cp's OEIS Frontend

A280992 Squarefree triangular numbers that are products of consecutive primes.

%I A280992 #72 Jun 10 2019 11:13:30
%S A280992 1,3,6,15,105,210,255255
%N A280992 Squarefree triangular numbers that are products of consecutive primes.
%C A280992 No more terms up to the 5000000th triangular number.
%C A280992 If a(8) exists, it's divisible by a prime p > prime(2000) = 17389. - _David A. Corneth_, Oct 21 2017
%e A280992 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.
%e A280992 1 (the empty product) is a term, so is 3 (the product of just one triangular number).
%p A280992 # reuses code of A097889 and A061304
%p A280992 isA280992 := proc(n)
%p A280992     isA097889(n) and isA061304(n) ;
%p A280992 end proc:
%p A280992 for t from 0 do
%p A280992     n := t*(t+1)/2 ;
%p A280992     if isA280992(t) then
%p A280992         print(t) ;
%p A280992     end if;
%p A280992 end do: # _R. J. Mathar_, Oct 20 2017
%t A280992 Select[PolygonalNumber@ Range[10^5], And[NoneTrue[#[[All, -1]], # > 1 &], Union@ Differences[PrimePi[#[[All, 1]] ] ] == {1}] &@ FactorInteger@ # &] (* _Michael De Vlieger_, Oct 06 2017 *)
%o A280992 (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)
%o A280992 search(start) = if(start < 4, if(start < 2, print1(1, ", ")); print1(3, ", ")); forcomposite(c=start, , if(is(c), print1(c, ", ")))
%o A280992 /* Start a search from 1 upwards as follows: */
%o A280992 search(1) \\ _Felix Fröhlich_, Oct 21 2017 [Corrected Jun 10, 2019]
%o A280992 (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));
%o A280992 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
%Y A280992 Cf. A000217, A061304, A097889.
%K A280992 nonn,more
%O A280992 1,2
%A A280992 _Rick L. Shepherd_, Jan 13 2017
%E A280992 1 and 3 prepended by _David A. Corneth_, Oct 21 2017