A369670 Integers k such that k, k + 1 and 2 k + 1 all have the same number of prime factors, counted with multiplicity.
2, 25, 34, 38, 57, 93, 118, 133, 145, 171, 177, 201, 205, 213, 218, 298, 332, 334, 361, 381, 387, 394, 446, 501, 507, 604, 633, 694, 698, 710, 805, 842, 865, 878, 898, 902, 921, 1004, 1075, 1084, 1105, 1114, 1130, 1141, 1172, 1182, 1226, 1285, 1293, 1358, 1412, 1445, 1465, 1513, 1557, 1587, 1592
Offset: 1
Keywords
Examples
a(3) = 34 is a term because 34 = 2 * 17, 34 + 1 = 35 = 5 * 7 and 2 * 34 + 1 = 69 = 3 * 23 all have 2 prime factors.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Maple
filter:= proc(n) local t; uses numtheory; t:= bigomega(n); t = bigomega(n+1) and t = bigomega(2*n+1) end proc: select(filter, [$1..10000]);
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Mathematica
s = {}; Do[If[PrimeOmega[k] == PrimeOmega[k + 1] == PrimeOmega[2 k + 1],AppendTo[s, k]], {k, 1000}]; s Select[Range[2000],Length[Union[PrimeOmega[{#,#+1,2#+1}]]]==1&] (* Harvey P. Dale, Mar 17 2025 *)