A242963 -id:A242963 - cp's OEIS frontend

A009005 All natural numbers except 1, 2 and 4.

3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

David W. Wilson

Short legs of Pythagorean triangles.
Also the natural numbers n such that there is an open knight's tour of the 4 X n chessboard. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 02 2002
Also perimeters of integral triangles; equivalently, numbers n = a + b + c, where a + b > c, a + c > b, b + c > a, and a, b, c are integers (necessarily a, b, c, and n are positive). - Rick L. Shepherd, Aug 04 2013
Along with 0, possible scores in rugby union, where points can be scored with a penalty goal or drop goal (3 points), try (5 points) and converted try (7 points). - Charles R Greathouse IV, Sep 10 2016
Also the positive integers n such that floor(t(n+2)/(t(n+2) - sigma(n+2))) = 1, where t(n) = n*(n+1)/2 = A000217(n) and sigma(n) = A000203(n) (see A242963). - Lorenzo Sauras Altuzarra, Jan 29 2020

IBM Ponder This, December 2001
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000203 (sigma function), A000217 (triangular numbers), A242963.

Join[{3}, Range[5, 100]] (* Paolo Xausa, Sep 13 2024 *)

a(n)=n+4-(n==1) \\ Charles R Greathouse IV, Sep 02 2011

O.g.f.: x(3-x-x^2)/(1-x)^2. a(n)=A009056(n), n>1. - R. J. Mathar, May 26 2008

A242962 a(1) = a(2) = 0; for n >= 3: a(n) = (n*(n+1)/2) mod antisigma(n) = A000217(n) mod A024816(n).

Original entry on oeis.org

0, 0, 0, 1, 6, 3, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84

Offset: 1

Jaroslav Krizek, May 29 2014

nonn

A000217(n) = triangular numbers, A024816(n) = sum of numbers less than n which do not divide n.

a(n) = sigma(n) = A000203(n) for n = 5 and n>= 7 (see A242963).

			a(6) = 3 because A000217(6) mod A024816(6) = 21 mod 9 = 3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000217, A002961, A024816, A242963, A243117, A243118.

[((n*(n+1)div 2) mod (n*(n+1)div 2-SumOfDivisors(n))): n in [3..1000]]

Array[If[# < 3, 0, Mod[PolygonalNumber@ #, Total@ Complement[Range@ #, Divisors@ #]]] &, 65] (* Michael De Vlieger, Jan 28 2020 *)

A243117 Possible values of A242962 in increasing order.

Original entry on oeis.org

0, 1, 3, 6, 8, 12, 13, 14, 15, 18, 20, 24, 28, 30, 31, 32, 36, 38, 39, 40, 42, 44, 48, 54, 56, 57, 60, 62, 63, 68, 72, 74, 78, 80, 84, 90, 91, 93, 96, 98, 102, 104, 108, 110, 112, 114, 120, 121, 124, 126, 127, 128, 132, 133, 138, 140, 144, 150, 152, 156, 158

Offset: 1

Jaroslav Krizek, May 29 2014

nonn

A242962(n) = (n*(n+1)/2) mod antisigma(n) = A000217(n) mod A024816(n); where A000217(n) = triangular numbers, A024816(n) = sum of numbers less than n which do not divide n.
Complement of A243118.
a(n) = A002191(n+1) for n >= 5; where A002191 = possible values of sigma(n).

			0 is in the sequence because there is a number m such that A242962(m) = 0; m = 3.

Cf. A000203, A002191, A242962, A242963, A243118.

A243118 Numbers n such that A242962(x) = n has no solution.

Original entry on oeis.org

2, 4, 5, 7, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 58, 59, 61, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 92, 94, 95, 97, 99, 100, 101, 103, 105, 106

Offset: 1

Jaroslav Krizek, May 29 2014

nonn

A242962(n) = (n*(n+1)/2) mod antisigma(n) = A000217(n) mod A024816(n); where A000217(n) = triangular numbers, A024816(n) = sum of numbers less than n which do not divide n.
Union of A007369 and numbers 4 and 7.
Complement of A243117.
a(n) = A007369(n-2) for n >= 5, where A007369 = numbers n such that sigma(x) = n has no solution.

			4 is in the sequence because there is no x whose A242962(x) = 4.

Cf. A000203, A007369, A242962, A242963, A243117.

cp's OEIS Frontend

A009005 All natural numbers except 1, 2 and 4.

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A242962 a(1) = a(2) = 0; for n >= 3: a(n) = (n*(n+1)/2) mod antisigma(n) = A000217(n) mod A024816(n).

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A243117 Possible values of A242962 in increasing order.

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A243118 Numbers n such that A242962(x) = n has no solution.

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