A177222
Numbers k that are the products of two distinct primes, such that 2*k + 1 and 4*k + 3 are also products of two distinct primes.
Original entry on oeis.org
38, 46, 106, 129, 133, 145, 201, 203, 235, 291, 298, 334, 335, 381, 407, 417, 458, 489, 497, 538, 579, 583, 597, 623, 626, 649, 685, 689, 694, 707, 758, 767, 781, 815, 898, 899, 921, 926, 959, 995, 1073, 1079, 1082, 1094, 1099, 1139, 1142, 1157, 1214, 1226
Offset: 1
38 is a term because 38 = 2*19, 2*38 + 1 = 77 = 7*11, and 4*38 + 1 = 155 = 5*31.
Cf.
A006881,
A111153,
A177210,
A177211,
A177212,
A177213,
A177214,
A177215,
A177216,
A177217,
A177220,
A177221.
-
f[n_] := Last/@FactorInteger[n] == {1,1}; lst = {}; Do[If[f[n] && f[2*n+1] && f[4*n+3], AppendTo[lst, n]], {n, 1000}]; lst
A177223
Numbers k that are the products of two distinct primes such that 2*k+1, 4*k+3 and 8*k+7 are also products of two distinct primes.
Original entry on oeis.org
145, 203, 291, 298, 407, 497, 649, 707, 758, 815, 899, 926, 959, 995, 1079, 1094, 1139, 1142, 1157, 1313, 1403, 1415, 1461, 1497, 1538, 1639, 1658, 1691, 1857, 1934, 1945, 1991, 2123, 2159, 2217, 2234, 2315, 2603, 2629, 2807, 2991, 3215, 3254, 3279, 3305
Offset: 1
145 is a term because 145 = 5*29, 2*145 + 1 = 291 = 3*97, 4*145 + 1 = 583 = 11*53, and 8*145 + 1 = 1167 = 3*389.
Cf.
A006881,
A111153,
A177210,
A177211,
A177212,
A177213,
A177214,
A177215,
A177216,
A177217,
A177220,
A177221,
A177222.
-
f[n_]:=Last/@FactorInteger[n]=={1,1}; lst={};Do[If[f[n]&&f[2*n+1]&&f[4*n+3]&&f[8*n+7],AppendTo[lst,n]],{n,0,2*7!}];lst
tdpQ[n_]:=With[{c={n, 2n+1, 4n+3,8n+7}},PrimeNu[c]==PrimeOmega[c]=={2,2,2,2}]; Select[Range[3500],tdpQ] (* Harvey P. Dale, Jan 11 2025 *)
A378297
Squarefree semiprimes k that remain squarefree semiprimes for exactly two iterations of the map k -> 2*k+1.
Original entry on oeis.org
38, 46, 106, 129, 133, 201, 235, 334, 335, 381, 417, 458, 489, 538, 579, 583, 597, 623, 626, 685, 689, 694, 767, 781, 898, 921, 1073, 1082, 1099, 1214, 1226, 1227, 1234, 1285, 1299, 1315, 1385, 1486, 1514, 1517, 1546, 1603, 1631, 1646, 1799, 1817, 1819, 1841
Offset: 1
38 is a term because 38 (semiprime) gives 2*38+1 (77, semiprime) gives 2*77+1 (155, semiprime). But 155 gives 2*155+1 (311, not semiprime), so the chain has length three (38, 77, 155).
921 is a term because 921 (semiprime) gives 2*921+1 (1843, semiprime) gives 2*1843+1 (3687, semiprime). But 3687 gives 2*3687+1 (7375, not semiprime), so the chain has length three (921, 1843, 3687).
-
s[n_] := -1 + Length@ NestWhileList[2*# + 1 &, n, FactorInteger[#][[;; , 2]] == {1, 1} &]; Select[Range[2000], s[#] == 3 &] (* Amiram Eldar, Dec 17 2024 *)
A379372
Sphenic numbers k such that 2*k+1 is also a sphenic number.
Original entry on oeis.org
322, 357, 370, 402, 430, 442, 610, 654, 790, 822, 826, 874, 885, 942, 1045, 1054, 1105, 1130, 1182, 1222, 1342, 1358, 1414, 1510, 1578, 1582, 1677, 1702, 1738, 1742, 1767, 1798, 1802, 1810, 1842, 1947, 2014, 2035, 2086, 2185, 2222, 2247, 2282, 2334, 2365, 2397
Offset: 1
322 is a term because 322 = 2*7*23 (sphenic) and 2*322+1 = 645 = 3*5*43 (sphenic).
-
sphenicQ[n_] := FactorInteger[n][[;; , 2]] == {1, 1, 1}; Select[Range[3000], And @@ sphenicQ /@ {#, 2*# + 1} &] (* Amiram Eldar, Dec 21 2024 *)
Showing 1-4 of 4 results.
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