A326235 -id:A326235

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A326231

Numbers n such that N = (5n)^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

See A326233, A326234 for m^3 and A326235, A326236 for m^6.

LINKS

M. F. Hasler, Table of n, a(n) for n = 1..10000

A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

FORMULA

a(n) = A326232(n+1)/5.

PROG

(PARI) select( is(n)=!for(s=1, 2, ispseudoprime(150*n^2+(-1)^s)||return), [1..10^3])

CROSSREFS

Cf. A002822, A326232 ({1} U {5*a(n)}), A326233 (analog for m^3), A326234, A326235 (analog for m^6), A326230 (least twin rank n^k > 1 for given k).

KEYWORD

nonn

AUTHOR

M. F. Hasler and Antonie Dinculescu, Jun 14 2019

STATUS

approved

A326232

Numbers k such that N = k^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

See A326234 and A326233 for k^3, A326236 and A326235 for k^6.

LINKS

A. Dinculescu, Table of n, a(n) for n = 1..10001

A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

PROG

(PARI) select( is(n)=!for(s=1, 2, ispseudoprime(6*n^2+(-1)^s)||return), [1..5000])

CROSSREFS

Cf. A002822, A326231 (a(n)/5, n>1), A326233, A326234 (analog for k^3), A326235, A326236 (analog for k^6), A326230 (least twin rank m^n for given n).

KEYWORD

nonn

AUTHOR

M. F. Hasler and Antonie Dinculescu, Jun 14 2019

STATUS

approved

A326234

Numbers n such that N = n^3 is a twin rank (A002822: 6N +- 1 are twin primes).

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

See A326232 and A326231 for the case n^2, A326236 and A326235 for n^6.

LINKS

A. Dinculescu, Table of n, a(n) for n = 1..14709

A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

FORMULA

a(n) = 7*A326233(n-1), n >= 2.

PROG

(PARI) select( is(n)=!for(s=1, 2, ispseudoprime(6*n^3+(-1)^s)||return), [1..10^5])

CROSSREFS

Cf. A002822, A326233 (a(n)/7, n>1), A326231, A326232 (analog for n^2), A326235, A326236 (analog for n^6), A326230 (least twin rank n^k > 1 for given k).

KEYWORD

nonn

AUTHOR

M. F. Hasler and Antonie Dinculescu, Jun 14 2019

STATUS

approved

A326236

Numbers k such that N = k^6 is a twin rank (cf. A002822: 6N +- 1 are twin primes).

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

See A326232 and A326231 for m^2, A326234 and A326233 for m^3.

LINKS

M. F. Hasler, Table of n, a(n) for n = 1..10001 (3667 terms from A. Dinculescu).

A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

FORMULA

a(n) = 35*A326235(n-1), n >= 2.

PROG

(PARI) select( is(n)=!for(s=1, 2, ispseudoprime(6*n^6+(-1)^s)||return), [1..10^5])

CROSSREFS

Cf. A002822, A326235 (a(n)/35, n>1), A326231, A326232 (analog for n^2), A326233, A326234 (analog for n^3), A326230 (least twin rank n^k for given k).

KEYWORD

nonn

AUTHOR

M. F. Hasler and Antonie Dinculescu, Jun 14 2019

STATUS

approved

A326233

Numbers n such that N = (7n)^3 is a twin rank (A002822: 6N +- 1 are twin primes).

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

See A326231, A326232 for m^2 and A326235, A326236 for m^6.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

FORMULA

a(n) = A326234(n+1)/7.

MAPLE

filter:= proc(n) local m;

m:= (7*n)^3;

isprime(6*m+1) and isprime(6*m-1)

end proc:

select(filter, [$1..3000]); # Robert Israel, Jun 17 2019

PROG

(PARI) select( is(n)=!for(s=1, 2, ispseudoprime(6*(7*n)^3+(-1)^s)||return), [1..10^4])

CROSSREFS

Cf. A002822, A326234 ({1} U 7*{a(n)}), A326231 (analog for n^2), A326232, A326235 (analog for n^6), A326236, A326230 (least twin rank n^k > 1 for given k).

KEYWORD

nonn

AUTHOR

M. F. Hasler and Antonie Dinculescu, Jun 14 2019

STATUS

approved

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