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 1-6 of 6 results.

A175974 Primes p such that A175933(p)=0.

Original entry on oeis.org

2, 149, 191, 331, 457, 499, 727, 743, 761, 821, 877, 881, 991, 1009, 1171, 1213, 1277, 1399, 1409, 1451, 1471, 1543, 1567, 1787, 1801, 1871, 1889, 1949, 1973, 2039, 2053, 2063, 2153, 2179, 2531, 2671, 2837, 2861, 3023, 3203, 3449, 3457, 3779, 4001, 4201
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 01 2010

Keywords

Crossrefs

A175933(n) is the number of ways of writing n=m+k with m prime and k primorial.

Extensions

Corrected (761 inserted, 821 inserted, 1451 inserted etc) by R. J. Mathar, Nov 03 2010

A175940 Number of ways of writing n=p+f with p a prime and f a factorial.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 2, 0, 2, 1, 1, 0, 1, 1, 3, 1, 1, 0, 2, 1, 2, 0, 0, 0, 2, 1, 2, 0, 0, 0, 1, 1, 2, 1, 1, 0, 2, 0, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 2, 0, 1, 1, 1, 0, 1, 1, 3, 0, 0, 0, 1, 1, 2, 0, 0, 0, 2, 0, 1, 1, 1, 0, 0, 1, 3, 1, 1
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 25 2010

Keywords

Comments

Number of partitions of n into the sum of a prime number and a factorial number. Number of decompositions of n into an unordered sum of a prime number and a factorial number.

Examples

			a(29)=2 because 29 has two prime + factorial representations, 5+4! and 23+3!.

Links

Crossrefs

Cf. A000142, A062602, A175931, A175933.

Programs

  • Maple
    a:= proc(n) local t,k;
       t:= 0;
       for k while k! < n do
         if isprime(n-k!) then t:= t+1 fi
       od;
       t
end proc:
seq(a(n), n=1..100); # Robert Israel, Oct 13 2014
  • Mathematica
    a[n_] := Module[{t = 0, k}, For[k = 1, k! < n, k++, If[PrimeQ[n - k!] , t++]]; t];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 02 2023, after Robert Israel *)
  • PARI
    a(n) = c=0;for(i=1,n,if(isprime(n-i!),c++));c
vector(100,n,a(n)) \\ Derek Orr, Oct 13 2014

Extensions

Edited and entries checked by D. S. McNeil, Nov 26 2010

A172366 Number of partitions of prime(n) into the sum of two semiprimes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 1, 2, 3, 2, 2, 4, 4, 3, 5, 6, 4, 6, 5, 5, 7, 4, 7, 6, 7, 7, 5, 5, 8, 8, 8, 8, 9, 10, 9, 7, 13, 10, 10, 11, 13, 10, 10, 10, 11, 15, 18, 13, 10, 13, 14, 16, 12, 18, 12, 15, 12, 11
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 20 2010

Keywords

Examples

			a(8)=2 because prime(8)=19=4(semiprime)+15(semiprime)=9(semiprime)+10(semiprime).

Links

Crossrefs

Cf. A129363, A175933,

Programs

  • Mathematica
    Table[Total[If[PrimeOmega[#]=={2,2},1,0]&/@Table[{x-n,n},{n,x/2}]],{x, Prime[ Range[60]]}] (* Harvey P. Dale, Dec 21 2015 *)

Formula

a(n) = A072931(A000040(n)).

Extensions

Corrected and extended by D. S. McNeil, Nov 20 2010

A175957 Numbers that cannot be written as the sum of a prime number and a primorial number.

Original entry on oeis.org

1, 2, 10, 16, 22, 26, 27, 28, 34, 36, 40, 46, 50, 51, 52, 56, 57, 58, 64, 66, 70, 76, 78, 82, 86, 87, 88, 92, 93, 94, 96, 100, 106, 112, 116, 117, 118, 120, 121, 122, 123, 124, 125, 126, 130, 134, 135, 136, 142, 144, 146, 147, 148, 149, 154, 156, 160, 162, 166, 170
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 29 2010

Keywords

Crossrefs

Cf. A118954, A175933.

Programs

  • Mathematica
    fQ[n_] := Block[{pp = Select[ FoldList[Times, 1, Prime@ Range@ 20], # <= n &]}, Last@ Union@ PrimeQ[n - pp] == False]; Select[ Range@ 170, fQ] (* Robert G. Wilson v, Nov 05 2010 *)

Formula

{k : A175933(k)=0}. - R. J. Mathar, Nov 03 2010

Extensions

Corrected (90 removed) and extended by R. J. Mathar and Robert G. Wilson v, Nov 03 2010

A181674 Number of partitions of n into a primorial and a semiprime.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 0, 2, 2, 2, 0, 0, 2, 3, 1, 0, 0, 1, 1, 1, 2, 1, 0, 1, 3, 2, 0, 0, 1, 1, 0, 2, 2, 3, 1, 0, 3, 4, 2, 0, 0, 2, 2, 0, 1, 1, 0, 1, 2, 3, 1, 0, 2, 2, 2, 1, 2, 1, 1, 0, 3, 3, 1, 1, 1, 2, 1, 1, 2, 0, 0, 0, 2, 2, 0, 1, 2, 1, 1, 0, 2, 1, 1, 1, 3, 4, 1, 0, 1, 3, 2, 1, 3, 2, 2, 0, 2, 1, 1, 0, 0, 1, 0
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 04 2010

Keywords

Comments

Number of ways of writing n=k+m where k primorial and m semiprime. The number of decompositions of n into an unordered sum of primorial number and semiprime number.
Conjecture: Average order is Theta(log log n). - Charles R Greathouse IV, May 25 2011

Examples

			a(10)=2 because 10 = 6(primorial)+4(semiprime) = 1(primorial)+9(semiprime).

Crossrefs

Cf. A001358, A002110, A100949, A175933.

A377887 a(n) is the number of ways of writing prime(n) as k-q with q a prime and k a primorial.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 4, 3, 3, 3, 1, 3, 4, 4, 5, 4, 5, 3, 4, 4, 5, 4, 4, 7, 8, 6, 5, 7, 4, 5, 9, 5, 6, 5, 9, 7, 4, 5, 8, 5, 8, 6, 7, 5, 9, 6, 5, 7, 6, 4, 3, 9, 6, 5, 12, 8, 5, 6, 7, 7, 6, 9, 8, 7, 13, 4, 8, 4, 6, 6, 7, 4, 7, 11, 5, 8, 8, 9, 6, 7, 7, 6, 12, 4, 10, 11, 11, 7, 8, 11
Offset: 1

Views

Author

Daniel D Gibson, Nov 10 2024

Keywords

Comments

Except at n=2, the largest k which must be considered is the product of the first n-1 primes, since if k includes p=prime(n) itself then q = k-p is divisible by p and so not prime.

Examples

			For n=4, p = prime(4) = 7 can be written as p = 30 - 23, where 30 is a primorial and 23 is prime, but no other primorials satisfy this condition, so a(3) = 1.

Crossrefs

Cf. A002110, A175933.

Programs

Showing 1-6 of 6 results.

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