Displaying 1-10 of 90 results found.
Primes q such that 6*q-1 and 6*q+1 are twin primes. Proper subset of A002822.
+20
14
2, 3, 5, 7, 17, 23, 47, 103, 107, 137, 283, 313, 347, 373, 397, 443, 467, 577, 593, 653, 773, 787, 907, 1033, 1117, 1423, 1433, 1613, 1823, 2027, 2063, 2137, 2153, 2203, 2287, 2293, 2333, 2347, 2677, 2903, 3257, 3307, 3407, 3413, 3593, 3623, 3673, 3923
COMMENTS
Conjecture: a(n) ~ n*log(n)*log(n*log(n))*log(log(n)). - Carl R. White, Nov 16 2023
PROG
(PARI) forprime(p=2, 9999, if(isprime(6*p+1) & isprime(6*p-1), print(p))) \\ David Radcliffe, Apr 02 2016 (Python) from sympy import *; print([p for p in primerange(2, 9999) if isprime(6*p-1) and isprime(6*p+1)]) # David Radcliffe, Apr 02 2016
Numbers n such that N = (5n)^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
+20
8
1, 2, 7, 12, 14, 15, 42, 48, 77, 86, 89, 99, 118, 131, 146, 161, 163, 167, 201, 208, 209, 246, 278, 286, 306, 334, 343, 370, 378, 384, 400, 404, 420, 422, 449, 462, 481, 483, 499, 509, 537, 551, 568, 587, 590, 609, 651, 652, 667, 684, 730, 755, 761, 806, 817, 825, 827, 848, 867, 870, 882, 916, 931, 980, 982, 992
COMMENTS
Dinculescu notes that if N = m^2 > 1 is a twin rank (i.e., in A002822), then m is always a multiple of 5, and if N = m^3 > 1, then m is a multiple of 7, cf. A326234. He asks whether there are other pairs (a, b) different from (5, 2) and (7, 3) such that all twin ranks m^b > 1 are of the form m = a*n. (Of course (5, 2) and (7, 3) imply that (5, 2k), (7, 3k) and (35, 6k) is such a pair for any k.) This sequence lists the n for (a, b) = (5, 2), see A326232 for the numbers m.
PROG
(PARI) select( is(n)=!for(s=1, 2, ispseudoprime(150*n^2+(-1)^s)||return), [1..10^3])
Numbers k such that N = k^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
+20
8
1, 5, 10, 35, 60, 70, 75, 210, 240, 385, 430, 445, 495, 590, 655, 730, 805, 815, 835, 1005, 1040, 1045, 1230, 1390, 1430, 1530, 1670, 1715, 1850, 1890, 1920, 2000, 2020, 2100, 2110, 2245, 2310, 2405, 2415, 2495, 2545, 2685, 2755, 2840, 2935, 2950, 3045, 3255, 3260, 3335, 3420, 3650, 3775, 3805
COMMENTS
Dinculescu notes that when k^2 > 1 is a twin rank (i.e., in A002822), then k is always a multiple of 5, and if k^3 > 1 is a twin rank, it is divisible by 7. See A326231 for the terms > 1 divided by 5.
PROG
(PARI) select( is(n)=!for(s=1, 2, ispseudoprime(6*n^2+(-1)^s)||return), [1..5000])
Numbers n such that N = n^3 is a twin rank ( A002822: 6N +- 1 are twin primes).
+20
8
1, 28, 42, 168, 203, 287, 308, 518, 1043, 1057, 1512, 1603, 1638, 1680, 1757, 1988, 2905, 3367, 3927, 4018, 4928, 5033, 5145, 5257, 5292, 5432, 5733, 6762, 7182, 7210, 7798, 8715, 10213, 10318, 10668, 10745, 11088, 12243, 13552, 14245, 14588, 14707, 15155, 15323, 15687, 15722, 15757
COMMENTS
Dinculescu notes that when n^2 or n^3 is a twin rank > 1 (i.e., in A002822), then n is a multiple of 5, resp. 7. It is unknown whether there exist other pairs (a, b) different from (5, 2) and (7, 3) such that n^b => a | n. (Of course (5, 2k) and (7, 3k) and (35, 6k) is a solution for any k.) See A326233 for the terms > 1 divided by 7.
PROG
(PARI) select( is(n)=!for(s=1, 2, ispseudoprime(6*n^3+(-1)^s)||return), [1..10^5])
Numbers k such that N = k^6 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
+20
8
1, 1820, 2590, 4795, 5565, 8330, 8470, 10640, 10710, 15960, 16730, 19145, 24535, 26460, 34580, 37065, 41510, 42630, 43505, 48230, 59675, 69160, 84910, 90860, 99540, 103320, 112560, 114205, 117600, 127120, 129220, 131670, 143290, 152740, 161105, 164115, 170030, 175105, 181195, 185045
COMMENTS
Dinculescu notes that when N = m^2 (resp. m^3) > 1 is a twin rank (i.e., in A002822), then m is a multiple of 5 (resp. of 7), cf. A326232 and A326234. Thus, when N = m^6, then m is a multiple of 35. See A326235 for a(n)/35, n > 1.
PROG
(PARI) select( is(n)=!for(s=1, 2, ispseudoprime(6*n^6+(-1)^s)||return), [1..10^5])
Least k > 1 such that k^n is a twin rank (cf. A002822: 6*k^n +- 1 are twin primes).
+20
7
2, 5, 28, 70, 2, 1820, 110, 1850, 2520, 220, 2023, 9415, 647, 2880, 2562, 3895, 2, 51240, 525, 3750, 147, 2350, 355, 4480, 2588, 3370, 38157, 1185, 1473, 12530, 4338, 1540, 1988, 535, 102, 22606, 13773, 18895, 16373, 2635, 20428, 76300, 23037, 29005, 11078
COMMENTS
Dinculescu observes that when k^2 > 1 is a twin rank (i.e., in A002822) then 5 | k (k is divisible by 5), and if k^3 is a twin rank, then 7 | k; cf. A326232 & A326234. It is unknown whether there are other pairs (a, b) such that a | n whenever n^b > 1 is a twin rank. (Of course 2 | b => 5 | a and 3 | b => 7 | a, so we aren't interested in pairs (a, b) which are consequence of this.)
PROG
(PARI) a(n)=for(k=2, oo, ispseudoprime(6*k^n-1)&&ispseudoprime(6*k^n+1)&&return(k))
Numbers n such that N = (7n)^3 is a twin rank ( A002822: 6N +- 1 are twin primes).
+20
6
4, 6, 24, 29, 41, 44, 74, 149, 151, 216, 229, 234, 240, 251, 284, 415, 481, 561, 574, 704, 719, 735, 751, 756, 776, 819, 966, 1026, 1030, 1114, 1245, 1459, 1474, 1524, 1535, 1584, 1749, 1936, 2035, 2084, 2101, 2165, 2189, 2241, 2246, 2251, 2301, 2305, 2384, 2511, 2541, 2710, 2865, 2955, 2990
COMMENTS
Dinculescu notes that if m^3 > 1 is a twin rank (i.e., in A002822), then m is always a multiple of 7. (Indeed, 6m^3 + 1 == 0 (mod 7) if m == 1, 2 or 4 (mod 7), and 6m^3 - 1 == 0 (mod 7) for m == 3, 5 or 6 (mod 7).) He asks whether there are other pairs (a, b) different from (5, 2) and (7, 3) such that all twin ranks m^b > 1 are of the form m = a*n. (Of course (5, 2) and (7, 3) imply that (5, 2k), (7, 3k) and (35, 6k) is also such a pair for any k >= 1.)
This sequence lists these m/7 for (a, b) = (7, 3), see A326234 for the numbers m.
MAPLE
filter:= proc(n) local m;
m:= (7*n)^3;
isprime(6*m+1) and isprime(6*m-1)
end proc:
PROG
(PARI) select( is(n)=!for(s=1, 2, ispseudoprime(6*(7*n)^3+(-1)^s)||return), [1..10^4])
Numbers k such that N = (35k)^6 is a twin rank ( A002822: 6N +- 1 are twin primes).
+20
6
52, 74, 137, 159, 238, 242, 304, 306, 456, 478, 547, 701, 756, 988, 1059, 1186, 1218, 1243, 1378, 1705, 1976, 2426, 2596, 2844, 2952, 3216, 3263, 3360, 3632, 3692, 3762, 4094, 4364, 4603, 4689, 4858, 5003, 5177, 5287, 5361, 5426, 5999, 6054, 6285, 6347, 6417, 6457, 6639, 6862, 7269, 7500
COMMENTS
Dinculescu notes that if N = m^2 > 1 is a twin rank (i.e., in A002822), then m is a multiple of 5, and if N = m^3 > 1, then m is a multiple of 7, cf. A326231 and A326233. Thus, when N = m^6, then m is a multiple of 35, and here we list these m/35.
PROG
(PARI) select( is(n)=!for(s=1, 2, ispseudoprime(6*(35*n)^6+(-1)^s)||return), [1..10^4])
EXTENSIONS
a(1..10^4) independently computed using Mathematica and PARI/GP, by A. D. and M. F. Hasler, Jun 19 2019
Numbers k such that k and k+3 are in A002822.
+20
2
2, 7, 30, 100, 107, 135, 172, 217, 322, 352, 452, 562, 590, 667, 707, 917, 940, 975, 1092, 1127, 1222, 1470, 1570, 1950, 2282, 2545, 2772, 2865, 2930, 3007, 3087, 3682, 3770, 3840, 3945, 4447, 4452, 4477, 5142, 5555, 5600, 5625, 5635, 6262, 6442, 7520, 8232
EXAMPLE
2 is a term because 2 and 5 are in A002822.
MAPLE
isA002822 := proc(n) isprime(6*n-1) and isprime(6*n+1) ; end proc:
isA173233 := proc(n) isA002822( n ) and isA002822(n+3 ) ; end proc:
for n from 1 to 3000 do if isA173233(n) then printf("%d, ", n) ; end if; end do: # R. J. Mathar, May 02 2010
MATHEMATICA
SequencePosition[Table[If[AllTrue[6n+{1, -1}, PrimeQ], 1, 0], {n, 9000}], {1, _, _, 1}][[;; , 1]] (* Harvey P. Dale, Jul 05 2023 *)
EXTENSIONS
Definition and sequence corrected (667, 707, 1097 etc inserted) by R. J. Mathar, May 02 2010
Numbers n such that 6n is in A002822 but n is not.
+20
2
63, 65, 88, 98, 102, 133, 157, 163, 185, 193, 198, 203, 208, 210, 233, 245, 250, 262, 310, 340, 380, 387, 413, 437, 457, 462, 473, 478, 483, 493, 507, 508, 515, 530, 585, 600, 627, 635, 640, 647, 658, 662, 677, 718, 742, 765, 772, 793, 795, 830, 847, 857
COMMENTS
To use Dinculescu's terminology (see links): non-ranks n such that 6n is a twin-rank.
EXAMPLE
Take n = 63; then 6n = 378 and 36n = 2268; now 379, 2267, and 2269 are prime, but 377 = 13 x 29.
PROG
(Magma) IsInA2822:=func<n|IsPrime(6*n-1)and IsPrime(6*n+1)>;
[n:n in[1..10^3]|not IsInA2822(n)and IsInA2822(6*n)];
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