A118534 -id:A118534 - 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

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

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

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

A119403 Primes p=prime(i) of level (1,10), i.e., such that A118534(i)=prime(i-10).

Original entry on oeis.org

745757, 1103639, 1583369, 1895359, 2124049, 3327419, 4234537, 4437779, 5071973, 6287647, 7702573, 8470927, 8675923, 9493151, 9750079, 10868203, 11213843, 14244173, 14796253, 14978893, 15611909, 16489273, 17528681, 18280771, 19125163, 19403831, 19631411, 21975167

Offset: 1

Rémi Eismann and Fabien Sibenaler, Jul 25 2006

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,10): 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(353166) - prime(353165) = 5072057 - 5071973 = 5071973 - 5071889 = prime(353165) - prime(353165-10) and prime(353165) has level 1 in A117563, so prime(353165)=5071973 has level (1,10).

Fabien Sibenaler, Table of n, a(n) for n = 1..10000

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

lista(nn) = my(c=11, v=primes(11)); forprime(p=37, nn, if(2*v[c]-p==v[c=c%11+1], print1(precprime(p-1), ", ")); v[c]=p); \\ Jinyuan Wang, Jun 18 2021

More terms from Fabien Sibenaler, Oct 20 2006

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

A119404 Primes p=prime(i) of level (1,9), i.e., such that A118534(i)=prime(i-9).

Original entry on oeis.org

678659, 855739, 1403981, 2366543, 2744783, 2830657, 3027539, 3317033, 4525909, 4676851, 5341463, 5819563, 7087123, 7181897, 8815663, 9324257, 9878929, 9976937, 10403251, 10440641, 10447457, 10766411, 10787377, 11829151, 11881957, 12539389, 14026433, 14087179

Offset: 1

Rémi Eismann and Fabien Sibenaler, Jul 25 2006

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,9): 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(780815) - prime(780814) = 11882071 - 11881957 = 11881957 - 11881843 = prime(780814) - prime(780814-9) and prime(780814) has level 1 in A117563, so prime(780814)=11881957 has level (1,9).

Fabien Sibenaler, Table of n, a(n) for n = 1..10000

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

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

A125576 Primes p=prime(i) of level (1,15), i.e., such that A118534(i)=prime(i-15).

Original entry on oeis.org

264426203, 295902073, 361949821, 704544167, 1075639757, 1259347393, 1290546427, 1301756207, 1335396547, 1370742383, 1460811643, 1497078991, 1514647247, 1643839649, 1783137281, 2142070103, 2424093281, 2471124197, 2494743721, 2577014057, 2706824389, 2951139253

Offset: 1

Rémi Eismann and Fabien Sibenaler, Jan 27 2007

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,15): 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(16042282) - prime(16042281) = 295902247 - 295902073 = 295902073 - 295901899 = prime(16042281) - prime(16042281-15) and prime(16042281) has level 1 in A117563, so prime(16042281)=295902073 has level (1,15).

Fabien Sibenaler, Table of n, a(n) for n = 1..100

Cf. A117078, A117563, A006562 (primes of level (1,1)), A117876, A118464, A118467, A119402, A119403, A119404.

lista(nn) = my(c=16, v=primes(16)); forprime(p=59, nn, if(2*v[c]-p==v[c=c%16+1], print1(precprime(p-1), ", ")); v[c]=p); \\ Jinyuan Wang, Jun 18 2021

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

Terms a(5) and beyond from b-file by Andrew Howroyd, Feb 05 2018

A125623 Primes p=prime(i) of level (1,16), i.e., such that A118534(i)=prime(i-16).

Original entry on oeis.org

356604959, 613768081, 709208323, 950803363, 979872743, 1174872271, 1186433617, 1625945609, 1796767963, 1840621901, 2348698453, 2547482281, 3385901059, 3446679371, 3512406283, 3735873397, 4080198391, 4106437259, 4319987921, 4695419887, 5285414713, 5288810297

Offset: 1

Rémi Eismann and Fabien Sibenaler, Jan 27 2007

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,16): 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(48470200) - prime(48470199) = 950803519 - 950803363 = 950803363 - 950803207 = prime(48470199) - prime(48470199-16) and prime(48470199) has level 1 in A117563, so prime(48470199) = 950803363 has level (1,16).

Fabien Sibenaler, Table of n, a(n) for n = 1..60

Cf. A117078, A117563, A006562 (primes of level (1,1)), A117876, A118464, A118467, A119402, A119403, A119404.

lista(nn) = my(v=primes(17)); forprime(p=61, nn, if(2*v[17]-p==v[1], print1(v[17], ", ")); v=concat(v[2..17], p)); \\ Jinyuan Wang, Jun 18 2021

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

A125565 Primes p=prime(i) of level (1,12), i.e., such that A118534(i)=prime(i-12).

Original entry on oeis.org

15014557, 27001043, 29602093, 50234633, 87028433, 91814759, 94529221, 103336843, 112840309, 113774329, 113961299, 114887657, 115528969, 118974901, 129235273, 144352123, 146127721, 160370491, 163559197, 169274999, 188168059, 188895919, 191829409, 198823447

Offset: 1

Rémi Eismann, Jan 27 2007

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,12): 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(5316164) - prime(5316163) = 91814831 - 91814759 = 91814759 - 91814687 = prime(5316163) - prime(5316163-12) and prime(5316163) has level 1 in A117563, so prime(5316163)=91814759 has level (1,12).

Fabien Sibenaler, Table of n, a(n) for n = 1..1800

Cf. A006562 (primes of level (1,1)), A117078, A117563, A006562, A117876, A118464, A118467, A119402, A119403, A119404.

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

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

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

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A119403 Primes p=prime(i) of level (1,10), i.e., such that A118534(i)=prime(i-10).

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A119404 Primes p=prime(i) of level (1,9), i.e., such that A118534(i)=prime(i-9).

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A125576 Primes p=prime(i) of level (1,15), i.e., such that A118534(i)=prime(i-15).