A069902 -id:A069902 - 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)

A074399 a(n) is the largest prime divisor of n(n+1).

Original entry on oeis.org

2, 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, 67, 23, 23, 71, 71, 73, 73, 37, 19

Offset: 1

N. J. A. Sloane, Nov 29 2002

Størmer shows that a(n) tends to infinity with n. Pólya generalized this result to other polynomials.
Kotov shows that a(n) >> log log n. - Charles R Greathouse IV, Mar 26 2012
Keates and Schinzel give effective constants for the above; in particular the latter shows that lim inf a(n)/log log n >= 2/7. - Charles R Greathouse IV, Nov 12 2012
Erdős conjectures ("on very flimsy probabilistic grounds") that for every e > 0, a(n) < (log n)^(2+e) infinitely often, while a(n) < (log n)^(2-e) only finitely often. - Charles R Greathouse IV, Mar 11 2015

S. V. Kotov, The greatest prime factor of a polynomial (in Russian), Mat. Zametki 13 (1973), pp. 515-522.
K. Mahler, Über den größten Primteiler spezieller Polynome zweiten Grades, Archiv for mathematik og naturvidenskab 41:6 (1934), pp. 3-26.
Georg Pólya, Zur arithmetischen Untersuchung der Polynome, Math. Zeitschrift 1 (1918), pp. 143-148.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
P. Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975), Congress. Numer. XVI , pp. 25-44, Utilitas Math., Winnipeg, Man., 1976.
M. Keates, On the greatest prime factor of a polynomial (1968), pp. 301-303.
Hector Pasten, The largest prime factor of n^2+1 and improvements on subexponential ABC, arXiv:2312.03566 [math.NT] (2024)
A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13:2 (1967-1968), pp. 177-236.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter udgivne af Videnskabsselskabet i Christiania: Mathematisk-naturvidenskabelig Klasse (1897).

With A037464, the bisections of A076605.
Essentially the same as A069902.
Positions of primes <= p: A085152 (p=5), A085153 (p=7), A252494 (p=11), A252493 (p=13), A252492 (p=17).
Last position of each prime: A002072.

Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 3, 160, 2}]
Table[FactorInteger[n(n+1)][[-1,1]],{n,80}] (* Harvey P. Dale, Sep 28 2021 *)

gpf(n)=my(f=factor(n)[,1]); f[#f]
a(n)=if(n<3, n+1, max(gpf(n),gpf(n+1))) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = Max (A006530(2n), A006530(2n+2)).

Pasten proves that a(n) >> (log log n)^2/(log log log n), see Corollary 1.5. - Charles R Greathouse IV, Oct 14 2024

Extended by Robert G. Wilson v, Dec 02 2002

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

A164314 Largest prime factor of n^2 - 2.

Original entry on oeis.org

2, 7, 7, 23, 17, 47, 31, 79, 7, 17, 71, 167, 97, 223, 127, 41, 23, 359, 199, 439, 241, 31, 41, 89, 337, 727, 23, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 31, 47, 73, 881, 1847, 967, 17, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 47, 3023, 1567, 191

Offset: 2

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A002583, A014442, A069902.

a:= n-> max(numtheory[factorset](n^2-2)):
seq(a(n), n=2..60);  # Alois P. Heinz, Jul 22 2017

Table[FactorInteger[n^2 - 2][[-1, 1]], {n, 2, 57}] (* Michael De Vlieger, Jul 22 2017 *)

a(n) = vecmax(factor(n^2-2)[,1]); \\ Michel Marcus, Jul 22 2017

a(n) = A006530(A008865(n)).

Offset corrected by R. J. Mathar, Aug 21 2009

A253254 Largest prime factor of the n-th 11-gonal number.

Original entry on oeis.org

11, 5, 29, 19, 47, 7, 13, 37, 83, 23, 101, 13, 17, 5, 137, 73, 31, 41, 173, 13, 191, 23, 19, 109, 227, 59, 7, 127, 263, 31, 281, 29, 23, 11, 317, 163, 67, 43, 353, 181, 53, 43, 389, 199, 37, 47, 17, 31, 443, 113, 461, 53, 479, 61, 71, 23, 103, 131, 41, 271, 31, 7, 569, 17, 587, 149, 17, 307, 89, 79

Offset: 2

Gionata Neri, May 31 2015

nonn

a(A024907(n)) = A061238(n).

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A006530, A051682, A069902 (similar, with triangular numbers).

gpf:= n -> max(numtheory:-factorset(n));
seq(gpf(n*(9*n-7)/2),n=2..100); # Robert Israel, Jun 24 2015

a(n) = my(f = factor(n*(9*n-7)/2)); f[#f~,1]; \\ Michel Marcus, May 31 2015

a(n) = A006530(A051682(n)).

A321069 Greatest prime factor of n^3+2.

Original entry on oeis.org

3, 5, 29, 11, 127, 109, 23, 257, 43, 167, 43, 173, 733, 1373, 307, 683, 983, 2917, 2287, 4001, 157, 71, 283, 223, 5209, 47, 127, 3659, 24391, 587, 9931, 113, 433, 6551, 809, 569, 307, 27437, 433, 10667, 439, 239, 1559, 223, 91127, 16223, 4153, 457, 39217, 62501

Offset: 1

Charles R Greathouse IV, Oct 27 2018

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown, The largest prime factor of x^3+2, Proceedings of the London Mathematical Society, 82:3 (2000), pp. 554-596.
Christopher Hooley, On the greatest prime factor of a cubic polynomial, Journal für die reine und angewandte Mathematik, 303 (1978), pp. 21-50.
A. J. Irving, The largest prime factor of x^3+2, arXiv:1412.0024 [math.NT], 2014.

Greatest prime factors of polynomials: A006530 (n), A076565 (2n+1), A076566 (3n+3), A076567 (4n+6), A164314 (n^2-2), A076605 (n^2-1), A014442 (n^2+1), A069902 (n^2+n), A074399 (n^2+n), A199423 (2n^2+n), A089619 (2n^2+2n+1), A037464 (4n^2-1), A253254 (9n^2-7n), A093074 (n^3-n), A081257 (n^3-1), A081256 (n^3+1), A321069(n^3+2), A281793 (n^3+n^2+n+1), A281793 (n^4-1), A096172 (n^4+1), A190136 (n^4 + 6n^3 + 11n^2 + 6n), A140538 (2n^4+1), A240548 (n^5+1), A281794 (n^5+n^3+n^2+1), A240549 (n^6+1), A240550 (n^7+1), A240551 (n^8+1), A240552 (n^9+1), A240553 (n^10+1).

[Maximum(PrimeDivisors(n^3 + 2)): n in [1..60]]; // Vincenzo Librandi, Oct 27 2018

Table[FactorInteger[n^3 + 2] [[-1, 1]], {n, 80}] (* Vincenzo Librandi, Oct 27 2018 *)

a(n) = vecmax(factor(n^3+2)[,1]); \\ Michel Marcus, Oct 27 2018

A233004 Pt(n) mod n!, where Pt(n) is product of first n positive triangular numbers (A000217).

Original entry on oeis.org

0, 1, 0, 12, 60, 540, 0, 20160, 181440, 907200, 19958400, 359251200, 1556755200, 32691859200, 0, 10461394944000, 177843714048000, 1600593426432000, 60822550204416000, 608225502044160000, 38318206628782080000, 702500454861004800000, 12926008369442488320000

Offset: 1

Alex Ratushnyak, Dec 03 2013

nonn

Pt(n) = n!*(n+1)! / 2^n.

Pt(n) mod n! = 0 if and only if 2^n divides (n+1)!, that is, n+1 is a power of 2. Thus indices of zeros are of the form 2^k-1.

Cf. A000142, A000217.
Cf. A006472 (triangular factorial, essentially equal to Pt(n)).
Cf. A067667 (Pt(n)/n! for n's of the form 2^k-1).
Cf. A069902, A007917.

f=t=1
for n in range(1,33):
  t*=n*(n+1)//2
  f*=n
  print(t%f, end=', ')

cp's OEIS Frontend

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

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A074399 a(n) is the largest prime divisor of n(n+1).

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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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A164314 Largest prime factor of n^2 - 2.

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A253254 Largest prime factor of the n-th 11-gonal number.

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A321069 Greatest prime factor of n^3+2.

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