A039963 -id:A039963 - cp's OEIS frontend

A361670 Squarefree part of the n-th triangular number.

1, 3, 6, 10, 15, 21, 7, 1, 5, 55, 66, 78, 91, 105, 30, 34, 17, 19, 190, 210, 231, 253, 69, 3, 13, 39, 42, 406, 435, 465, 31, 33, 561, 595, 70, 74, 703, 741, 195, 205, 861, 903, 946, 110, 115, 1081, 282, 6, 1, 51, 1326, 1378, 159, 165, 385, 399, 1653, 1711, 1770, 1830, 1891, 217, 14, 130, 2145, 2211, 2278

Offset: 1

R. J. Mathar, Mar 20 2023

a(n) / A083481(n) is either 2 or 1/2 depending on A136480(n) being even or odd, which is indicated by A039963(n).

a(n) = 1 for n>0 in A001108. - Michel Marcus, Mar 22 2023

Cf. A000217, A007913, A083481 (of oblong), A361671 (of tetrahedral).

Cf. A001108, A039963.

a:= n-> mul(i[1]^irem(i[2], 2), i=ifactors(n*(n+1)/2)[2]):
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 20 2023

a(n) = core(n*(n+1)/2); \\ Michel Marcus, Mar 22 2023

from sympy.ntheory.factor_ import core
def A361670(n): return core(n*(n+1)>>1) # Chai Wah Wu, Mar 20 2023

a(n) = A007913(A000217(n)).

A092444 a(n+1) = 11*a(n) - a(n-1) - 3, a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 7, 73, 793, 8647, 94321, 1028881, 11223367, 122428153, 1335486313, 14567921287, 158911647841, 1733460204961, 18909150606727, 206267196469033, 2250030010552633, 24544062919609927, 267734662105156561

Offset: 0

N. J. A. Sloane, Sep 19 2008, based on emails from M. F. Hasler and Jim Nastos on Apr 25 2008

nonn

The old entry with this sequence number was a duplicate of A039963.

The simultaneous equations (p+1)(p+2) == -1 (mod q), (q+1)(q+2) == -1 (mod p), where p and q are odd, have solutions {3, 3}, {3, 21}, {7, 73}, {21, 507}, {73, 793}, {793, 8647} and suggested this recurrence.

Vincenzo Librandi, Table of n, a(n) for n = 0..980
Index entries for linear recurrences with constant coefficients, signature (12,-12,1).
Index entries for sequences that are fixed points of mappings

nxt[{a_,b_}]:={b,11b-a-3}; NestList[nxt,{1,1},20][[;;,1]] (* or *) LinearRecurrence[{12,-12,1},{1,1,7},20] (* Harvey P. Dale, Jul 06 2025 *)

a(n) = 2*(A004190(n)-10*A004190(n-1))/3+1/3. G.f.: (1-11x+7x^2)/((1-x)(1-11x+x^2)). [From R. J. Mathar, Sep 20 2008]

More terms from R. J. Mathar, Sep 20 2008

A128810 Triangle formed by reading triangle A064189 mod 2 .

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1

Offset: 0

Philippe Deléham, Apr 09 2007

Also triangle formed by reading triangles A091965, A108149, A110877, A125906, A126954 mod 2 .

			Triangle begins:
1;
1, 1;
0, 0, 1;
0, 1, 1, 1;
1, 0, 1, 0, 1;
1, 0, 1, 0, 1, 1;
1, 0, 1, 0, 0, 0, 1;
1, 0, 1, 1, 0, 1, 1, 1;
1, 0, 0, 0, 0, 0, 1, 0, 1;
1, 1, 0, 0, 0, 1, 1, 0, 1, 1;
0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1 ;...

Sum_{k, 0<=k<=n}T(n,k)*x^k=A039963(n), A097357(n+1), A110565(n+1) for x=0,1,2 respectively . T(n,k)= (T(n-1,k-1)+T(n-1,k)+T(n-1,k+1)) mod 2, T(0,0)=1, T(n,k)=0 if k<0 or if k>n .

A354292 Primes p such that for all m, M(m) is not divisible by p^2 where M(n) is the n-th Motzkin number A001006.

Original entry on oeis.org

5, 13, 31, 37, 61, 79, 97, 103

Offset: 1

Michel Marcus, May 23 2022

E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, Journal de Théorie des Nombres de Bordeaux, 27(1):245-288, 2015.
Armin Straub, On congruence schemes for constant terms and their applications, arXiv:2205.09902 [math.NT], 2022. See Theorem 3.3 p. 13.

Cf. A001006, A001248, A354293.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

A375863 a(1) = 0 and a(n) = A050603(n-1)^2 for n > 0. Lexicographically earliest nonnegative sequence of integers such that the Gilbreath transform of a(1..n) gives floor(log_2(n)).

Original entry on oeis.org

0, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 16, 16, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 25, 25, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 16, 16, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 36, 36, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 16, 16, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 25

Offset: 1

Thomas Scheuerle, Sep 02 2024

nonn

			The first row is the sequence itself. The rows below are the absolute differences
of each previous row:
0, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 16, 16, ...
 1, 0, 3, 0, 3, 0, 8, 0, 8, 0, 3, 0, 3, 0, 15, 0, ...
  1, 3, 3, 3, 3, 8, 8, 8, 8, 3, 3, 3, 3, 15, 15, ...
   2, 0, 0, 0, 5, 0, 0, 0, 5, 0, 0, 0, 12, 0, ...
    2, 0, 0, 5, 5, 0, 0, 5, 5, 0, 0, 12, 12, ...
     2, 0, 5, 0, 5, 0, 5, 0, 5, 0, 12, 0, ...
      2, 5, 5, 5, 5, 5, 5, 5, 5, 12, 12, ...
       3, 0, 0, 0, 0, 0, 0, 0, 7, 0, ...
        3, 0, 0, 0, 0, 0, 0, 7, 7, ...
         3, 0, 0, 0, 0, 0, 7, 0, ...
          3, 0, 0, 0, 0, 7, 7, ...
           3, 0, 0, 0, 7, 0, ...
            3, 0, 0, 7, 7, ...
             3, 0, 7, 0, ...
              3, 7, 7, ...
               4, 0, ...
                4, ...
The main diagonal is floor(log_2(n)), where n = 1 in the first row and n = 2 in the second etc. .

Cf. A050603, A039963 (Gilbreath transform of floor(log_2(n))).

a(n) = if(n == 1, 0,valuation(n-(n-2)%2, 2)^2)

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A361670 Squarefree part of the n-th triangular number.

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A092444 a(n+1) = 11*a(n) - a(n-1) - 3, a(0)=a(1)=1.

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A128810 Triangle formed by reading triangle A064189 mod 2 .

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A354292 Primes p such that for all m, M(m) is not divisible by p^2 where M(n) is the n-th Motzkin number A001006.

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A375863 a(1) = 0 and a(n) = A050603(n-1)^2 for n > 0. Lexicographically earliest nonnegative sequence of integers such that the Gilbreath transform of a(1..n) gives floor(log_2(n)).

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PARI