A131462 -id:A131462 - cp's OEIS frontend

A131458 Residues of 3^(2^(p(n)-1)-1) for Mersenne numbers with prime indices.

0, 6, 30, 126, 1565, 8190, 131070, 524286, 7511964, 89777599, 2147483646, 20166585982, 840455563322, 4787976306682, 5519162753736, 2617809209727498, 334169564069012755, 2305843009213693950, 47306781863857413639

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 13 2007, Jul 20 2007

nonn

Mp is prime iff 3^(2^(p(n)-1)-1) is congruent to (-1) Mod Mp. Thus M7 = 127 is prime because 3^63 Mod 127 = 126 (=127-1) while M11 = 2047 is composite because 3^1023 Mod 2047 <> 2046.

			a(5) = 3^(2^(11-1)-1) Mod 2^11-1 = 3^1023 Mod 2047 = 1565

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131459, A131460, A131461, A131462, A131463.

a(n) = 3^(2^(p(n)-1)-1) Mod 2^p(n)-1

A131459 Residues of 3^(2^(p(n)-1)) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 4, 28, 124, 601, 8188, 131068, 524284, 5758678, 269332797, 2147483644, 60499757946, 322343434415, 5567835897839, 16557488261208, 7853427629182494, 426047939903614778, 2305843009213693948, 141920345591572240917

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 13 2007, Jul 20 2007

nonn

Mp is prime iff 3^(2^(p(n)-1)) is congruent to (-3) Mod Mp. Thus M7 = 127 is prime because 3^64 Mod 127 = 124 (=127-3) while M11 = 2047 is composite because 3^1024 Mod 2047 <> 2044.

			a(5) = 3^(2^(11-1)) Mod 2^11-1 = 3^1024 Mod 2047 = 601

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131460, A131461, A131462, A131463.

a(n) = 3^(2^(p(n)-1)) Mod 2^p(n)-1

A131460 Residues of 3^(2^(p(n)-1)+1) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 5, 22, 118, 1803, 8182, 131062, 524278, 498820, 271127480, 2147483638, 44060320367, 967030303245, 7907414671310, 49672464783624, 5545884378065500, 125222315103997360, 2305843009213693942, 130613131595363896897

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 13 2007, Jul 20 2007

nonn

Mp is prime iff 3^(2^(p(n)-1)+1) is congruent to (-9) Mod Mp. Thus M7 = 127 is prime because 3^65 Mod 127 = 118 (=127-9) while M11 = 2047 is composite because 3^1025 Mod 2047 <> 2038.

			a(5) = 3^(2^(11-1)+1) Mod 2^11-1 = 3^1025 Mod 2047 = 1803

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131459, A131461, A131462, A131463.

a(n) = 3^(2^(p(n)-1)+1) Mod 2^p(n)-1

A131461 Residues of 3^(2^p(n)-2) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 1, 1, 1, 1013, 1, 1, 1, 5884965, 65165529, 1, 103888408793, 474639880182, 4112907695371, 72685811469476, 5155089749987738, 440411515280180314, 1, 95591506202441271281, 69291880649932219827

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 20 2007

nonn

M_p is prime iff 3^(M_p-1) is congruent to 1 mod M_p. Thus M_7 = 127 is prime because 3^126 mod 127 = 1 while M_11 = 2047 is composite because 3^2046 mod 2047 <> 1.

			a(5) = 3^(2^11-2) mod 2^11-1 = 3^2046 mod 2047 = 1013

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131459, A131460, A131462, A131463.

a(n) = 3^(2^p(n)-2) mod 2^p(n)-1

A131463 Residues of 3^(2^p(n)) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 2, 9, 9, 929, 9, 9, 9, 2633043, 49618850, 9, 110361958311, 2072735666087, 1831797169511, 91222349803976, 1359811476184687, 504939123701081904, 9, 122453792873589376894, 623626925849389978443

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 20 2007

nonn

M_p is prime iff 3^(M_p+1) is congruent to 9 mod M_p. Thus M_7 = 127 is prime because 3^128 mod 127 = 9 while M_11 = 2047 is composite because 3^2048 mod 2047 <> 9.

			a(5) = 3^(2^11) mod 2^11-1 = 3^2048 mod 2047 = 929

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131459, A131460, A131461, A131462.

a(n) = 3^(2^p(n)) mod 2^p(n)-1

cp's OEIS Frontend

A131458 Residues of 3^(2^(p(n)-1)-1) for Mersenne numbers with prime indices.

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A131459 Residues of 3^(2^(p(n)-1)) for Mersenne numbers with prime indices.

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A131460 Residues of 3^(2^(p(n)-1)+1) for Mersenne numbers with prime indices.

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A131461 Residues of 3^(2^p(n)-2) for Mersenne numbers with prime indices.

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A131463 Residues of 3^(2^p(n)) for Mersenne numbers with prime indices.

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