A002381 -id:A002381 - cp's OEIS frontend

A002855 {m + n: m in A002382, n in A002381}.

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 35, 37, 38, 40, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 54, 56, 57, 58, 60, 63, 64, 65, 66, 67, 68, 70, 71, 73, 75, 77, 78, 79, 81, 84, 85, 86, 87, 88, 89, 91, 92, 95, 96, 98, 99, 100

Offset: 1

N. J. A. Sloane

nonn

H. Gupta, On a conjecture of Chowla, Proc. Indian Acad. Sci., 5A (1937), 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000
H. Gupta, On a conjecture of Chowla, Proc. Indian Acad. Sci., 5A (1937), 381-384. [Annotated scanned copy]

Cf. A002381, A002382.

Extended by Ray Chandler, Jul 31 2019

A093722 Integers of the form (k^2 - 1) / 120.

Original entry on oeis.org

0, 1, 3, 7, 8, 14, 20, 29, 31, 42, 52, 66, 69, 85, 99, 118, 122, 143, 161, 185, 190, 216, 238, 267, 273, 304, 330, 364, 371, 407, 437, 476, 484, 525, 559, 603, 612, 658, 696, 745, 755, 806, 848, 902, 913, 969, 1015, 1074, 1086, 1147, 1197, 1261, 1274, 1340

Offset: 1

Michael Somos, Apr 13 2004

This is "one-fifteenth of triangular numbers (integers only)". - Vladimir Joseph Stephan Orlovsky, Mar 04 2009

The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^n)/( (1 - q^(10*n-2))*(1 - q^(10*n-8)) ) = 1 - q - q^3 + q^7 + q^8 - q^14 - q^20 + + - - ... . - Peter Bala, Dec 26 2024

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A002381, A057538, A113430, A214429, A379210.

A093722 := proc(q) local n;
for n from 0 to q do
 if type(sqrt(120*n+1), integer) then print(n);
fi; od; end:
A093722(1500); # Peter Bala, Dec 26 2024

Select[Table[(n^2-1)/120,{n,0,700}],IntegerQ] (* Harvey P. Dale, Nov 26 2010 *)

{a(n) = (((n\4 * 3 + n%4) * 10 + (-1)^(n\2))^2 - 1) / 120 } /* Michael Somos, Oct 17 2006 */

|A113430(n-1)| is the characteristic function of the numbers in A093722.
a(-1 - n) = a(n). a(n) = (A057538(n) * 2 - 1) / 120.
G.f.: -x^2*(1+2*x+4*x^2+x^3+4*x^4+x^6+2*x^5) / ( (1+x)^2*(x^2+1)^2*(x-1)^3 ). - R. J. Mathar, Jun 09 2013
From Peter Bala, Dec 26 2024: (Start)
a(n) is quasi-polynomial in n
a(4*n) = n*(15*n + 1)/2; a(4*n+1) = (3*n + 1)*(5*n + 2)/2;
a(4*n+2) = (3*n + 2)*(5*n + 3)/2; a(4*n+3) = (n + 1)*(15*n + 14)/2.
For 0 <= k <= 3, a(4*n+k) = (N_k(n)^2 - 1)/120, where N_0(n) = 30*n + 1, N_1(n) = 30*n + 11, N_2(n) = 30*n + 19 and N_3(n) = 30*n + 29. (End)

More terms from Harvey P. Dale, Nov 26 2010

Offset corrected to 1 by Ray Chandler, Jul 29 2019

A002382 Numbers of the form (p^2 - 49)/120 where p is prime.

Original entry on oeis.org

0, 1, 2, 4, 11, 15, 18, 23, 37, 44, 57, 78, 88, 95, 106, 134, 156, 205, 221, 232, 249, 310, 323, 414, 429, 452, 550, 576, 639, 667, 715, 785, 816, 837, 946, 1003, 1038, 1122, 1159, 1222, 1313, 1562, 1635, 1740, 1786, 1817, 1976, 2108, 2279, 2493, 2585, 2641

Offset: 1

N. J. A. Sloane

nonn

Primes p corresponding to a(n) are found in A003631(n+2) = A042993(n+3) = A097957(n+1). - Ray Chandler, Jul 29 2019

H. Gupta, On a conjecture of Chowla, Proc. Indian Acad. Sci., 5A (1937), 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
H. Gupta, On a conjecture of Chowla, Proc. Indian Acad. Sci., 5A (1937), 381-384. [Annotated scanned copy]

Cf. A002381, A002855, A003631, A042993, A097957.

Select[(Prime[Range[150]]^2-49)/120,IntegerQ] (* Harvey P. Dale, Jan 19 2014 *)

More terms from James Sellers, May 03 2000

cp's OEIS Frontend

A002855 {m + n: m in A002382, n in A002381}.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A093722 Integers of the form (k^2 - 1) / 120.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A002382 Numbers of the form (p^2 - 49)/120 where p is prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions