A268674 a(1) = 1, after which, for odd numbers: a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1, and for even numbers: a(n) = a(A000265(n)).

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 8, 7, 19, 2, 9, 11, 10, 5, 23, 6, 29, 1, 12, 13, 15, 4, 31, 17, 14, 3, 37, 8, 41, 7, 16, 19, 43, 2, 25, 9, 18, 11, 47, 10, 21, 5, 20, 23, 53, 6, 59, 29, 22, 1, 27, 12, 61, 13, 24, 15, 67, 4, 71, 31, 26, 17, 35, 14, 73, 3, 28, 37, 79, 8, 33, 41, 30, 7

Offset: 1

Author

Antti Karttunen, Feb 11 2016

Keywords

• nonn

Comments

For odd numbers n > 1, a(n) tells which term is on the immediately preceding row of A083221, in the same column where n itself is.

The sequence offers a left inverse for A250469 that is slightly easier to compute than A250470.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..32769
• Index entries for sequences generated by sieves

Crossrefs

Left inverse of A250469.

Cf. A000040, A000265, A001248, A055396, A078898, A083221.

Cf. also A064989.

Differs from A250470 for the first time at n=42, where a(42)=8, while A250470(42) = 10.

Programs

Mathematica

(* b = A250469 *) b[1] = 1; b[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[1, 1]]; For[ k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[ FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1 + 2 == k2, Return[m2]]]];
a[1] = a[2] = 1; a[n_?EvenQ] := a[n] = a[n/2]; a[n_] := a[n] = For[k = 1, True, k++, If[b[k] == n, Return[k]]];
Array[a, 100] (* Jean-François Alcover, Mar 14 2016 *)

Formula

a(1) = 1, after which, a(n) = a(A000265(n)) if n is even, otherwise for odd n, a(n) = A083221(A055396(n)-1, A078898(n)).

Other identities. For all n >= 1:

a(A250469(n)) = n. [This works as a left inverse for sequence A250469.]

a(2n) = a(n). [The even bisection gives the whole sequence back.]

a(2n-1) = A250470(2n-1). [Matches with A250470 on odd numbers.]

a(A000040(n+1)) = A000040(n). [Maps each odd prime to the preceding prime.]

a(A001248(n+1)) = A001248(n). [Maps each square of an odd prime to the square of the preceding prime.]