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A280992 Squarefree triangular numbers that are products of consecutive primes. 0
1, 3, 6, 15, 105, 210, 255255 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 21 01:33 EST 2019. Contains 329349 sequences. (Running on oeis4.)