A097889 -id:A097889 - cp's OEIS frontend

A334175 Numbers that can be written as a product of two or more consecutive primorial numbers.

2, 12, 180, 360, 6300, 37800, 75600, 485100, 14553000, 69369300, 87318000, 174636000, 14567553000, 15330615300, 437026590000, 2622159540000, 4951788741900, 5244319080000, 35413721343000, 2163931680210300, 7436881482030000, 148702215919257000, 223106444460900000

Offset: 1

Ilya Gutkovskiy, Apr 17 2020

nonn

			     2 = prime(0)# * prime(1)#;
    12 = prime(1)# * prime(2)#;
   180 = prime(2)# * prime(3)#;
   360 = prime(1)# * prime(2)# * prime(3)#;
  6300 = prime(3)# * prime(4)#,
  where prime(k)# is the product of the first k primes.

Amiram Eldar, Table of n, a(n) for n = 1..1318
Eric Weisstein's World of Mathematics, Primorial.
Index entries for sequences related to primorial numbers.

Cf. A002110, A006939, A025487, A097889, A129912, A322804, A334174.

A254859 Numbers that are both a sum and a product of two or more consecutive primes.

Original entry on oeis.org

15, 30, 77, 143, 210, 221, 323, 1001, 2310, 4199, 5767, 7429, 9797, 10403, 11021, 12317, 20711, 22499, 23707, 25591, 28891, 30030, 33263, 34571, 36863, 38021, 46189, 47053, 75067, 77837, 79523, 82861, 82919, 89951, 95477, 99221, 104927, 111547, 116939, 136891, 141367, 145157, 146969, 154433

Offset: 1

Altug Alkan, May 05 2016

			15 is a term because 15 = 3 + 5 + 7 = 3*5.
30 is a term because 30 = 13 + 17 = 2*3*5.
77 is a term because 77 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 7*11.

Michael S. Branicky, Table of n, a(n) for n = 1..4381 (terms 1..644 from Amiram Eldar)
Michael S. Branicky, Python program

Intersection of A050936 and A097889.

np = NextPrime; pro[n_] := Block[{e, f}, {f, e} = Transpose@ FactorInteger@ n; Length@ f > 1 && Union@ e == {1} && np@ Most@ f == Rest@ f]; seq[lim_] := Union[Reap[Block[{p = 2, q, s}, While[2 p < lim, q = np@p; s = p+q; While[s <= lim, If[pro@s, Sow@s]; q = np@q; s += q]; p = np@p]]][[2, 1]]]; seq[10^5] (* Giovanni Resta, May 05 2016 *)

Python
```
# see link
```

A280992 Squarefree triangular numbers that are products of consecutive primes.

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 13 2017

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

			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).

Cf. A000217, A061304, A097889.

# 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

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

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]

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

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

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A334175 Numbers that can be written as a product of two or more consecutive primorial numbers.

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A254859 Numbers that are both a sum and a product of two or more consecutive primes.

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A280992 Squarefree triangular numbers that are products of consecutive primes.

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