A349406 a(n) = A349405(n)/A348779(n).

19, 3, 5, 3, 2, 2, 7, 3, 3, 2, 3, 5, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 3, 2, 5, 2, 5, 2, 2, 2, 3, 7, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 11, 3, 2, 2, 2, 2, 2, 41, 2, 2, 3, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 5, 3, 5, 3, 3, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 3, 2, 2

Offset: 1

Author

Michael De Vlieger, Nov 16 2021

Keywords

• nonn

Comments

Ratio of progenitor and prime in A347113.

Let s = A347113, j = s(i)+1, and k = s(i+1). For prime k, j is a squarefree semiprime pq, p < q.

The first 3 primes in s have k = p, while all others observed for i <= 2^19 have k = q. This sequence thus lists the other prime factor r of j such that r*k = j.

The quasi-linear striations k < n are arranged according to this sequence (see color-coded log-log scatterplot). - Michael De Vlieger, Nov 17 2021

Examples

			s(6)+1 = 95 -> s(7) = 5; a(1) = 95/5 = 19.
s(7)+1 = 6 -> s(8) = 2; a(2) = 6/2 = 3.
s(10)+1 = 15, -> s(11) = 3; a(3) = 15/3 = 5.
s(18)+1 = 39, -> s(19) = 13; a(4) = 39/13 = 3, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Notes on ratio j/k in A347113, listing numbers of certain ratios seen given 2^19 terms of A347113.
• Michael De Vlieger, Extended table of n, a(n) for n = 1..23082 (all terms resulting from 2^19 terms of A347113)
• Michael De Vlieger, Log-log scatterplot of A347113(n), n=1..256 indicating primes in green, labeling them with the ratio j/k = a(n) = (A347113(n-1)+1)/A347113(n). Red dots represent records A347307 in A347113, and blue represent local minima A347756 in A347113.
• Michael De Vlieger, Log-log scatterplot of a(n), for n=1..2^16 showing primes k=A347113(A348784) listed in A348779, color coded according to the ratio j/k, which is always prime. Color function: red = 2, orange = 3, chartreuse = 5, small green = 7, large green = 11, large cyan = 13, large blue = 17, large indigo = 19, large purple = 29, large magenta = 41.

Crossrefs

Cf. A006530, A020639, A347113, A347313, A348779, A349405.

Programs

Mathematica

c[] = 0; j = m = 2; m = {1}~Join~Reap[Do[If[IntegerQ@ Log2[i], While[c[m] > 0, m++]]; Set[k, m]; While[Or[c[k] > 0, k == j, GCD[j, k] == 1], k++]; Sow[k]; Set[c[k], i]; j = k + 1, {i, 900}]][[-1, -1]]; Map[(#1 + 1)/#2 & @@ m[[# - 1 ;; #]] &, Position[m, ?PrimeQ][[All, 1]]]