A117563 -id:A117563 - cp's OEIS frontend

A162175 Primes classified by weight.

11, 17, 29, 41, 59, 67, 71, 79, 83, 89, 101, 103, 107, 109, 137, 149, 167, 179, 191, 193, 197, 227, 229, 239, 241, 251, 269, 277, 281, 283, 311, 331, 347, 349, 359, 367, 379, 383, 409, 419, 431, 433, 439, 443, 449, 461, 463, 467, 487, 491, 499, 503, 521, 557

Offset: 1

Rémi Eismann, Jun 27 2009

nonn

Conjecture: primes classified by level are rarefying among prime numbers.
A000040(n) = 2, 3, 7, A162174(n), a(n). - Rémi Eismann, Jun 27 2009
By definition, primes classified by weight have a prime gap g(n) < sqrt(p(n)) (or more precisely, for primes classified by weight, we have A001223(n) <= sqrt(A118534(n)) - 1 ). So by definition, prime numbers classified by weight follow Legendre's conjecture and Andrica's conjecture - Rémi Eismann, Aug 26 2013

			For prime(5)=11, A117078(5)=3 <= A117563(5)=3 ; prime(5)=11 is classified by weight. For prime(170)=1013, A117078(170)=19 <= A117563(170)=53 ; prime(170)=1013 is classified by weight.

R. Eismann, Table of n, a(n) for n = 1..10000
Remi Eismann, Decomposition of natural numbers into weight * level + jump and application to a new classification of prime numbers
OEIS Wiki, Decomposition into weight * level + jump

Cf. A117078, A117563, A000040, A162174.

If for prime(n), A117078(n) (the weight) <= A117563(n) (the level) and A117078(n) <> 0 then prime(n) is classified by weight. If for prime(n), A117078(n) (the weight) > A117563(n) (the level) then prime(n) is classified by level.

A179620 a(n) = largest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 4, 7, 8, 8, 10, 13, 14, 14, 16, 19, 20, 20, 23, 24, 25, 26, 26, 28, 31, 32, 33, 34, 34, 37, 38, 38, 40, 43, 44, 44, 47, 48, 49, 50, 50, 53, 54, 55, 56, 56, 58, 61, 62, 63, 64, 64, 67, 68, 68, 70, 73, 74, 75, 76, 76, 79, 80, 80, 83, 84, 85, 86, 86, 89

Offset: 1

Rémi Eismann, Jan 09 2011

a(n) = A002808(n) - A073783(n) if A002808(n) - A073783(n) > A073783(n), 0 otherwise.

A002808(n): composite numbers; A073783(n): first difference of composite numbers.

			For n = 1 we have A002808(n) = 4, A002808(n+1) = 6; there is no k such that 6 - 4 = 2 = (4 mod k), hence a(1) = 0.
For n = 3 we have A002808(n) = 8, A002808(n+1) = 9; 7 is the largest k such that 9 - 8 = 1 = (8 mod k), hence a(3) = 7; a(3) = A002808(3) - A073783(3) = 8 - 1 = 7.
For n = 24 we have A002808(n) = 36, A002808(n+1) = 38; 34 is the largest k such that 38 - 36 = 2 = (36 mod k), hence a(24) = 34; a(24) = A002808(24) - A073783(24) = 34.

Rémi Eismann, Table of n, a(n) for n = 1..10000

Cf. A002808, A073783, A130882, A179621, A118534, A117078, A117563, A001223.

A184218 a(n) = largest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377

Offset: 1

Rémi Eismann, Jan 10 2011

From the definition, a(n) = A000217(n) - (n + 1) if A000217(n) - (n + 1) > (n + 1), or 0 otherwise, where A000217 are the triangular numbers.

			For n = 3 we have A000217(3) = 6, A000217(4) = 10; there is no k such that 10 - 6 = 4 = (6 mod k), hence a(3) = 0.
For n = 5 we have A000217(5) = 15, A000217(6) = 21; 9 is the largest k such that 21 - 15 = 6 = (15 mod k), hence a(5) = 9; a(5) = A000217(5) - (5 + 1) = 15 - 6 = 9.
For n = 24 we have A000217(24) = 300, A000217(25) = 325; 275 is the largest k such that 325 - 300 = 25 = (300 mod k), hence a(24) = 275; a(24) = A000217(24) - (24 + 1) = 275.

G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 1..10000 [Originally computed by Remi Eismann]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. essentially the same as A000096, A000217, A000027, A130703, A184219, A118534, A117078, A117563, A001223.

[0,0,0,0] cat [(n+1)*(n-2)/2: n in [5..60]]; // Vincenzo Librandi, Jun 22 2016

Join[{0, 0, 0, 0}, LinearRecurrence[{3, -3, 1}, {9, 14, 20}, 100]] (* G. C. Greubel, Jun 22 2016 *)
lim = 10^4; Table[SelectFirst[Reverse@ Range@ lim, Function[k, PolygonalNumber[n + 1] == # + Mod[#, k] &@ PolygonalNumber@ n]], {n, 53}] /. {k_ /; MissingQ@ k -> 0, k_ /; k == lim -> 0} (* Michael De Vlieger, Jun 30 2016, Version 10.4 *)

a(n) = (n+1)*(n-2)/2 = A000096(n-2) for n >= 5 and a(n) = 0 for n <= 4. - M. F. Hasler, Jan 10 2011
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 7.
G.f.: x^5*(5*x^2 - 13*x + 9)/(1 - x)^3. (End)

A184220 a(n) = largest k such that A000290(n+1) = A000290(n) + (A000290(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 14, 23, 34, 47, 62, 79, 98, 119, 142, 167, 194, 223, 254, 287, 322, 359, 398, 439, 482, 527, 574, 623, 674, 727, 782, 839, 898, 959, 1022, 1087, 1154, 1223, 1294, 1367, 1442, 1519, 1598, 1679, 1762

Offset: 1

Rémi Eismann, Jan 10 2011

From the definition, a(n) = A000290(n) - A005408(n) if A000217(n) - A005408(n) > A005408(n), 0 otherwise, where A000290 are the squares and A005408 are the gaps between squares: 2n + 1.

			For n = 3 we have A000290(3) = 9, A000290(4) = 16; there is no k such that 16 - 9 = 7 = (9 mod k), hence a(3) = 0.
For n = 5 we have A000290(5) = 25, A000290(6) = 36; 14 is the largest k such that 36 - 25 = 11 = (25 mod k), hence a(5) = 14; a(5) = A000290(5) - A005408(5) = 25 - 11 = 14.
For n = 25 we have A000217(25) = 625, A000217(26) = 676; 574 is the largest k such that 676 - 625 = 51 = (625 mod k), hence a(25) = 574; a(25) = A000290(25) - A005408(25) = 574.

Rémi Eismann, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. essentially the same as A008865, A000290, A005408, A133150, A184221, A118534, A117078, A117563, A001223.

a(n) = (n-1)^2-2 = A008865(n-1) for n >= 5 and a(n) = 0 for n <= 4.

A106752 Numbers of prime factors of k, k defined in A117078.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Rémi Eismann, Jun 22 2007, Feb 14 2008

nonn

a(n) = 0 only for n = 1, 2 and 4.

			For a(1), k=0 thus a(1)=0,
For a(3), k=3 thus a(3)=1,
For a(11), k=25=5*5 thus a(11)=2.

Remi Eismann, Table of n, a(n) for n = 1..10000

Cf. A117078, A117563, A118534, A118144, A118123.

a(n) = numbers of factors of A117078(n). A117078(n) : smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

A117877 Least p=prime(k) for which A118123(k)=n.

Original entry on oeis.org

2, 5, 11, 17, 67, 101, 109, 107, 227, 569, 499, 821, 1163, 2153, 1489, 1487, 1579, 4111, 6841, 10739, 5783, 21383, 4729, 3467, 34183, 58741, 19319, 22283, 22279, 22277, 16069, 16067, 17333, 91583, 20479, 20477, 82223, 158363, 31189, 70877, 45061

Offset: 0

Rémi Eismann and Robert G. Wilson v, May 14 2006

nonn

			a(0)=2 because no k exists and it is the least of the three {2, 3 & 7} in A117563 or A117078.
a(1)=5 because 5 + 5 (mod 3) = 7,
a(2)=11 because 11 + 11 (mod 3) = 11 + 11 (mod 9) = 13.
a(3)=17 because 17 + 17 (mod 3) = 17 + 17 (mod 5) = 17 + 17 (mod 15) = 19,
a(4)=67 because 67 + 67 (mod 7) = 67 + 67 (mod 9) = 67 + 67 (mod 21) = 67 + 67 (mod 63) = 71,
a(5)=101 because 101 + 101 (mod 3) = 101 + 101 (mod 9) = 101 + 101 (mod 11) = 101 + 101 (mod 33) = 101 + 101 (mod 99), etc.

Cf. A118123, A117078, A117563.

f[n_] := Block[{p = Prime@n, np = Prime[n + 1]}, Length@ Select[ Divisors[2p - np], # >= np - p &]]; t = Table[0, {50}]; Do[ a = f@n; If[a < 50 && t[[a + 1]] == 0, t[[a + 1]] = n; Print[{a, n, Prime@n}]], {n, 100000}]

A117877(n)={ for( k=n+1, 1e9, n==A118123(k) & return(prime(k)))}

Edited by M. F. Hasler, Nov 07 2009

A118741 Primes for which the weight as defined in A117078 is 7.

Original entry on oeis.org

67, 83, 167, 193, 251, 277, 433, 487, 503, 587, 601, 613, 727, 823, 907, 1063, 1217, 1231, 1553, 1663, 1777, 1861, 1873, 1973, 1987, 2083, 2281, 2293, 2351, 2377, 2393, 2797, 2897, 3217, 3343, 3541, 3847, 4073, 4283, 4451, 4507, 4591, 4813, 4871, 5081

Offset: 1

Rémi Eismann, May 22 2006; May 27 2006; May 04 2007

nonn

The gap as defined in A001223 of this prime numbers is 4 or 6.

The prime numbers in this sequence are of the form (14i-1) (if gap=6) or (14i-3) (if gap=4) with i=(level(n)+1)/2, level(n) defined in A117563.

Remi Eismann, Table of n, a(n) for n=1..10000

Cf. A117078, A117563, A118922, A118924.

f[n_] := Block[{a, p = Prime[n], np = Prime[n + 1]}, a = Min[Select[Divisors[2*p - np], #1 > np - p & ]]; If[a == Infinity, 0, a]]; Prime@ Select[ Range@695, f@# == 7 &] (* Robert G. Wilson v, May 26 2006 *)

A117078 : a(n) = smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists. prime(n) for which k=7.

More terms from Robert G. Wilson v, May 26 2006

A119593 Primes for which the weight as defined in A117078 is 7 and the gap as defined in A001223 is 4.

Original entry on oeis.org

67, 193, 277, 487, 613, 823, 907, 1663, 1873, 2083, 2293, 2377, 2797, 3217, 3343, 3847, 4813, 5233, 5527, 5653, 5737, 6577, 6997, 7207, 7753, 8677, 8803, 9433, 11113, 11617

Offset: 1

Rémi Eismann, Jun 01 2006, May 04 2007

The prime numbers in this sequence are of the form (14i-3) with i=(level(n)+1)/2, level(n) defined in A117563.

Remi Eismann, Table of n, a(n) for n=1..10000

Cf. A117078, A001223, A117563, A001359, A118359, A074822, A118922, A118924, A119504.

forprimestep(p=67,1e4,14, t=p%5; if((t==2 || t==3) && isprime(p+4), print1(p", "))) \\ Charles R Greathouse IV, Sep 17 2022

p=67; forprime(q=p+2,1e4, if(q-p==4 && (p%70==53 || p%70==67), print1(p", ")); p=q) \\ Charles R Greathouse IV, Sep 17 2022

Primes p such that (1) p = 53 or 67 mod 70 and (2) p+4 is prime. - Charles R Greathouse IV, Sep 17 2022

a(n) = Omega(n log^2 n). - Charles R Greathouse IV, Sep 17 2022

A119594 Primes for which the weight as defined in A117078 is 9 and the gap as defined in A001223 is 4.

Original entry on oeis.org

13, 103, 463, 643, 877, 967, 1093, 1597, 1867, 1993, 2137, 2857, 3037, 3163, 3253, 3613, 3793, 4153, 4513, 4783, 5413, 5503, 5647, 6007, 6043, 6547, 6907, 7537, 7573, 7933, 8167, 8293, 9157, 9337, 9463, 9787

Offset: 1

Rémi Eismann, Jun 01 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (18i-5) with i=(level(n)+1)/2, level(n) defined in A117563.

			a(1)=13 because of 17=13+mod(13;9) and 17-13=4.
18*1-5=13, level=1
a(2)=103 because of 107=103+mod(103;9) and 107-103=4
18*((11+1)/2)-5=103, level=11

Remi Eismann, Table of n, a(n) for n=1..10000

Cf. A117078, A001223, A117563, A001359, A118359, A074822, A118922, A118924, A119504.

A133346 a(n) = smallest k such that prime(n+2) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 7, 11, 0, 15, 21, 21, 31, 7, 11, 35, 9, 17, 17, 61, 9, 21, 23, 23, 77, 7, 19, 97, 101, 91, 19, 13, 41, 25, 127, 47, 139, 21, 17, 31, 11, 167, 13, 37, 11, 61, 25, 39, 7, 13, 73, 9, 227, 25, 239, 35, 15, 9, 29, 271, 269, 37, 25, 7, 61, 59, 27, 21, 13, 11, 113, 113

Offset: 1

Rémi Eismann, Oct 20 2007

nonn

			For n = 1 we have prime(n) = 2, prime(n+2) = 5; there is no k such that 5 - 2 = 3 = (2 mod k), hence a(1) = 0.
For n = 6 we have prime(n) = 13, prime(n+2) = 19; 7 is the smallest k such that 19 - 13 = 6 = (13 mod k), hence a(6) = 7.
For n = 30 we have prime(n) = 113, prime(n+2) = 131; 19 is the smallest k such that 131 - 113 = 18 = (113 mod k), hence a(30) = 19.

Remi Eismann, Table of n, a(n) for n = 1..10000

Cf. A000040, A117078, A117563, A001223, A118534, A020639, A090369, A090368, A130533, A130650, A130703, A130889, A130882.

cp's OEIS Frontend

A162175 Primes classified by weight.

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A179620 a(n) = largest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists.

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A184218 a(n) = largest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

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A184220 a(n) = largest k such that A000290(n+1) = A000290(n) + (A000290(n) mod k), or 0 if no such k exists.

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A106752 Numbers of prime factors of k, k defined in A117078.

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A117877 Least p=prime(k) for which A118123(k)=n.

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A118741 Primes for which the weight as defined in A117078 is 7.

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A119593 Primes for which the weight as defined in A117078 is 7 and the gap as defined in A001223 is 4.

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