A129369 -id:A129369 - cp's OEIS frontend

A128372 a(n) = least k such that the remainder when 32^k is divided by k is n.

31, 3, 29, 6, 201, 13, 25, 9, 23, 11, 183, 22, 19, 159, 17, 20, 45, 49, 169, 502, 209, 42, 35, 50, 91919, 27, 3265, 36, 1159, 98, 75197, 33, 95, 66, 2817, 38, 1385, 58, 25187, 82, 32727, 982, 55, 117, 7031, 91, 2517, 52, 46528545441593, 57, 503981, 92, 135, 194

Offset: 1

Alexander Adamchuk, Feb 27 2007

Values a(50), ..., a(149) are relatively small again (starting 57, 503981, 92, 135, 194, 576353, 87, 125, 1902, 6019, 323, 43335727, 69, ...). - Hagen von Eitzen, Jun 04 2009

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found

Cf. A128361, A128362, A128363, A128364, A128365, A128366, A128367, A128368, A129369, A128370, A128371. Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128158, A128159, A128160.

Cf. A128149, A128150, A128172.

t = Table[0, {10000} ]; k = 1; While[ k < 4000000000, a = PowerMod[32, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 06 2009 *)

Incorrect comment removed by Hagen von Eitzen, Jul 19 2009

a(49) found by Hagen von Eitzen, Jul 20 2009

A128371 a(n) = least k such that the remainder when 31^k is divided by k is n.

Original entry on oeis.org

2, 29, 7, 29787, 13, 113413, 51, 23, 11, 3309, 38, 19, 21, 17, 22, 115, 118, 37237, 261, 60212617, 94, 29769, 134, 51205605391, 26, 35, 209, 549, 466, 1558391, 37, 5033228393, 58, 39, 926, 565, 57, 1561, 922, 119, 46, 2512157, 111, 949, 76, 85

Offset: 1

Alexander Adamchuk, Feb 27 2007

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found [Superseded by Cariboni table]
Fausto A. C. Cariboni, Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found, Sep 14 2016 [With 123 new terms, this supersedes the earlier table from Robert G. Wilson v et al.]

Cf. A128361, A128362, A128363, A128364, A128365, A128366, A128367, A128368, A129369, A128370, A128372.
Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128158, A128159, A128160.
Cf. A128149, A128150, A128172.

t = Table[0, {10000} ]; k = 1; While[ k < 4750000000, a = PowerMod[31, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 06 2009 *)

More terms from Ryan Propper, Mar 24 2007

a(494) = 14353729267 = 64609 * 222163. a(498) = 9547024387, a(540) = 29711794103. - Daniel Morel, Jun 17 2010. a(618) = 15150617101, a(750) = 13728669221. - Daniel Morel, Jun 28 2010

A128370 a(n) = least k such that the remainder of 30^k divided by k is n.

Original entry on oeis.org

29, 7, 26997, 13, 8471, 33, 23, 11, 721, 55, 19, 39, 17, 886, 21, 26, 803, 98, 13289, 22, 51, 878, 1141, 146, 35, 38, 111, 218, 515267673651961, 31, 3212679202339, 56, 267, 866, 4367, 42, 10129, 862, 57, 86, 42691, 13479, 949, 214, 95, 77, 7633, 52, 1469, 170, 429, 68, 2791229, 94, 215, 422, 3849, 842, 9773, 140

Offset: 1

Alexander Adamchuk, Feb 27 2007

nonn

Robert G. Wilson v et al., Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found

Cf. A128361, A128362, A128363, A128364, A128365, A128366, A128367, A128368, A129369, A128371, A128372.
Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128158, A128159, A128160.
Cf. A128149, A128150, A128172.

t = Table[0, {10000} ]; k = 1; While[ k < 5100000000, a = PowerMod[30, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 06 2009 *)

Terms a(29) onward from Max Alekseyev, Mar 22 2012

A129368 a(n) = Sum_{k=floor((n+1)/2)..n} binomial(2*k,k).

Original entry on oeis.org

1, 2, 8, 26, 96, 342, 1266, 4678, 17548, 66098, 250854, 956034, 3660190, 14059866, 54176466, 209290554, 810370944, 3143964294, 12219099594, 47564314774, 185410843594, 723668533278, 2827767496998, 11061197519166

Offset: 0

Paul Barry, Apr 11 2007

Partial sums of A129369.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A129369.

[(&+[Binomial(2*(n-k),n-k): k in [0..Floor(n/2)]]): n in [0..60]]; // G. C. Greubel, Jan 31 2024

Table[Sum[Binomial[2k,k],{k,Floor[(n+1)/2],n}],{n,0,30}] (* Harvey P. Dale, Aug 13 2012 *)

[sum(binomial(2*(n-k),n-k) for k in range(1+(n//2))) for n in range(61)] # G. C. Greubel, Jan 31 2024

G.f.: (1/(1-x))*( 1/sqrt(1-4*x) - x/sqrt(1-4*x^2) ).

a(n) = Sum_{k=0..floor(n/2)} C(2*(n-k), n-k).

cp's OEIS Frontend

A128372 a(n) = least k such that the remainder when 32^k is divided by k is n.

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A128371 a(n) = least k such that the remainder when 31^k is divided by k is n.

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Extensions

A128370 a(n) = least k such that the remainder of 30^k divided by k is n.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A129368 a(n) = Sum_{k=floor((n+1)/2)..n} binomial(2*k,k).

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