A117563 -id:A117563 - cp's OEIS frontend

A117078 a(n) is the smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

0, 0, 3, 0, 3, 9, 3, 5, 17, 3, 25, 11, 3, 13, 41, 47, 3, 11, 7, 3, 67, 5, 7, 9, 31, 3, 9, 3, 5, 33, 41, 25, 3, 43, 3, 29, 151, 53, 7, 167, 3, 19, 3, 7, 3, 17, 199, 73, 3, 5, 227, 3, 11, 7, 251, 257, 3, 53, 7, 3, 13, 31, 101, 3, 103, 101, 13, 109, 3, 5, 347, 9, 19, 367, 5, 13, 127, 131, 131, 19, 3

Offset: 1

Rémi Eismann, Apr 18 2006, Dec 10 2006, Feb 14 2008

nonn

There is a unique decomposition of the primes: provided the weight a(n) is > 0, we have prime(n) = weight * level + gap, or A000040(n)=a(n)*A117563(n)+A001223(n).
a(n) is the smallest divisor of A118534(n) greater than A001223(n) (gap).
a(n) == 0 (mod 2) only for n = {1, 2 or 4}. - Robert G. Wilson v, May 05 2006
a(n) = 0 only for primes 2, 3 and 7. Conjecture: 2, 3 and 7 are the only primes for which log(A000040(n)) < sqrt(A001223(n)).
a(n) > 0 if and only if 2*prime(n+1) < 3*prime(n). - Thomas Ordowski, Nov 25 2013

			For n = 1 we have prime(n) = 2, prime(n+1) = 3; there is no k such that 3 - 2 = 1 = (2 mod k), hence a(1) = 0.
For n = 3 we have prime(n) = 5, prime(n+1) = 7; 3 is the smallest k such that 7 - 5 = 2 = (5 mod k), hence a(3) = 3.
For n = 19 we have prime(n) = 67, prime(n+1) = 71; 7 is the smallest k such that 71 - 67 = 4 = (67 mod k), hence a(19) = 7.

Remi Eismann, Table of n, a(n) for n = 1..10000
Rémi Eismann, Decomposition into weight * level + jump and application to a new classification of primes, arXiv:0711.0865 [math.NT], 2007-2010.
Fabien Sibenaler, Program in assembly that gives the decomposition of a prime number [prime = weight * level + gap, or A000040(n) = A117078(n) * A117563(n) + A001223(n)]

Cf. A074822 (k=5), A118534, A117563.

f[n_] := Block[{a, p = Prime@n, np = Prime[n + 1]}, a = Min@ Select[ Divisors[2p - np], # > np - p &]; If[a == Infinity, 0, a]]; Array[f, 80] (* Robert G. Wilson v, May 08 2006 *)

{m=78; for(n=1,m,p=prime(n);d=prime(n+1)-p; k=0; j=1; while(k==0&&j

Edited and corrected by Don Reble and Klaus Brockhaus, Apr 21 2006

A118534 a(n) is the largest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 3, 0, 9, 9, 15, 15, 17, 27, 25, 33, 39, 39, 41, 47, 57, 55, 63, 69, 67, 75, 77, 81, 93, 99, 99, 105, 105, 99, 123, 125, 135, 129, 147, 145, 151, 159, 161, 167, 177, 171, 189, 189, 195, 187, 199, 219, 225, 225, 227, 237, 231, 245, 251, 257, 267, 265, 273, 279

Offset: 1

Rémi Eismann, Apr 18 2006, Feb 14 2008

a(n) = prime(n) - g(n) or A000040(n) - A001223(n) if prime(n) - g(n) > g(n), 0 otherwise.
a(n) = 0 only for primes 2, 3 and 7.
Under the twin prime conjecture prime(n+1)-prime(n) = 2 infinitely often, and from that we can conclude that k=prime(n)-2 infinitely often. [Roderick MacPhee, Jul 24 2012]
a(n) = A062234(n) for 5 <= n <= 1000. - Georg Fischer, Oct 28 2018

			n=5: prime(5) = 11, prime(6) = 13, 13 = 11 + (11 mod 3) = 11 + (11 mod 9), so A117078(5) = 3, a(5) = 9 and A117563(5) = 9/3 = 3. Thus 11 has level 3 and so is a member of A117873.

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

Cf. A062234, A117078; essentially the same as A117563.

a[n_] := If[n == 1 || n == 2 || n == 4, 0, 2Prime[n] - Prime[n + 1]]; Array[a, 62] (* Robert G. Wilson v, May 09 2006 *)

Edited by N. J. A. Sloane, May 07 2006

More terms from Robert G. Wilson v, May 09 2006

A117876 Primes p=prime(k) of level (1,2), i.e., such that A118534(k) = prime(k-2).

Original entry on oeis.org

23, 47, 73, 233, 353, 647, 1097, 1283, 1433, 1453, 1493, 1613, 1709, 1889, 2099, 2161, 2383, 2621, 2693, 2713, 3049, 3533, 3559, 3923, 4007, 4133, 4643, 4793, 4937, 5443, 5743, 6101, 7213, 7309, 7351, 7561, 7621, 7829, 8179, 8237, 8719, 8849, 9109, 9343, 9467

Offset: 1

Rémi Eismann, May 02 2006

nonn

If prime(k) has level 1 in A117563, and if 2*prime(k) - prime(k+1) = prime(k-i), then we say that prime(k) has level (1,i). Sequence gives primes of level (1,2).

The prime p(4)=7 cannot be decomposed into weight*level+gap (<=> A117563(4)=0 <=> A118534(4)=0 <=> A117078(4)=0). For all other primes, an equivalent definition would be: Primes p(k) such that 2*p(k) - p(k+1) = p(k-2). - Rémi Eismann and M. F. Hasler, Nov 08 2009

			29 = 2*23 - 17, 2179 = 2*2161 - 2143, 5749 = 2*5743 - 5737.

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

Cf. A000040, A066495, A117078, A117563, A118534.

With[{m = 2}, Prime@ Select[Range[m + 1, 1200], If[MemberQ[{1, 2, 4}, #], 0, 2 Prime[#] - Prime[# + 1]] == Prime[# - m] &]] (* Michael De Vlieger, Jul 16 2017 *)

for(n=5,9999, 2*prime(n)-prime(n+1) == prime(n-2) & print1(prime(n),",")) \\ M. F. Hasler, Nov 08 2009

is_A117876(p)={ isprime(p) & isprime(d=2*p-nextprime(p+2)) & d == precprime(precprime(p-2)-2) & p>7 } \\ M. F. Hasler, Nov 08 2009

(define (A117876 n) (A000040 (A066495 (+ 1 n)))) ;; Antti Karttunen, Nov 30 2013

a(n) = A000040(A066495(n+1)). - Antti Karttunen, Nov 30 2013

Edited by N. J. A. Sloane, May 14 2006
More terms from Rémi Eismann, May 25 2006
Definition corrected and terms double-checked by M. F. Hasler, Nov 08 2009

A074822 Primes p such that p + 4 is prime and p == 9 (mod 10).

Original entry on oeis.org

19, 79, 109, 229, 349, 379, 439, 499, 739, 769, 859, 1009, 1279, 1429, 1489, 1549, 1579, 1609, 1999, 2239, 2269, 2389, 2539, 2659, 2689, 2749, 3019, 3079, 3319, 3529, 3919, 4129, 4519, 4639, 4729, 4789, 4969, 4999, 5479, 5569, 5689, 5779, 5839, 6199

Offset: 1

Roger L. Bagula, Sep 30 2002

From Rémi Eismann, May 14 2006; May 04 2007: (Start)
Also primes for which k is equal to 5 in A117078. Examples: prime(9) = prime(8) + (prime(8) mod 5) = 19 + (19 mod 5)=23; prime(23) = prime(22) + (prime(22) mod 5) = 79 + (79 mod 5)=83; prime(1359) = prime(1358) + (prime(1358) mod 5) = 11239+ (11239 mod 5)=11243.
The prime numbers in this sequence are of the form (10i-1) with i=(level(n)+1)/2, level(n) defined in A117563.
Consider A117078: a(n) = smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists. Sequence gives values of prime(n) for which k=5. (End)
p is the lesser member of cousin primes (p,p+4) such that p == 9 (mod 10). - Muniru A Asiru, Jul 03 2017

Remi Eismann, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cousin Primes

Cf. A001223, A117078, A117563.

Intersection of A023200 and A030433.

Prime[ Select[ Range[1000], Prime[ # ] + 4 == Prime[ # + 1] && Mod[ Prime[ # ], 10] == 9 & ]]
Transpose[Select[Partition[Prime[Range[820]],2,1],Last[#]-First[#] == 4 && Mod[ First[ #],10]==9&]][[1]] (* Harvey P. Dale, Oct 20 2011 *)

is(n)=n%30==19 && isprime(n+4) && isprime(n) \\ Charles R Greathouse IV, Jul 12 2017

list(lim)=my(v=List(),p=19); forprime(q=23,lim+4, if(q-p==4 && p%30==19, listput(v,p)); p=q); Vec(v) \\ Charles R Greathouse IV, Jul 12 2017

Edited by Robert G. Wilson v and N. J. A. Sloane, Oct 03 2002

Entry revised by N. J. A. Sloane, Feb 24 2007

A162174 Primes classified by level.

Original entry on oeis.org

5, 13, 19, 23, 31, 37, 43, 47, 53, 61, 73, 97, 113, 127, 131, 139, 151, 157, 163, 173, 181, 199, 211, 223, 233, 257, 263, 271, 293, 307, 313, 317, 337, 353, 373, 389, 397, 401, 421, 457, 479, 509, 523, 541, 547, 563, 571, 593, 607, 619, 647, 653, 661, 673, 691

Offset: 1

Rémi Eismann, Jun 27 2009

nonn

Conjecture : primes classified by level are rarefying among prime numbers.

A000040(n) = 2, 3, 7, A162175(n), a(n) [From Rémi Eismann, Jun 27 2009]

			For prime(3)=5, A117078(3)=3 > A117563(3)=1 ; prime(3)=5 is classified by level. For prime(172)=1021, A117078(172)=337 > A117563(172)=3 ; prime(172)=1021 is classified by level.

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

Cf. A117078, A117563, A000040.

Cf. A162175. [From Rémi Eismann, Jun 27 2009]

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

If for prime(n), A117078(n) (the weight) <= A117563(n) (the level) and A117078(n) <> 0 then prime(n) is classified by weight. [From Rémi Eismann, Jun 27 2009]

A118464 Primes p=prime(i) of level (1,5), i.e., such that A118534(i) = prime(i-5).

Original entry on oeis.org

13933, 23633, 28229, 49223, 71363, 79633, 81239, 90547, 96857, 97613, 108827, 115363, 117443, 126781, 130657, 133733, 153533, 157679, 176819, 186799, 197389, 206651, 221327, 222199, 228139, 246947, 266297, 272203, 276049, 279221, 282493, 290627, 292493, 296299

Offset: 1

Rémi Eismann, May 04 2006

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,5): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

			prime(5061) = 49223 has level (1,5): prime(5062) = 49253 = 2*prime(5061) - prime(5061-5) = 2*prime(5061) - prime(5056).

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

Cf. A117078, A117563, A118534, A125830, A162174.

With[{m = 5}, Prime@ Select[Range[m + 1, 3*10^4], If[MemberQ[{1, 2, 4}, #], 0, 2 Prime[#] - Prime[# + 1]] == Prime[# - m] &]] (* Michael De Vlieger, Jul 16 2017 *)

lista(nn) = my(c=6, v=primes(6)); forprime(p=17, nn, if(2*v[c]-p==v[c=c%6+1], print1(precprime(p-1), ", ")); v[c]=p); \\ Jinyuan Wang, Jun 18 2021

Edited by N. J. A. Sloane, May 14 2006
More terms from Rémi Eismann, May 21 2006
Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009

A118467 Primes p = prime(i) of level (1,3), i.e., such that A118534(i) = prime(i-3).

Original entry on oeis.org

619, 1069, 1459, 1499, 1759, 1789, 2861, 3331, 3931, 4177, 4801, 4831, 5419, 6229, 6397, 8431, 8893, 9067, 9631, 11003, 11131, 11789, 12619, 14251, 15331, 15889, 16661, 17683, 17939, 18269, 18553, 19219, 19391, 19507, 20029, 20759, 22039, 22159, 22171, 22549

Offset: 1

Rémi Eismann, May 24 2006

nonn

If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

			prime(115) - prime(114) = 631 - 619 = 619 - 607 = prime(114) - prime(114-3).

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

Subsequence of A125830 and A162174.

Cf. A006562 (primes of level (1,1)), A117078, A117563, A117876, A118464.

Select[Partition[Prime[Range[2600]],5,1],#[[5]]-#[[4]]==#[[4]]-#[[1]]&][[All,4]] (* Harvey P. Dale, Aug 28 2021 *)

Definition and comment reworded, following author's suggestions, by M. F. Hasler, Nov 30 2009

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.

A119402 Primes p=prime(i) of level (1,11), i.e., such that A118534(i)=prime(i-11).

Original entry on oeis.org

576791, 3361517, 9433859, 10460719, 11630503, 11707537, 12080027, 19743677, 28716287, 33384517, 34961923, 36627659, 37776967, 38087983, 40794049, 45650359, 49152757, 52230229, 53152907, 53240927, 55036789, 56167103, 56177783, 57717749, 58804483, 71849423, 76119269

Offset: 1

Rémi Eismann and Fabien Sibenaler, Jul 25 2006

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,11): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

			prime(240963) - prime(240962) = 3361601 - 3361517 = 3361517 - 3361433 = prime(240962) - prime(240962-11) and prime(240962) has level 1 in A117563, so prime(240962)=3361517 has level (1,11).

Fabien Sibenaler, Table of n, a(n) for n = 1..4000

Cf. A006562 (primes of level (1,1)), A117078, A117563, A006562, A117876, A118464, A118467.

More terms from Fabien Sibenaler, Oct 20 2006

Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009

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.

cp's OEIS Frontend

A117078 a(n) is the smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

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A118534 a(n) is the largest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

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A117876 Primes p=prime(k) of level (1,2), i.e., such that A118534(k) = prime(k-2).

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A074822 Primes p such that p + 4 is prime and p == 9 (mod 10).

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A162174 Primes classified by level.

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A118464 Primes p=prime(i) of level (1,5), i.e., such that A118534(i) = prime(i-5).

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A118467 Primes p = prime(i) of level (1,3), i.e., such that A118534(i) = prime(i-3).

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