A369093 Numbers k >= 1 such that sigma(k) divides the sum of the triangular numbers T(k) and T(k+1), where sigma(k) = A000203(k) is the sum of the divisors of k.

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 119, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Author

Claude H. R. Dequatre, Jan 13 2024

Keywords

• nonn
• easy

Comments

k is a term if (k^2+k)/2 + ((k+1)^2+k+1)/2 = k^2+2*k+1 = (k+1)^2 is divisible by sigma(k).

Trivial case: If k is prime, then sigma(k) = k+1 and (k+1)^2 is divisible by k+1, thus all primes are terms of this sequence.

Table with the percentage of primes <= 10^k compared with the number of terms and the number of primes <= 10^k, for k = 2..8:

.

| k | #terms <= 10^k | #primes <= 10^k | %primes <= 10^k |

| 2 | 27 | 25 | 92.59 |

| 3 | 175 | 168 | 96.00 |

| 4 | 1248 | 1229 | 98.48 |

| 5 | 9627 | 9592 | 99.64 |

| 6 | 78565 | 78498 | 99.91 |

| 7 | 664707 | 664579 | 99.98 |

| 8 | 5761724 | 5761455 | 99.99 |

.

The percentage of primes increases asymptotically as 10^k increases.

Conjecture: The asymptotic density of primes in this sequence is 1.

Contains terms like 2, 399, 935, 1539,.. which are not in A210494. Does not contain terms like 775, 819, 3335, 6815,.. which are in A210494. - R. J. Mathar, Jan 18 2024

Examples

			3 is a term since (3+1)^2 = 4^2 = 16 is divisible by sigma(3) = 4.
35 is a term since (35+1)^2 = 36^2 = 1296 is divisible by sigma(35) = 48.
42 is not a term since (42+1)^2 = 43^2 = 1849 is not divisible by sigma(42) = 96.
		

Crossrefs

Cf. A000203, A000217.

Cf. A090777, A356410, A369096, A369097.

Subsequence: A000040.

Programs

Maple

isA369093 := proc(k)
    if modp((k+1)^2, numtheory[sigma](k)) = 0 then
        true;
    else
        false;
    end if;
end proc:
A369093 := proc(n)
    option remember ;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA369093(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
[seq(A369093(n),n=1..100)] ; # R. J. Mathar, Jan 18 2024

PARI

isok(n) = my(x=(n+1)^2,y=sigma(n));!(x%y);