A392184 - OEIS

A392184

a(n) is the least k for which omega(k)*omega(k + 1)*omega(k + 2) = n where omega = A001221, or -1 if no such k exists.

%I #43 Mar 27 2026 21:16:57

%S 1,2,4,29,10,2309,28,1138829,20,130,2308,239378649509,68,

%T 461282657605769,570569,2728,208,3338236629672919864889,154,

%U 11465419967969569966774409,3568,570568,200560490129,447252622660353534197972753024069,713,77140,304250263527208,644

%N a(n) is the least k for which omega(k)*omega(k + 1)*omega(k + 2) = n where omega = A001221, or -1 if no such k exists.

%H Sean A. Irvine, <a href="/A392184/b392184.txt">Table of n, a(n) for n = 0..43</a>

%H Brian Hayes, <a href="http://bit-player.org/2021/does-having-prime-neighbors-make-you-more-composite">Does having prime neighbors make you more composite?</a>, Bit-Player Article, Nov 04 2021.

%F From _Vaclav Kotesovec_, Jan 03 2026: (Start)

%F For p>2 prime, A002110(p) - 1 <= a(p) <= A075590(p) - 1.

%F Conjecture: a(p) = A075590(p) - 1. (End)

%e k | omega(k)*omega(k + 1)*omega(k + 2) = n

%e -----------------------------------------------------------

%e 1 | 0 * 1 * 1 = 0

%e 2 | 1 * 1 * 1 = 1

%e 4 | 1 * 1 * 2 = 2

%e 29 | 1 * 3 * 1 = 3

%e 10 | 2 * 1 * 2 = 4

%e 2309 | 1 * 5 * 1 = 5

%e 28 | 2 * 1 * 3 = 6

%e 1138829 | 1 * 7 * 1 = 7

%e 20 | 2 * 2 * 2 = 8

%e 130 | 3 * 1 * 3 = 9

%e 2308 | 2 * 1 * 5 = 10

%o (Magma) [Min([k: k in [1..10000] | #PrimeDivisors(k)*#PrimeDivisors(k+1)*#PrimeDivisors(k+2) eq n]): n in [0..6]];

%o (PARI) isok(k,n) = omega(k)*omega(k+1)*omega(k+2) == n;

%o a(n) = my(k=1); while (! isok(k,n), k++); k; \\ _Michel Marcus_, Feb 26 2026

%Y Cf. A001221, A002110, A075590, A391216, A006549, A088071, A173037, A264734, A329364, A392232, A393108.

%K nonn,more

%O 0,2

%A _Juri-Stepan Gerasimov_, Jan 02 2026

%E a(11)-a(18) from _Vaclav Kotesovec_, Jan 03 2026