90, 98, 107, 108, 112, 116, 124, 132, 134, 140, 147, 153, 155, 162, 164, 167, 170, 171, 174, 179, 180, 183, 184, 199, 203, 204, 209, 219, 220, 225, 230, 234, 244, 245, 247, 248, 249

Offset: 1

Jonathan Vos Post, Feb 13 2006

			a(1) = 15 because T(15) = TriangularNumber(15) = 15*(15+1)/2 = 120 = 2^3 * 3 * 5 is a 5-almost prime.
a(2) = 24 because T(24) = 24*(24+1)/2 = 300 = 2^2 * 3 * 5^2 is a 5-almost prime.
a(3) = 27 because T(27) = 27*(27+1)/2 = 378 = 2 * 3^3 * 7 is a 5-almost prime.
a(4) = 31 because T(27) = 31*(31+1)/2 = 496 = 2^4 * 31 is a 5-almost prime.
a(17) = 84 because T(27) = 84*(84+1)/2 = 3570 = 2 * 3 * 5 * 7 * 17 is a 5-almost prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Eric Weisstein's World of Mathematics, Triangular Number.
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A000217, A001222, A014614, A069904.

Select[Range[250],PrimeOmega[(#(#+1))/2]==5&] (* Harvey P. Dale, Sep 14 2012 *)
Flatten[Position[Accumulate[Range[700]], ?(PrimeOmega[#]== 5 &)]] (* _Vincenzo Librandi, Apr 09 2014 *)

{a(n)} = {k such that A001222(A000217(k)) = 5}. {a(n)} = {k such that k*(k+1)/2 has exactly 5 prime factors, with multiplicity}. {a(n)} = {k such that A000217(k) is an element of A014614}.

{ m : A069904(m) = 5 }. - Alois P. Heinz, Aug 05 2019

Corrected and extended by Harvey P. Dale, Apr 02 2011

A114437 Indices of 6-almost prime triangular numbers.

Original entry on oeis.org

32, 48, 96, 99, 104, 111, 119, 120, 125, 152, 161, 168, 176, 188, 189, 195, 200, 208, 223, 231, 239, 240, 252, 260, 264, 275, 299, 300, 303, 304, 315, 336, 342, 343, 344, 352, 359, 363, 374, 377, 391, 392, 395, 400

Offset: 1

Jonathan Vos Post, Feb 14 2006

			a(1) = 48 because T(48) = TriangularNumber(48) = 48*(48+1)/2 = 1176 = 2^3 * 3 * 7^2 is a 6-almost prime.
a(2) = 96 because T(96) = 96*(96+1)/2 = 4656 = 2^4 * 3 * 97 is a 6-almost prime.
a(18) = 200 because T(200) = 200*(200+1)/2 = 20100 = 2^2 * 3 * 5^2 * 67 is a 6-almost prime.
a(29) = 300 because T(300) = 300*(300+1)/2 = 45150 = 2 * 3 * 5^2 * 7 * 43 is a 6-almost prime.
a(38) = 363 because T(363) = 363*(363+1)/2 = 45150 = 66066 = 2 * 3 * 7 * 11^2 * 13 is a 6-almost prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Triangular Number.

Cf. A000217, A001222, A046306, A069904.

Select[Range[400],PrimeOmega[(#(#+1))/2]==6&] (* Harvey P. Dale, Mar 29 2012 *)
Flatten[Position[Accumulate[Range[800]], ?(PrimeOmega[#]== 6 &)]] (* _Vincenzo Librandi, Apr 09 2014 *)

isA114437(n)=bigomega(n*(n+1)/2)==6 /* Michael B. Porter, Mar 30 2012 */

{a(n)} = {k such that A001222(A000217(k)) = 6}. {a(n)} = {k such that k*(k+1)/2 has exactly 6 prime factors, with multiplicity}.
{a(n)} = {k such that A000217(k) is an element of A046306}.
{ m :  A069904(m) = 6 }. - Alois P. Heinz, Aug 05 2019

Corrected by Harvey P. Dale, Mar 29 2012

A101745 Indices of triangular numbers which are 10-almost primes. Indices of A101744.

Original entry on oeis.org

255, 384, 511, 575, 639, 728, 767, 896, 1088, 1295, 1376, 1407, 1599, 1700, 1727, 1792, 1919, 1920, 2015, 2024, 2375, 2431, 2672, 2815, 2880, 2915, 2944, 2975, 3104, 3159, 3199, 3327, 3375, 3392, 3456, 3583, 3744, 3999, 4031, 4032, 4160, 4223, 4256

Offset: 1

Jonathan Vos Post, Dec 14 2004

			a(1) = 255 because that is the smallest index of a triangular number which is also a 10-almost prime; specifically T(255) = 255*(255+1)/2 = 32640 = 2^7 * 3 * 5 * 17.

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Cf. A101744, A000217, A046314, A076550, A069904.

F:=List([1..4300],n->Length(Factors(n*(n+1)/2)));; a:=Filtered([1..Length(F)],i->F[i]=10); # Muniru A Asiru, Dec 22 2018

[n: n in [2..4500] | &+[d[2]: d in Factorization((n*(n+1)))] eq 11]; // Vincenzo Librandi, Dec 22 2018

BigOmega[n_Integer]:=Plus@@Last[Transpose[FactorInteger[n]]]; Do[ t=n*(n+1)/2; If[BigOmega[t]==10, Print[n, " ", t];];, {n, 2, 5000}]; (* Ray Chandler, Dec 14 2004 *)
Flatten[Position[Accumulate[Range[5000]],?(PrimeOmega[#]==10&)]] (* _Harvey P. Dale, May 12 2011 *)

a(n)*(a(n)+1)/2 has exactly 10 prime factors.

{ m : A069904(m) = 10 }. - Alois P. Heinz, Aug 05 2019

More terms from Ray Chandler, Dec 14 2004

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

cp's OEIS Frontend

A164977 Numbers m such that the set {1..m} has only one nontrivial decomposition into subsets with equal element sum.

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A069902 Largest prime factor of n(n+1)/2, the n-th triangular number.

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A069901 Smallest prime factor of n-th triangular number.

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A069903 Number of distinct prime factors of n-th triangular number.

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A108815 Indices of triangular numbers which are products of 3 primes.

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A114435 Indices of 4-almost prime triangular numbers.

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A114436 Indices of 5-almost prime triangular numbers.

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