A280992 Squarefree triangular numbers that are products of consecutive primes.

1, 3, 6, 15, 105, 210, 255255

Offset

1,2

Comments

No more terms up to the 5000000th triangular number.

If a(8) exists, it's divisible by a prime p > prime(2000) = 17389. - David A. Corneth, Oct 21 2017

Links

Table of n, a(n) for n=1..7.

Example

The triangular number 255255 = 714*715/2 is a term because 255255 = 3*5*7*11*13*17 is a product of distinct consecutive primes.
1 (the empty product) is a term, so is 3 (the product of just one triangular number).

Maple

# reuses code of A097889 and A061304
isA280992 := proc(n)
    isA097889(n) and isA061304(n) ;
end proc:
for t from 0 do
    n := t*(t+1)/2 ;
    if isA280992(t) then
        print(t) ;
    end if;
end do: # R. J. Mathar, Oct 20 2017

Mathematica

Select[PolygonalNumber@ Range[10^5], And[NoneTrue[#[[All, -1]], # > 1 &], Union@ Differences[PrimePi[#[[All, 1]] ] ] == {1}] &@ FactorInteger@ # &] (* Michael De Vlieger, Oct 06 2017 *)

Prog

(PARI) is(n) = my(f=factor(n)[, 1]); for(k=1, #f-1, if(f[k+1]!=nextprime(f[k]+1), return(0))); ispolygonal(n, 3) && issquarefree(n)
search(start) = if(start < 4, if(start < 2, print1(1, ", ")); print1(3, ", ")); forcomposite(c=start, , if(is(c), print1(c, ", ")))
/* Start a search from 1 upwards as follows: */
search(1) \\ Felix Fröhlich, Oct 21 2017 [Corrected Jun 10, 2019]
(PARI) uptoprime(n) = {my(prim = vector(n), i = 2, res = List([1]));  prim[1] = 2; forprime(p = 3, , prim[i] = prim[i - 1] * p; i++; if(i>n, break));
for(i=1, n, if(issquare(8 * prim[i] + 1), listput(res, prim[i])); for(j=1, i-1, c = prim[i]/prim[j]; if(issquare(8 * c + 1), listput(res, c)))); listsort(res); res} \\ David A. Corneth, Oct 21 2017

Crossrefs

Cf. A000217, A061304, A097889.

Sequence in context: A013277 A190187 A144549 * A013270 A013276 A110809

Adjacent sequences: A280989 A280990 A280991 * A280993 A280994 A280995

Keyword

nonn,more

Author

Rick L. Shepherd, Jan 13 2017

Extensions

1 and 3 prepended by David A. Corneth, Oct 21 2017

Status

approved