A356322 a(n) is the smallest number that starts a run of exactly n consecutive numbers in A126706, or -1 if no such number exists.

12, 44, 98, 3174, 844, 22020, 217070, 1092747, 8870024, 262315467, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548

Offset

1,1

Comments

Term a(n) begins a run of n consecutive nonsquarefree numbers m such that omega(m) > 1.

The run of m must occur between successive primes.

Links

Table of n, a(n) for n=1..15.

Rémy Sigrist, C program

Example

a(n) is the first term in the sequences shown below:
n: a(n)..a(n)+n-1
-----------------
1: {12}
2: {44, 45}
3: {98, 99, 100}
4: {3174, 3175, 3176, 3177}
5: {844, 845, 846, 847, 848}
6: {22020, 22021, 22022, 22023, 22024, 22025}
7: {217070, 217071, 217072, 217073, 217074, 217075, 217076}
...
There are 4 consecutive numbers m in A126706 starting from 844 and again from 2888, but since 848 and 2892, respectively, are also in A126706, these m ascribe to n = 5 instead. The range m = 3174..3177 has at most n = 4 numbers in A126706 and 3174 is the smallest number with that quality, hence a(4) = 3174.

Mathematica

j = 0; k[_] = False; Sort[Reap[Do[If[And[#2 > 1, #1 != #2] & @@ {PrimeOmega[n], PrimeNu[n]}, j++; If[! IntegerQ[c], Set[c, n]], If[j > 0, If[! k[j], Sow[{j, c}] ]; Set[{k[j], j}, {True, 0}]; Clear[c] ] ], {n, 2^16}] ][[-1, -1]] ][[All, -1]]

Prog

(C) See Links section.

Crossrefs

Cf. A001221, A001222, A001223, A005250, A013929, A024619, A126706.

Sequence in context: A296181 A007899 A343729 * A100156 A320998 A294521

Adjacent sequences: A356319 A356320 A356321 * A356323 A356324 A356325

Keyword

nonn,more

Author

Michael De Vlieger, Oct 28 2022

Extensions

a(10)-a(11) from Rémy Sigrist, Oct 29 2022

a(12)-a(14) from Martin Ehrenstein, Oct 30 2022

a(15) from Martin Ehrenstein, Nov 02 2022

Status

approved