A068084 -id:A068084 - cp's OEIS frontend

A188621 Smallest number k>1 such that k*(n-th triangular number) is also a triangular number.

3, 2, 6, 12, 3, 5, 42, 56, 14, 18, 8, 10, 33, 2, 27, 240, 60, 68, 15, 3, 13, 105, 61, 67, 138, 150, 47, 51, 24, 26, 930, 117, 21, 6, 40, 66, 315, 41, 7, 231, 35, 37, 118, 5, 83, 495, 220, 230, 564, 55, 28, 147, 663, 98, 10, 50, 92, 798, 221, 229, 885, 12, 741, 615

Offset: 1

Zak Seidov, Apr 06 2011

nonn

There is a sequence of triangular numbers >3 which are not divisible by any smaller triangular number > 1, primitive triangular numbers in that sense: 3, 10, 28, 55, 91, 136, 253.... whose indices are in A137281.

(This is apparently a subsequence of A060544. - R. J. Mathar, Apr 06 2011)

			a(1)=3 because A000217(1)=1, 3*1 is triangular and k*1 for 1<k<3 is not triangular.
a(2)=2 because A000217(2)=3, 2*3 is triangular and k*3 for 1<k<2 (empty condition) is not triangular.
a(3)=6 because A000217(3)=6, 6*6 is triangular and k*6 for 1<k<6 is not triangular.
a(1000)=153 because A000217(1000)=500500, 153*500500=76576500 is triangular and k*500500 for 1<k<153 is not triangular.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000217, A068084, A069752, A137281.

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8 n]]; Table[t = (n + 1)*n/2; k = 2; While[! TriangularQ[k*t], k++]; k, {n, 100}] (* T. D. Noe, Apr 06 2011 *)
snk[n_]:=Module[{k=2},While[!OddQ[Sqrt[8k*n+1]],k++];k]; snk/@Accumulate[ Range[ 70]] (* Harvey P. Dale, Apr 29 2018 *)

a(n) = A068084(n)/A000217(n).

A069752 Smallest k>n such that the triangular number n(n+1)/2 divides the triangular number k(k+1)/2.

Original entry on oeis.org

2, 3, 8, 15, 9, 14, 48, 63, 35, 44, 32, 39, 77, 20, 80, 255, 135, 152, 75, 35, 77, 230, 183, 200, 299, 324, 188, 203, 144, 155, 960, 351, 153, 84, 224, 296, 665, 246, 104, 615, 245, 258, 472, 99, 414, 1034, 704, 735, 1175, 374, 272, 636, 1377, 539, 175, 399, 551

Offset: 1

Benoit Cloitre, May 01 2002

Note that k <= n^2-1, with equality occurring only if n and n+1 are a prime and a power of 2 (in either order); that is, when n is a Mersenne prime or n+1 is a Fermat prime. - T. D. Noe, Apr 08 2011

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

Cf. A000217, A068084, A188621.

Clear[k]; Join[{2}, Table[Reduce[k*(k+1) == 0, k, Modulus -> n*(n+1)][[3, 2]], {n, 2, 100}]] (* T. D. Noe, Apr 08 2011 *)

for(s=1,1000,s1=s*(s+1);n=s+1; while(n*(n+1)%s1>0,n++); print1(n,","); ) \\ Zak Seidov, Apr 08 2011

A225787 Least prime number p such that p*triangular(n) is a triangular number, or 0 if no such p exists.

Original entry on oeis.org

2, 3, 2, 11, 19, 3, 5, 0, 71, 23, 109, 131, 17, 181, 2, 239, 271, 307, 0, 379, 3, 13, 127, 61, 67, 163, 701, 47, 811, 97, 37, 991, 0, 31, 0, 79, 83, 0, 41, 7, 0, 191, 37, 0, 5, 83, 541, 251, 2351, 613, 71, 0, 0, 2861, 743, 3079, 3191, 367, 0, 3539, 229, 0, 977, 0, 4159

Offset: 0

Alex Ratushnyak, May 16 2013

nonn

			Least prime p such that triangular(3)*p is a triangular number is p=11, so a(3) = 11.

Cf. A000217, A068084, A225502.

a(n) <= n^2 + n + 2. For n>0, a(n) <= n^n + n + 1.

A225789 Least triangular number of the form p*triangular(n) where p is a prime number, or 0 if no such triangular number exists.

Original entry on oeis.org

0, 3, 6, 66, 190, 45, 105, 0, 2556, 1035, 5995, 8646, 1326, 16471, 210, 28680, 36856, 46971, 0, 72010, 630, 3003, 32131, 16836, 20100, 52975, 246051, 17766, 329266, 42195, 17205, 491536, 0, 17391, 0, 49770, 55278, 0, 30381, 5460, 0, 164451, 33411, 0, 4950, 85905, 584821

Offset: 0

Alex Ratushnyak, May 16 2013

nonn

Cf. A000217, A068084, A225503, A225787.

a(n) = A225787(n) * A000217(n).

a(28) corrected by Mansour Qahwaji, Jul 25 2019

cp's OEIS Frontend

A188621 Smallest number k>1 such that k*(n-th triangular number) is also a triangular number.

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A069752 Smallest k>n such that the triangular number n(n+1)/2 divides the triangular number k(k+1)/2.

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A225787 Least prime number p such that p*triangular(n) is a triangular number, or 0 if no such p exists.

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A225789 Least triangular number of the form p*triangular(n) where p is a prime number, or 0 if no such triangular number exists.

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cp's OEIS Frontend

A188621 Smallest number k>1 such that k*(n-th triangular number) is also a triangular number.

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A069752 Smallest k>n such that the triangular number n*(n+1)/2 divides the triangular number k*(k+1)/2.

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A225787 Least prime number p such that p*triangular(n) is a triangular number, or 0 if no such p exists.

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A225789 Least triangular number of the form p*triangular(n) where p is a prime number, or 0 if no such triangular number exists.

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Extensions

A069752 Smallest k>n such that the triangular number n(n+1)/2 divides the triangular number k(k+1)/2.