cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A339194 Sum of all squarefree semiprimes with greater prime factor prime(n).

Original entry on oeis.org

0, 6, 25, 70, 187, 364, 697, 1102, 1771, 2900, 3999, 5920, 8077, 10234, 13207, 17384, 22479, 26840, 33567, 40328, 46647, 56248, 65653, 77786, 93411, 107060, 119583, 135248, 149439, 167240, 202311, 225320, 253587, 276332, 316923, 343676, 381039, 421192, 458749
Offset: 1

Views

Author

Gus Wiseman, Dec 02 2020

Keywords

Examples

			The triangle A339116 with row sums equal to this sequence begins (n > 1):
    6 = 6
   25 = 10 + 15
   70 = 14 + 21 + 35
  187 = 22 + 33 + 55 + 77

Crossrefs

A025129 gives sums of squarefree semiprimes by weight, row sums of A338905.
A143215 is the not necessarily squarefree version, row sums of A087112.
A339116 is a triangle of squarefree semiprimes with these row sums.
A339360 looks at all squarefree numbers, row sums of A339195.
A001358 lists semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd terms A046388.
A024697 is the sum of semiprimes of weight n.
A168472 gives partial sums of squarefree semiprimes.
A332765 gives the greatest squarefree semiprime of weight n.
A338898/A338912/A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338904 groups semiprimes by weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
Cf. A000040, A001221, A014342, A098350, A100484, A319613, A320656, A338901, A339003, A339114/A339115.

Programs

  • Mathematica
    Table[Sum[Prime[i]*Prime[j],{j,i-1}],{i,10}]
  • PARI
    a(n) = prime(n)*vecsum(primes(n-1)); \\ Michel Marcus, Jun 15 2024

Formula

a(n) = prime(n) * Sum_{k=1..n-1} prime(k) = prime(n) * A007504(n-1).
a(n) = A024447(n) - A024447(n-1).
a(n) = A034960(n) - A143215(n). - Marco Zárate, Jun 14 2024

A143215 a(n) = prime(n) * Sum_{i=1..n} prime(i).

Original entry on oeis.org

4, 15, 50, 119, 308, 533, 986, 1463, 2300, 3741, 4960, 7289, 9758, 12083, 15416, 20193, 25960, 30561, 38056, 45369, 51976, 62489, 72542, 85707, 102820, 117261, 130192, 146697, 161320, 180009, 218440, 242481, 272356, 295653, 339124, 366477
Offset: 1

Views

Author

Gary W. Adamson, Jul 30 2008

Keywords

Comments

Row sums of triangle A087112.
Sum of semiprimes (A001358) with greater prime factor prime(n). - Gus Wiseman, Dec 06 2020

Examples

			The series begins (4, 15, 50, 119, 308,...) since the primes = (2, 3, 5, 7, 11,...) and partial sum of primes = (2, 5, 10, 17, 28,...).
a(5) = 308 = 11 * 28.
a(4) = 119 = sum of row 4 terms of triangle A087112: (14 + 21 + 35 + 49).

Links

Crossrefs

Row sums of A087112.
The squarefree version is A339194, row sums of A339116.
Semiprimes grouped by weight are A338904, with row sums A024697.
Squarefree semiprimes grouped by weight are A338905, with row sums A025129.
Squarefree numbers grouped by greatest prime factor are A339195, with row sums A339360.
A001358 lists semiprimes.
A006881 lists squarefree semiprimes.
A332765 is the greatest semiprime of weight n.
A338898/A338912/A338913 give the prime indices of semiprimes.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A000040, A001222, A001748, A007504, A014342, A098350, A100484, A168472, A319613, A339003, A339114/A339115.

Programs

Formula

a(n) = A000040(n) * A007504(n).

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Sep 21 2009

A347047 Smallest squarefree semiprime whose prime indices sum to n.

Original entry on oeis.org

6, 10, 14, 21, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526
Offset: 3

Views

Author

Gus Wiseman, Aug 22 2021

Keywords

Comments

Compared to A001747, we have 21 instead of 22 and lack 2 and 4.
Compared to A100484 (shifted) we have 21 instead of 22 and lack 4.
Compared to A161344, we have 21 instead of 22 and lack 4 and 8.
Compared to A339114, we have 11 instead of 9 and lack 4.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Examples

			The initial terms and their prime indices:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   21: {2,4}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}

Crossrefs

The opposite version (greatest instead of smallest) is A332765.
These are the minima of rows of A338905.
The nonsquarefree version is A339114 (opposite: A339115).
A001358 lists semiprimes (squarefree: A006881).
A024697 adds up semiprimes by weight (squarefree: A025129).
A056239 adds up prime indices, row sums of A112798.
A246868 gives the greatest squarefree number whose prime indices sum to n.
A320655 counts factorizations into semiprimes (squarefree: A320656).
A338898, A338912, A338913 give the prime indices of semiprimes.
A338899, A270650, A270652 give the prime indices of squarefree semiprimes.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
A339362 adds up prime indices of squarefree semiprimes.
Cf. A001221, A087112, A089994, A098350, A176504, A338900, A338901, A338904, A338907/A338908, A339005, A339191.

Programs

  • Mathematica
    Table[Min@@Select[Table[Times@@Prime/@y,{y,IntegerPartitions[n,{2}]}],SquareFreeQ],{n,3,50}]
  • Python
    from sympy import prime, sieve
def a(n):
    p = [0] + list(sieve.primerange(1, prime(n)+1))
    return min(p[i]*p[n-i] for i in range(1, (n+1)//2))
print([a(n) for n in range(3, 58)]) # Michael S. Branicky, Sep 05 2021
Previous Showing 11-13 of 13 results.

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