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A369096
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Numbers k >= 2 such that omega(k) divides the sum of the triangular numbers T(k) and T(k+1), where omega(k) is the number of distinct primes dividing k (A001221).
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2
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2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 110, 111, 113, 115, 117, 119, 121, 123, 125, 127, 128, 129
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OFFSET
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1,1
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COMMENTS
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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 omega(k).
Trivial case: If k is prime, then omega(k) = 1 and (k+1)^2 is always divisible by 1, thus all primes are terms of this sequence.
Table with percentage of primes <= 10^k for k = 2..9:
| k | #terms <= 10^k | #primes <= 10^k | %primes <= 10^k |
| 2 | 55 | 25 | 45.45 |
| 3 | 506 | 168 | 33.20 |
| 4 | 4832 | 1229 | 25.43 |
| 5 | 46675 | 9592 | 20.55 |
| 6 | 456155 | 78498 | 17.21 |
| 7 | 4480617 | 664579 | 14.83 |
| 8 | 44081959 | 5761455 | 13.07 |
| 9 | 433916814 | 50847535 | 11.72 |
The percentage of primes decreases asymptotically as 10^k increases.
Conjecture: the asymptotic density of primes in this sequence is 0.
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LINKS
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EXAMPLE
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2 is a term since (2+1)^2 = 3^2 = 9 is divisible by omega(2) = 1.
15 is a term since (15+1)^2 = 16^2 = 256 is divisible by omega(15) = 2.
12 is not a term since (12+1)^2 = 13^2 = 169 is not divisible by omega(12) = 2.
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MAPLE
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isA369096 := proc(k)
if modp((k+1)^2, A001221(k)) = 0 then
true;
else
false;
end if;
end proc:
option remember ;
if n = 1 then
2;
else
for a from procname(n-1)+1 do
if isA369096(a) then
return a;
end if;
end do:
end if;
end proc:
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PROG
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(PARI) isok(n)=my(x=(n+1)^2, y=omega(n)); !(x%y);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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