A117078 -id:A117078 - cp's OEIS frontend

A117563 a(n) = A118534(n)/A117078(n) unless A117078(n) = 0 in which case a(n) = 0.

0, 0, 1, 0, 3, 1, 5, 3, 1, 9, 1, 3, 13, 3, 1, 1, 19, 5, 9, 23, 1, 15, 11, 9, 3, 33, 11, 35, 21, 3, 3, 5, 45, 3, 49, 5, 1, 3, 23, 1, 59, 9, 63, 27, 65, 11, 1, 3, 75, 45, 1, 79, 21, 35, 1, 1, 89, 5, 39, 93, 21, 9, 3, 103, 3, 3, 25, 3, 115, 69, 1, 39, 19, 1, 75, 29, 3, 3, 3, 21, 139, 3, 143, 61, 87

Offset: 1

Rémi Eismann, Apr 29 2006, Feb 14 2008

nonn

a(n) is the "level" of prime(n).
There is a unique decomposition of the primes: provided the level a(n) is > 0, we have prime(n) = weight * level + gap, or A000040(n)=A117078(n)*a(n)+A001223(n).
a(n) = 0 only for primes 2, 3 and 7.
A118534(n) = prime(n) - g(n) or A000040(n) - A001223(n) if prime(n) - g(n) > g(n), 0 otherwise.

			a(7)=15/3=5; a(14)=39/13=3; a(16)=47/47=1; a(18)=55/11=5; a(29)=105/5=11.

Remi Eismann, Table of n, a(n) for n = 1..10000
Remi Eismann, Java program to decompose a prime as weight*level + gap, or A117078(n)*A117563(n) + A001223(n).
Rémi Eismann, Decomposition into weight * level + jump and application to a new classification of primes, arXiv:0711.0865 [math.NT], 2007-2010.

Cf. A117078, A118534.

a34[n_] := If[n == 1 || n == 2 || n == 4, 0, 2 Prime[n] - Prime[n+1]];
a78[n_] := Block[{a, p = Prime[n], np = Prime[n+1]}, a = Min[Select[ Divisors[2p - np], # > np - p& ]]; If[a == Infinity, 0, a]];
a[n_] := If[a78[n] == 0, 0, a34[n]/a78[n]];
Array[a, 85] (* Jean-François Alcover, Nov 02 2018, after Robert G. Wilson v in A118534  *)

More terms from Robert G. Wilson v, May 05 2006

Edited by N. J. A. Sloane, May 14 2006

A118922 Primes for which the weight as defined in A117078 is 9 and the gap as defined in A001223 is 8.

Original entry on oeis.org

89, 359, 449, 683, 701, 719, 1439, 1979, 2213, 2609, 2663, 2699, 2843, 2879, 3041, 3221, 3491, 4751, 5399, 5813, 6029, 6389, 6983, 7019, 7919, 8171, 8369, 8513, 9539, 10151, 10169, 10259, 10313, 10781, 10979, 11321, 11519, 11681, 12149, 12203

Offset: 1

Rémi Eismann, May 25 2006, May 04 2007

nonn

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

			a(1) = 89 because of prime(25) = prime(24) + (prime(24)mod 9) = 97
g(n) = 8

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

Cf. A117078, A001223, A001359, A117563, A001359, A074822, A118359, A118924, A119504, A119597, A119596, A119595.

A118924 Primes for which the weight as defined in A117078 is 53 and the gap as defined in A001223 is 52.

Original entry on oeis.org

19609, 547171, 3099757, 3282289, 3401221, 4286851, 4648099, 5544859, 5622769, 5731207, 5868901, 6387559, 6581857, 6949147, 6985081, 7382899, 7412791, 7675141, 7697401, 8203021, 8366791, 9190411, 9649921, 9990499, 9994951

Offset: 1

Rémi Eismann, May 25 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (106i-1) with i=(level(n)+1)/2, level(n) defined in A117563.

			Prime(2226) = prime(2225) + (prime(2225) mod 53) = 19609 + (19609 mod 53) = 19661
g(n) = 19661 - 19609 = 53 - 1 = 52

Cf. A117078, A001223, A001359, A117563, A074822, A118922, A118359, A118924, A119504, A119597, A119596, A119595, A118380.

A118359 Primes for which the weight as defined in A117078 is 7 and the gap as defined in A001223 is 6.

Original entry on oeis.org

83, 167, 251, 433, 503, 587, 601, 727, 1063, 1217, 1231, 1553, 1777, 1861, 1973, 1987, 2281, 2351, 2393, 2897, 3541, 4073, 4283, 4451, 4507, 4591, 4871, 5081, 5431, 5557, 5641, 5683

Offset: 1

Rémi Eismann, May 24 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (14i-1) with i=(level(n)+1)/2, level(n) defined in A117563. level(n) is not multiple of 3.

			prime(24) = prime (23) + prime(23)mod(7) = prime (23) + prime(23)mod(77)
89 = 83 + 83mod(7) = 83 + 83mod(77)
k=7, level = 77/7 = 11

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

Cf. A117078, A117563, A001359, A074822, A118922, A118924, A119504, A119597, A119596, A119595.

A119504 Primes for which the weight as defined in A117078 is 23.

Original entry on oeis.org

631, 773, 2467, 2833, 3121, 3203, 3347, 3617, 4219, 4733, 4909, 4951, 5273, 6619, 7027, 7129, 7529, 8263, 8783, 9049, 9413, 9643, 9649, 10891, 11483, 11719, 12541, 13093, 13183, 13841, 14243, 14293, 14851, 15121, 15629, 15667, 15671, 15761

Offset: 1

Rémi Eismann, May 27 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (56i-23+gap) with i=(level(n)+1)/2, level(n) defined in A117563.

			a(1) = prime(115) = 631 because prime(116) = prime(115) + (prime(115) mod 53) = 641
g(n) = 641 - 631 = 10
Prime(115) + 23 - 10 = 644, 644/46 = 14

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

Cf. A117078, A118359, A001359, A118380, A118924, A118922.

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=23.

A119595 Primes for which the weight as defined in A117078 is 15 and the gap as defined in A001223 is 8.

Original entry on oeis.org

743, 1193, 1523, 1733, 2003, 2243, 2273, 3623, 4583, 4943, 5573, 5693, 6143, 6203, 6473, 7673, 8573, 8933, 9803, 10103, 11243, 11813, 12413, 12503, 13163, 14423, 14843, 15053, 15233, 15383, 16103

Offset: 1

Rémi Eismann, Jun 01 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (30i-7) with i=(level(n)+1)/2, level(n) defined in A117563.

			a(1)=743 because of 751=743+mod(743;15) and g(n)=751-743=8
30*((49+1)/2)-7=743
a(2)=1193 because of 1201=1193+mod(1193;15) and g(n)=1201-1193=8
30*((79+1)/2)-7=1193

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

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

A119596 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 10.

Original entry on oeis.org

241, 1627, 2089, 4201, 4663, 4861, 5323, 6247, 6379, 6709, 8821, 9283, 9679, 10141, 12253, 12517, 12781, 13441, 15091, 15289, 15619, 17599, 17929, 19249, 19447, 19843, 21757, 23539, 26839, 28687, 33703, 34429, 34693, 35089, 35353, 36343

Offset: 1

Rémi Eismann, Jun 01 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (22i-1) with i=(level(n)+1)/2, level(n) defined in A117563.

			a(1)=241 because of 251=241+mod(241;11) and 251-241=10.
22*((21+1)/2)-1=241, level=21
a(2)=1627 because of 1637=1627+mod(1627;11) and 1637-1627=10
22*((147+1)/2)-1=1627, level=147

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

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

A119597 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 6.

Original entry on oeis.org

61, 677, 941, 1117, 1601, 2063, 2371, 3691, 3911, 4021, 5297, 5407, 6067, 6353, 6991, 7541, 7717, 8311, 8641, 8663, 9103, 9851, 10973, 11897, 12491, 12953, 13591, 13613, 13723, 14537, 15131, 15263, 15307, 15461, 15901, 16363

Offset: 1

Rémi Eismann, Jun 01 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (22i-5) with i=(level(n)+1)/2, level(n) defined in A117563.

			a(1)=61 because of 67=61+mod(61;11) and 67-61=6.
22*((5+1)/2)-5=61, level=5
a(2)=677 because of 683=677+mod(677;11) and 683-677=6
22*((61+1)/2)-5=677, level=5

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

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

A118380 Primes for which the weight as defined in A117078 is 15 and the gap as defined in A001223 is 14.

Original entry on oeis.org

839, 1409, 2039, 2819, 2939, 3779, 4139, 4889, 5309, 5669, 5939, 6719, 8039, 8609, 10739, 11369, 11909, 12329, 13049, 13499, 13859, 14159, 14489, 14519, 14639, 14669

Offset: 1

Rémi Eismann, May 24 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (30i-1) with i=(level(n)+1)/2, level(n) defined in A117563.

			Prime(147) = prime(146) + prime(146)mod(15) = 839 + 839 mod(15) = 853 839 mod (5) = 4.

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

Cf. A117078, A074822.

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.

cp's OEIS Frontend

A117563 a(n) = A118534(n)/A117078(n) unless A117078(n) = 0 in which case a(n) = 0.

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A118922 Primes for which the weight as defined in A117078 is 9 and the gap as defined in A001223 is 8.

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A118924 Primes for which the weight as defined in A117078 is 53 and the gap as defined in A001223 is 52.

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A118359 Primes for which the weight as defined in A117078 is 7 and the gap as defined in A001223 is 6.

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A119504 Primes for which the weight as defined in A117078 is 23.

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A119595 Primes for which the weight as defined in A117078 is 15 and the gap as defined in A001223 is 8.

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A119596 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 10.

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A119597 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 6.

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A118380 Primes for which the weight as defined in A117078 is 15 and the gap as defined in A001223 is 14.

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

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