A113935 -id:A113935 - cp's OEIS frontend

A023581 Sum of exponents in prime-power factorization of p(n)+3.

1, 2, 3, 2, 2, 4, 3, 2, 2, 5, 2, 4, 3, 2, 3, 4, 2, 6, 3, 2, 3, 2, 2, 3, 4, 4, 2, 3, 5, 3, 3, 2, 4, 2, 4, 3, 6, 2, 3, 5, 3, 4, 2, 4, 5, 2, 2, 2, 3, 4, 3, 3, 3, 2, 4, 3, 5, 2, 5, 3, 3, 4, 3, 2, 3, 7, 2, 4, 4, 6, 3, 2, 3, 4, 2, 2, 5, 6, 3, 3, 2, 4, 3, 3, 3, 2, 3, 4, 5, 2, 3, 2, 4, 3, 2, 3, 9, 3, 2

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

[n eq 1 select 1 else (&+[p[2]: p in Factorization(NthPrime(n) + 3)]): n in [1..100]]; // G. C. Greubel, May 21 2019

Table[PrimeOmega[3 + Prime[n]], {n, 1, 100}] (* G. C. Greubel, May 21 2019 *)

a(n) = bigomega(prime(n) + 3); \\ Michel Marcus, Sep 30 2013

[sloane.A001222(nth_prime(n)+3) for n in (1..100)] # G. C. Greubel, May 21 2019

a(n) = A001222(A113935(n)). - Michel Marcus, Sep 30 2013

A023580 Sum of distinct prime divisors of prime(n)+3.

Original entry on oeis.org

5, 5, 2, 7, 9, 2, 7, 13, 15, 2, 19, 7, 13, 25, 7, 9, 33, 2, 14, 39, 21, 43, 45, 25, 7, 15, 55, 18, 9, 31, 20, 69, 14, 73, 21, 20, 7, 85, 24, 13, 22, 25, 99, 9, 7, 103, 109, 115, 30, 31, 61, 13, 63, 129, 20, 28, 19, 139, 14, 73, 26, 39, 38, 159, 81, 7, 169, 24, 14

Offset: 1

Clark Kimberling

nonn

Cf. A008472, A113935.

Total[First/@FactorInteger[#]]&/@(Prime@Range@100+3) (* Giorgos Kalogeropoulos, May 05 2021 *)

from sympy import primefactors, prime
def a(n): return sum(primefactors(prime(n) + 3))
print([a(n) for n in range(1, 70)]) # Michael S. Branicky, May 05 2021

a(n) = A008472(A113935(n)). - Michel Marcus, May 05 2021

A370887 Square array read by descending antidiagonals: T(n,k) is the number of subgroups of the elementary abelian group of order A000040(k)^n for n >= 0 and k >= 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 5, 1, 2, 6, 16, 1, 2, 8, 28, 67, 1, 2, 10, 64, 212, 374, 1, 2, 14, 116, 1120, 2664, 2825, 1, 2, 16, 268, 3652, 42176, 56632, 29212, 1, 2, 20, 368, 19156, 285704, 3583232, 2052656, 417199, 1, 2, 22, 616, 35872, 3961832, 61946920, 666124288

Offset: 0

Miles Englezou, Mar 05 2024

As an elementary abelian group G of order p^n is isomorphic to an n-dimensional vector space V over the finite field of characteristic p, T(n,k) is also the number of subspaces of V.

V defined as above, T(n,k) is also the sum of the Gaussian binomial coefficients (n,r), 0 <= r < n, for a prime q number, since (n,r) counts the number of r-dimensional subspaces of V. The sequences of these sums for a fixed prime q number correspond to the columns of T(n,k).

			T(1,1) = 2 since the elementary abelian of order A000040(1)^1 = 2^1 has 2 subgroups.
T(3,5) = 2*T(2,5) + (A000040(5)^(3-1)-1)*T(1,5) = 2*14 + ((11^2)-1)*2 = 268.
First 6 rows and 8 columns:
n\k|   1     2       3        4          5           6            7            8
----+---------------------------------------------------------------------------
 0 |   1     1       1        1          1           1            1            1
 1 |   2     2       2        2          2           2            2            2
 2 |   5     6       8       10         14          16           20           22
 3 |  16    28      64      116        268         368          616          764
 4 |  67   212    1120     3652      19156       35872        99472       152404
 5 | 374  2664   42176   285704    3961832    10581824     51647264     99869288
 6 |2825 56632 3583232 61946920 3092997464 13340150272 141339210560 377566978168

Cf. A000040, A113935, A006116, A006117, A006119, A006121, A015197.

# produces an array A of the first (7(7+1))/2 terms. However computation quickly becomes expensive for values > 7.
LoadPackage("sonata");    # sonata package needs to be loaded to call function Subgroups. Sonata is included in latest versions of GAP.
N:=[1..7];; R:=[];; S:=[];;
for i in N do
    for j in N do
        if j>i then
            break;
        fi;
        Add(R,j);
    od;
    Add(S,R);
    R:=[];;
od;
A:=[];;
for n in N do
    L:=List([1..Length(S[n])],m->Size(Subgroups(ElementaryAbelianGroup( Primes[Reversed(S[n])[m]]^(S[n][m]-1)))));
    Add(A,L);
od;
A:=Flat(A);

T(n,k)=polcoeff(sum(i=0,n,x^i/prod(j=0,i,1-primes(k)[k]^j*x+x*O(x^n))),n)

T(n,k) = 2*T(n-1,k) + (A000040(k)^(n-1)-1)*T(n-2,k).
T(0,k) = 1.
T(1,k) = 2.
T(2,k) = A000040(k) + 3 = A113935(k).
T(3,k) = 2*(A000040(k)^3 + (A000040(k)-2))/(A000040(k)-1).

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A023581 Sum of exponents in prime-power factorization of p(n)+3.

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A023580 Sum of distinct prime divisors of prime(n)+3.

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A370887 Square array read by descending antidiagonals: T(n,k) is the number of subgroups of the elementary abelian group of order A000040(k)^n for n >= 0 and k >= 1.

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