A071367
Numbers n such that n+0, n+1, n+2, n+3 and n+4 are, in some order, 1 * a prime, 2 * a prime, 3 * a prime, 4 * a prime and 5 * a prime.
Original entry on oeis.org
6, 211, 2305, 2731, 19441, 116131, 174595, 222931, 229945, 232051, 243091, 266401, 334315, 350785, 423481, 495265, 523945, 530545, 535915, 539401, 556705, 600601, 663601, 671035, 689131, 721891, 907195, 908041, 1105105, 1113961, 1289731
Offset: 1
211 is a term because 211=1*211, 212=4*53, 213=3*71, 214=2*107 and 215=5*43. The left factors are the integers 1 to 5; and the right factors are primes.
6 is a term because 6=2*3, 7=1*7, 8=4*2, 9=3*3, 10=5*2 where the left factors are the integers 1 to 5 and the right factors are primes. - _Sean A. Irvine_, Jul 14 2024
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a071367 n = a071367_list !! (n-1)
a071367_list = tail $ filter f [1..] where
f x = and $ map g [5, 4 .. 1] where
g k = sum (map h $ map (+ x) [0..4]) == 1 where
h z = if r == 0 then a010051' z' else 0
where (z', r) = divMod z k
-- Reinhard Zumkeller, Jul 31 2015
A071368
Numbers k such that k+0, k+1, k+2, k+3, k+4, and k+5 are, in some order, 1 * a prime, 2 * a prime, ... and 6 * a prime.
Original entry on oeis.org
18362, 2914913, 5516281, 6618242, 7224834, 9018353, 9339114, 10780554, 16831081, 17800553, 18164161, 18646202, 20239913, 29743561, 32464433, 32915513, 42464514, 43502033, 45652314, 51755761, 53464314, 62198634
Offset: 1
From _Reinhard Zumkeller_, Jul 31 2015: (Start)
18362 is in the sequence because 18362=2*9181, 18363=3*6121, 18364=4*4591, 18365=5*3673, 18366=6*3061 and 18367=1*18367. The left factors are the integers 1 to 6; and the right factors are primes.
5516281 is the smallest term also occurring in A071367:
5516281 + 0 = 1 * 5516281 = prime(381844) = a(3) = A071367(77);
5516281 + 1 = 2 * 2758141 = 2 * prime(200537);
5516281 + 2 = 3 * 1838761 = 3 * prime(137758);
5516281 + 3 = 4 * 1379071 = 4 * prime(105622);
5516281 + 4 = 5 * 1103257 = 5 * prime(85955);
5516281 + 5 = 6 * 919381 = 6 * prime(72692), not needed for A071367.
(End)
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a071368 n = a071368_list !! (n-1)
a071368_list = filter f [1..] where
f x = and $ map g [6, 5 .. 1] where
g k = sum (map h $ map (+ x) [0..5]) == 1 where
h z = if r == 0 then a010051' z' else 0
where (z', r) = divMod z k
-- Reinhard Zumkeller, Jul 31 2015
A071369
Numbers n such that n+0, n+1, ... and n+6 are, in some order, 1 * a prime, 2 * a prime, ... and 7 * a prime.
Original entry on oeis.org
2914913, 5516281, 6618241, 9018353, 10780553, 18164161, 20239913, 45652313, 51755761, 62198633, 81235441, 91986833, 158764313, 175472641, 191010953, 197375753, 215206201, 322030801, 322461713, 362007353, 513284401, 668745001
Offset: 1
2914913 is there because 2914913=1*2914913, 2914914=6*485819, 2914915=5*582983, 2914916=4*728729, 2914917=3*971639, 2914918=2*1457459 and 2914919=7*416417. The left factors are the integers 1 to 7; and the right factors are primes.
A071370
Numbers n such that n+0, n+1, ... and n+7 are, in some order, 1 * a prime, 2 * a prime, ... and 8 * a prime.
Original entry on oeis.org
10780552, 62198632, 884811061, 1457032501, 3573315892, 7321991041, 7391371681, 8557865812, 11434075381, 16893247141, 21599190901, 22487905441, 28044279892, 28273111012, 37923188932, 50238568801, 59635316161
Offset: 1
10780552 is there because 10780552=8*1347569, 10780553=7*1540079, 10780554=6*1796759, 10780555=5*2156111, 10780556=4*2695139, 10780557=3*3593519, 10780558=2*5390279 and 10780559=1*10780559. The left factors are the integers 1 to 8; and the right factors are primes.
A071372
Numbers n such that n+0, n+1, ... and n+9 are, in some order, 1 * a prime, 2 * a prime, ... and 10 * a prime.
Original entry on oeis.org
1676641682, 1829413730, 862353305089, 2394196081201, 7816812203762, 9089234694530, 10689865119781, 10988437006262, 13826845745989, 17242727247890, 21487725800102, 24653435773682, 28779837186662
Offset: 1
1676641682 is there because 1676641682=2*838320841, 1676641683=3*558880561, 1676641684=4*419160421, 1676641685=5*335328337, 1676641686=6*279440281, 1676641687=7*239520241, 1676641688=8*209580211, 1676641689=9*186293521, 1676641690=10*167664169 and 1676641691=1*1676641691. The left factors are the integers 1 to 10; and the right factors are primes.
A138588
a(n) = the least integer > n such that r(1)|a(n), r(2)|(a(n)+1), r(3)|(a(n)+2),... and r(n)|(a(n)+n-1), where (r(1),r(2),r(3),...,r(n)) is some permutation of (1,2,3,...,n).
Original entry on oeis.org
2, 3, 4, 6, 6, 20, 24, 48, 48, 110, 110, 110, 243, 403, 402, 2504, 2352, 12219, 25200, 60458, 14256, 95760, 120120, 582090, 582096, 186120, 3299404, 11060250, 28648620, 376576202, 9469950, 832431604, 832431603, 962161203, 1403352722
Offset: 1
Example, n = 7:
For all stretches of 7 consecutive integers, with the least integer m in each stretch such that m >=8 and m <= 19, there are at least 2 primes (each > 7) in the stretch. Now both primes cannot be divided by any positive integer <= 7 except 1. But there is only one 1 in the permutation (r(1),r(2),...,r(7)). So a(7) is > 19.
If the least integer in the stretch of 7 consecutive integers is 20, 21, or 22, then there is only one prime in the stretch, but there are two integers, 22 and 26, that aren't divisible by any integer <= 7 except 1 and 2. (And there is already a prime, 23, that needs to be divided by 1.)
So a(7) is > 22. If the least integer in the stretch of 7 consecutive integers is 23, then there are 2 primes in the stretch. But if the smallest integer of the stretch is 24, then we have 4|24, 5|25, 2|26, 3|27, 7|28, 1|29 and 6|30. And the sequence of 7 divisors (4,5,2,3,7,1,6) is a permutation of (1,2,3,4,5,6,7). So a(7) = 24.
A071371
Numbers n such that n+0, n+1, ... and n+8 are, in some order, 1 * a prime, 2 * a prime, ... and 9 * a prime.
Original entry on oeis.org
1829413731, 13096880161, 28273111011, 32480018341, 79089694311, 330780346261, 363500177041, 602794125781, 679251409201, 905780175301, 956731265701, 1010903523841, 1011470714101, 1086338816631, 1312670706051
Offset: 1
1829413731 is there because 1829413731=3*609804577, 1829413732=4*457353433, 1829413733=7*261344819, 1829413734=6*304902289, 1829413735=5*365882747, 1829413736=8*228676717, 1829413737=9*203268193, 1829413738=2*914706869 and 1829413739=1*1829413739. The left factors are the integers 1 to 9; and the right factors are primes.
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