A381031 The second smallest prime not dividing n minus the smallest prime not dividing n.

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

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..30030

Formula

a(n) = A380539(n) - A053669(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A381113 - A249270 = 2.239091... . - Amiram Eldar, Feb 14 2025

Example

For n = 1, the least prime not dividing it is 2, and the second least prime not dividing is 3, thus a(1) = 3-2 = 1.
For n = 3, the least nondividing prime is 2, the second least nondividing prime is 5, thus a(3) = 5-2 = 3.
For n = 6 = 2*3, the least nondividing prime is 5, and the second least nondividing prime is 7, thus a(6) = 7-5 = 2.

Mathematica

a[n_] := Module[{p = 1, q = 1, c = 0}, While[c < 2, p = NextPrime[p]; If[! Divisible[n, p], c++; If[c == 1, q = p]]]; p-q]; Array[a, 105] (* Amiram Eldar, Feb 14 2025 *)

Prog

(PARI) A381031(n) = { my(c=0, e=0); forprime(p=2, , if(n%p, c++; if(1==c, e=p, if(2==c, return(p-e))))); };

Crossrefs

Cf. A053669, A249270, A380539, A381113.

Sequence in context: A084311 A353486 A026490 * A053555 A324645 A124160

Adjacent sequences: A381028 A381029 A381030 * A381032 A381033 A381034

Keyword

nonn

Author

Antti Karttunen, Feb 12 2025

Status

approved