A307743 -id:A307743 - cp's OEIS frontend

A307641 Triangle T(i,j=1..i) read by rows which contain the naturally ordered prime-or-one factorization of the row number i.

1, 1, 2, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 5, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 1

I. V. Serov, Apr 19 2019

i=Product_{j=1..i} T(i,j). This is an adjusted formulation of the fundamental theorem of arithmetic with the fixed order of the prime-or-one factors, as well as with the regular length i of the factorization of i.
Remove all 1's except for n = 1 to get irregular triangle A307746.
A307723 is a quasi-logarithmic binary encoding of this triangle.

			Triangle begins:
  1,
  1, 2,
  1, 1, 3,
  1, 2, 1, 2,
  1, 1, 1, 1, 5,
  1, 2, 3, 1, 1, 1,
  1, 1, 1, 1, 1, 1, 7,
  1, 2, 1, 2, 1, 1, 1, 2,
  1, 1, 3, 1, 1, 1, 1, 1, 3,
  1, 2, 1, 1, 5, 1, 1, 1, 1, 1,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1,11,
  1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1,
  ...

I. V. Serov, Rows n=1..131 of triangle, flattened

Cf. A027746, A027748, A014963, A100995, A307662, A307723, A307742, A307743, A307746.

Table[Map[Which[PrimeNu@ # > 1, 1, And[PrimeQ@ #, Mod[n, #] == 0], #, Mod[n, #] == 0, FactorInteger[#][[1, 1]], True, 1] &, Range@ n], {n, 13}] // Flatten (* Michael De Vlieger, Apr 23 2019 *)

w(n) = my(t=isprimepower(n)); if (t, t, 0);
row(n) = vector(n, k, mnk = if ((n % k) == 0, k, 1); if (t=w(k), sqrtnint(mnk, t), 1)); \\ Michel Marcus, Apr 21 2019

T(i,j) = A307662(i,j)^w(j), where w(j)=0 if A100995(j)=0; otherwise w(j)=1/A100995(j), for 1 <= j <= n.

A307742 Quasi-logarithm A064097(n) of von Mangoldt's exponential function A014963(n).

Original entry on oeis.org

0, 1, 2, 1, 3, 0, 4, 1, 2, 0, 5, 0, 5, 0, 0, 1, 5, 0, 6, 0, 0, 0, 7, 0, 3, 0, 2, 0, 7, 0, 7, 1, 0, 0, 0, 0, 7, 0, 0, 0, 7, 0, 8, 0, 0, 0, 9, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 9, 0, 8, 0, 0, 1, 0, 0, 9, 0, 0, 0, 9, 0, 8, 0, 0, 0, 0, 0, 9, 0, 2, 0, 9, 0, 0, 0, 0, 0, 9

Offset: 1

I. V. Serov, Apr 26 2019

nonn

I. V. Serov, Table of n, a(n) for n = 1..10000

Cf. A064097, A014963, A008683, A307743.

qLog[n_] := qLog[n] = Module[{p, e}, If[n == 1, 0, Sum[{p, e} = pe; (1 + qLog[p-1])e, {pe, FactorInteger[n]}]]];
a[n_] := qLog[Exp[MangoldtLambda[n]]];
Array[a, 100] (* Jean-François Alcover, May 07 2019 *)

mang(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ A014963
ql(n) = if (n==1, 0, if(isprime(n),1+ql(n-1), sumdiv(n,p, if(isprime(p),ql(p)*valuation(n,p))))); \\ A064097
a(n) = ql(mang(n)); \\ Michel Marcus, Apr 26 2019

a(n) = A064097(A014963(n)).
a(n) = 1 + A064097(n-1) if n is prime.
a(n) = a(p) if n=p^k with k > 1.
a(n) = 0 if n is not a prime power or n = 1.
a(n) = -Sum_{d|n} A064097(d)*A008683(d) by Mobius inversion.

cp's OEIS Frontend

A307641 Triangle T(i,j=1..i) read by rows which contain the naturally ordered prime-or-one factorization of the row number i.

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