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.

Showing 1-3 of 3 results.

A107347 Number of even semiprimes strictly between prime(n) and 2*prime(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 9, 9, 9, 9, 10, 11, 11, 12, 13, 12, 13, 13, 14, 13, 14, 15, 15, 15, 16, 16, 17, 18, 19, 19, 20, 19, 18, 18, 19, 20, 21, 22, 23, 23, 23, 24, 25, 25, 25, 26, 27, 26, 27, 28, 28, 28, 28, 28, 29, 30, 30, 30, 31, 32, 32, 32, 32
Offset: 1

Views

Author

Giovanni Teofilatto, May 23 2005

Keywords

Links

Programs

  • Maple
    A107347 := n -> n-1-pi(ithprime(n)/2);
[seq(A107347(n),n=1..100)]; # N. J. A. Sloane, Nov 04 2020
  • Mathematica
    fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; f[n_] := Count[fQ /@ Range[Prime[n] + 1, Prime[n + 1] - 1], True]; Table[ f[n], {n, 78}] (* Robert G. Wilson v, May 28 2005 *)

Formula

a(n) = A294602(prime(n)). - R. J. Mathar, Dec 17 2017
a(n) = n-1-pi(prime(n)/2). - N. J. A. Sloane, Nov 04 2020

Extensions

Corrected and extended by Robert G. Wilson v, May 28 2005

A368058 Sum of the smaller parts of the partitions of n into two distinct parts with larger part prime.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 4, 6, 3, 4, 6, 8, 4, 6, 8, 10, 13, 16, 20, 24, 17, 20, 24, 28, 19, 22, 25, 28, 32, 36, 41, 46, 34, 38, 42, 46, 32, 36, 40, 44, 49, 54, 60, 66, 49, 54, 60, 66, 72, 78, 84, 90, 97, 104, 111, 118, 96, 102, 109, 116, 93, 100, 107, 114, 121, 128, 136, 144, 152, 160
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 09 2023

Keywords

Links

Crossrefs

Cf. A010051, A294111, A294487, A294602.

Programs

  • Maple
    N:= 100: # for a(1) .. a(N)
V:= Vector(N):
for i from 1 do
  p:= ithprime(i);
  if p >= N then break fi;
  m:= min(2*p-1,N);
  V[p+1..m]:= V[p+1..m] + <$1..m-p>
od:
convert(V,list); # Robert Israel, Jan 26 2024
  • Mathematica
    Table[Sum[i (PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[(n - 1)/2]}],{n, 100}]

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i * c(n - i), where c is the prime characteristic (A010051).
a(n) = n*A294602(n) - A294487(n). - Wesley Ivan Hurt, Dec 09 2023

A294487 Sum of the lengths of the distinct rectangles with prime length and integer width such that L + W = n, W < L.

Original entry on oeis.org

0, 0, 2, 3, 3, 5, 5, 12, 12, 7, 7, 18, 18, 24, 24, 24, 24, 41, 41, 60, 60, 49, 49, 72, 72, 59, 59, 59, 59, 88, 88, 119, 119, 102, 102, 102, 102, 120, 120, 120, 120, 161, 161, 204, 204, 181, 181, 228, 228, 228, 228, 228, 228, 281, 281, 281, 281, 252, 252, 311
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 31 2017

Keywords

Comments

Sum of the largest parts of the partitions of n into two distinct parts with largest part prime.

Examples

			a(14) = 24; the rectangles are 1 X 13 and 3 X 11 (7 X 7 is not considered since W < L). The sum of the lengths is then 13 + 11 = 24.

Links

Crossrefs

Cf. A010051, A243485, A294602, A368058.

Programs

  • Mathematica
    Table[ Sum[(n - i)*(PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[(n-1)/2]}], {n, 60}]
  • PARI
    a(n) = sum(i=1, (n-1)\2, (n-i)*isprime(n-i)); \\ Michel Marcus, Nov 08 2017

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} (n-i) * A010051(n-i).
a(n) = n*A294602(n) - A368058(n). - Wesley Ivan Hurt, Dec 09 2023
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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