cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A037202 Number of lines in Pratt certificate for n-th prime.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 2, 4, 6, 6, 6, 4, 4, 8, 8, 6, 8, 6, 8, 8, 4, 8, 6, 6, 4, 4, 6, 8, 4, 6, 8, 8, 4, 10, 6, 6, 8, 4, 8, 10, 8, 6, 8, 4, 6, 8, 10, 8, 8, 8, 8, 8, 6, 4, 2, 10, 10, 6, 10, 8, 12, 6, 6, 10, 8, 10, 10, 8, 12, 10, 6, 10, 10, 10, 8, 10, 6, 8, 4, 6, 10, 10, 12, 4, 8, 8, 6, 8, 10, 12, 10, 10
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

a(k) = 2 for k = 2, 3, 7, 55, 6543, (Fermat Primes, A019434, probably finite),
a(k) = 4 for k = 4, 5, 6, 8, 12, 13, 21, 25, 26, 29, 33, 38, 44, 54, 79, 84, 93, 106, 116, 136, 191, 211, 232, ...,
a(k) = 6 for k = 9, 10, 11, 16, 18, 23, 24, 27, 30, 35, 36, 42, 45, 53, 58, 62, 63, 71, 77, 80, 87, 96, 100, 108, ...,
a(k) = 8 for k = 14, 15, 17, 19, 20, 22, 28, 31, 32, 37, 39, 41, 43, 46, 48, 49, 50, 51, 52, 60, 65, 68, 75, ...,
a(k) = 10 for k = 34, 40, 47, 56, 57, 59, 64, 66, 67, 70, 72, 73, 74, 76, 81, 82, 89, 91, 92, 95, 97, 99, 103, ...,
a(k) = 12 for k = 61, 69, 83, 90, 101, 102, 109, 117, 124, 125, 127, 128, 132, 138, 146, 147, 149, 156, 160, 170, ...,
a(k) = 14 for k = 120, 144, 150, 161, 163, 175, 200, 210, 213, 219, 225, 228, 236, 239, 249, 261, 263, 277, 281, ...,
a(k) = 16 for k = 215, 266, 299, 314, 360, 363, 417, 430, 432, 441, 467, 471, 505, 511, 524, 552, 553, 562, 565, ...,
a(k) = 18 for k = 690, 748, 766, 819, 999, 1027, 1050, 1067, 1105, 1109, 1141, 1154, 1218, 1235, 1259, 1270, ...,
a(k) = 20 for k = 1144, 1393, 1424, 1576, 1719, 1743, 1974, 2133, 2171, 2176, 2205, 2234, 2248, 2259, 2265, 2279, ...,
a(k) = 22 for k = 2584, 3226, 3632, 3659, 3810, 3959, 4127, 4344, 4470, 4588, 4622, 4710, 4747, 4806, 4930, 4936, ...,
a(k) = 24 for k = 5626, 7067, 7324, 7372, 8321, 8670, 8811, 8846, 9237, 9411, 9463, 9605, 9946, 9947, 10518, ...,
a(k) = 26 for k = 13709, 13808, 14659, 16064, 16576, 16596, 18025, 18667, 19223, 19410, 20390, 20731, 20785, ...,
a(k) = 28 for k = 31275, 33607, 39612, 40203, 40648, 42337, 43025, 43312, 44144, 45293, 45335, 45627, 45971, ...,
a(k) = 30 for k = 63461, 63513, 76559, 76858, 81347, 81886, 83430, 86987, 87033, 88871, 94263, 95480, 98307, ...,
a(k) = 32 for k = 145767, 165128, 178829, 186560, 187204, 187472, 204062, 211266, 221035, 230569, 234817, ...,
a(k) = 34 for k = 340332, 356380, 384242, 411259, 458002, 461050, 465782, 467942, 493977, 496416, 514571, ...,
a(k) = 36 for k = 649190, 893950, 982792, 1011067, 1060268, 1071045, 1095110, 1109882, 1142688, 1142952, 1149206, ...,
a(k) = 38 for k = 1703684, 1946813, 2195880, 2198933, 2293897, 2396259, 2480547, 2481840, 2482402, 2493847, ...,
a(k) = 40 for k = 4218462, 4597652, 5001025, 5295255, 5430142, 5438440, 5618213, 5837583, 5860573, 5890121, ...,
etc.
First occurrence of 2k: 2, 4, 9, 14, 34, 61, 120, 215, 690, 1144, 2584, 5626, ..., . - Robert G. Wilson v, Jul 01 2014

References

  • E. Bach and Jeffrey Shallit, Algorithmic Number Theory, I, p. 270.

Links

Crossrefs

Cf. A037231, A034697.

Programs

  • Mathematica
    a[1] = 1; a[n_] := 1 + Plus @@ (a@ PrimePi@ # & /@ First /@ FactorInteger[ Prime@ n - 1]); Array[a, 92]

Formula

a(2)=1, a(n) = 1 + Sum a(p), p prime, p | n-1, where n runs through primes.

A130790 Number of nodes in the Lucas-Pratt primality tree rooted at prime(n).

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 1, 2, 3, 3, 3, 2, 2, 4, 4, 3, 4, 3, 4, 4, 2, 4, 3, 3, 2, 2, 3, 4, 2, 3, 4, 4, 2, 5, 3, 3, 4, 2, 4, 5, 4, 3, 4, 2, 3, 4, 5, 4, 4, 4, 4, 4, 3, 2, 1, 5, 5, 3, 5, 4, 6, 3, 3, 5, 4, 5, 5, 4, 6, 5, 3, 5, 5, 5, 4, 5, 3, 4, 2, 3, 5, 5, 6, 2, 4, 4, 3, 4, 5, 6, 5, 5, 2, 4, 5, 3, 5, 4, 5, 3
Offset: 1

Views

Author

R. J. Mathar, Jul 15 2007, Mar 07 2008

Keywords

Comments

The primality tree starts at an odd prime at the root node. The branches of each node are the odd primes that divide the value of (the node minus 1). The length of the longest branch is related to A126805. Following the branch with the largest label gives A083647.

Links

Crossrefs

Cf. A083647, A126805.
Cf. A037231 (primes which set a new record).

Programs

  • PARI
    LP(p) = my(f=factor(p-1)); if(p <= 2, 0, 1+vecsum(vector(#f~, k, LP(f[k,1]))));
a(n) = LP(prime(n)); \\ Daniel Suteu, Nov 03 2019

A244624 Consider the number of lines in the Pratt certificate for the n-th prime (A037202). This sequence shows where 2n first occurs.

Original entry on oeis.org

1, 2, 4, 9, 14, 34, 61, 120, 215, 690, 1144, 2584, 5626, 13709, 31275, 63461, 145767, 340332, 649190, 1703684, 4218462, 10675070, 22892978
Offset: 0

Views

Author

Joerg Arndt and Robert G. Wilson v, Jul 02 2014

Keywords

Comments

See comment section of A037202.
a(n) ~ 2*a(n-1).

Crossrefs

Cf. A037202, A037231.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = 1 + Plus @@ (a@ PrimePi@# & /@ First /@ FactorInteger[ Prime@ n - 1]); k = 1; t = Table[0, {1000}]; While[k < 1000000000000001, If[a@ k < 1001 && t[[a[k]/2]] == 0, t[[a[k]/2]] = k; Print[{a@k, k}]]; k++]; t

Formula

Also PrimePi( A037231 ).
Showing 1-3 of 3 results.

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