A184829 -id:A184829 - cp's OEIS frontend

A184831 a(n) = A184830(n)/A184829(n) unless A184829(n) = 0 in which case a(n) = 0.

0, 0, 1, 1, 1, 3, 1, 1, 3, 2, 5, 5, 3, 7, 1, 5, 9, 15, 3, 3, 13, 3, 15, 9, 1, 19, 2, 1, 9, 23, 1, 11, 1, 11, 9, 3, 33, 11, 35, 21, 7, 13, 41, 63, 25, 5, 45, 3, 49, 5, 1, 3, 55, 33, 1, 59, 9, 63, 27, 65, 11, 1, 3, 75, 45, 1, 79, 1, 5, 41, 85, 1, 1, 89, 5, 39, 93, 1, 57, 9

Offset: 1

Rémi Eismann, Jan 23 2011

nonn

a(n) is the "level" of prime powers.
The decomposition of prime powers into weight * level + gap is A000961(n) = A184829(n) * a(n) + A057820(n) if a(n) > 0.
A184830(n) = A000961(n) - A057820(n) if A000961(n) - A057820(n) > A057820(n), 0 otherwise.

			For n = 1 we have A184829(1) = 0, hence a(1) = 0.
For n = 3 we have A184830(3)/A184829(3)= 2 / 2 = 1; hence a(3) = 1.
For n = 24 we have A184830(24)/A184829(24)= 45 / 5 = 9; hence a(24) = 9.

Rémi Eismann, Table of n, a(n) for n = 1..10000

Cf. A000961, A057820, A184829, A184830, A117078, A117563, A001223, A118534.

A184830 a(n) = largest k such that A000961(n+1) = A000961(n) + (A000961(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 2, 3, 3, 6, 7, 7, 9, 10, 15, 15, 15, 21, 23, 25, 27, 30, 27, 33, 39, 39, 45, 45, 47, 57, 58, 61, 63, 69, 67, 77, 79, 77, 81, 93, 99, 99, 105, 105, 105, 117, 123, 126, 125, 125, 135, 129, 147, 145, 151, 159, 165, 165, 167, 177, 171, 189, 189, 195

Offset: 1

Rémi Eismann, Jan 23 2011

From the definition, a(n) = A000961(n) - A057820(n) if A000961(n) - A057820(n) > A057820(n), 0 otherwise where A000961 are the prime powers and A057820 are the gaps between prime powers.

			For n = 1 we have A000961(1) = 1, A000961(2) = 2; for all k >= 2, 2 = 1 + (1 mod k), hence the largest k does not exist and a(1) = 0.
For n = 3 we have A000961(3) = 3, A000961(4) = 4; 2 is the largest k such that 4 = 3 + (3 mod k), hence a(3) = 2.
For n = 24 we have A000961(24) = 49, A000961(25) = 53; 45 is the largest k such that 53 = 49 + (49 mod k), hence a(24) = 45.

Rémi Eismann, Table of n, a(n) for n = 1..9790

Cf. A000961, A057820, A184829, A184831, A117078, A117563, A001223, A118534.

A184830 := proc(n)
    if A000961(n) > 2*A057820(n) then
        A000961(n)-A057820(n) ;
    else
        0;
    end if;
end proc:
seq(A184830(n),n=1..40) ; # R. J. Mathar, Sep 23 2016

nmax = 10000;
ppmax = 12*nmax; (* increase prime power max coef 12 in case of overflow *)
A000961 = Join[{1}, Select[Range[2, ppmax], PrimePowerQ]];
A057820 = Differences[A000961];
a[n_] := If[A000961[[n]] > 2*A057820[[n]], A000961[[n]] - A057820[[n]], 0];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Aug 06 2023 *)

cp's OEIS Frontend

A184831 a(n) = A184830(n)/A184829(n) unless A184829(n) = 0 in which case a(n) = 0.

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