A060254 -id:A060254 - cp's OEIS frontend

A109998 Non-Cunningham primes: primes isolated from any Cunningham chain under any iteration of 2p+-1 or (p+-1)/2.

17, 43, 67, 71, 101, 103, 109, 127, 137, 149, 151, 163, 181, 197, 223, 241, 257, 269, 283, 311, 317, 349, 353, 373, 389, 401, 409, 433, 449, 461, 463, 487, 521, 523, 557, 569, 571, 599, 617, 631, 643, 647, 677, 701, 709, 739, 751, 769, 773, 787, 797, 821

Offset: 1

Alexandre Wajnberg, Sep 01 2005

The condition that neither 2p - 1 nor 2p + 1 be prime is equivalent to ((p-1) mod 3 = 0) or ((p+1) mod 3 = 0). For example, the prime p = 2^607 - 1 is not in this sequence because p + 1 mod 3 = 2. - Washington Bomfim, Oct 30 2009

			a(1) = 17 is here because 17 * 2 + 1 = 35, 17 * 2 - 1 = 33; (17+1)/2 = 9, (17-1)/2 = 8: four composite numbers.

Chris Caldwell's Prime Glossary, Cunningham chains.
Douglas S. Stones, On prime chains, arXiv:0908.2166 [math.NT] [From _Washington Bomfim_, Oct 30 2009, edited by _R. J. Mathar_, Mar 01 2010]

Cf. A005385, A005383, A060254, A005384, A005382, A068497, A059455, A059762, A057326, A023272, A023302, A059764, A057328, A023330, A005602, A064812, A005603.

nonCunninghamPrimes = {}; Do[p = Prime[n]; If[!PrimeQ[2p - 1] && !PrimeQ[2p + 1] && !PrimeQ[(p - 1)/2] && !PrimeQ[(p + 1)/2], AppendTo[nonCunninghamPrimes, p]], {n, 6!}]; nonCunninghamPrimes (* Vladimir Joseph Stephan Orlovsky, Mar 22 2009 *)

Corrected and extended by Ray Chandler, Sep 02 2005

Replaced link to cached arXiv URL with link to the abstract - R. J. Mathar, Mar 01 2010

A037174 Primes which are not the sum of consecutive composite numbers.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 47, 61, 73, 107, 167, 179, 313, 347, 421, 479, 719, 863, 1153, 1213, 1283, 1307, 1523, 3467, 3733, 4007, 4621, 4787, 5087, 5113, 5413, 7523, 7703, 9817, 10333, 12347, 12539, 13381, 17027, 18553, 19717, 19813, 23399, 26003, 31873, 36097, 38833

Offset: 1

Naohiro Nomoto

nonn

It seems reasonable that a(n)/A079149(n) has an asymptote that could be estimated. - Peter Munn, Aug 21 2023

T. D. Noe, Table of n, a(n) for n = 1..3492 (terms less than 2*10^9)

Subsequence of A079149.
With {1}, the complement of A133576.
Primes that are the sum of specific numbers of consecutive composite numbers: A060254 (2), A060328 (3), A060329 (4), A060330 (5), A060331 (6), A060332 (7), A060333 (8).
Cf. A050940, A197227.

N:= 5000:
primes,comps:= selectremove(isprime,{$2..N}):
M:= nops(comps):
X:= primes:
for n from 1 to floor(sqrt(2*N)) do
i:= 1;
T:= add(comps[k],k=1..n);
while T <= N do
X := X minus {T};
if i + n > M then break fi;
T := T + comps[i+n] - comps[i];
i := i+1;
od;
od:
X;
# Robert Israel, Jun 24 2008

More terms from Jud McCranie, Jul 12 2000

Corrected by T. D. Noe, Aug 15 2008

A096676 a(n) = (A096788(n)-1)/2.

Original entry on oeis.org

4, 7, 10, 16, 17, 19, 25, 31, 32, 34, 37, 40, 47, 49, 52, 55, 59, 62, 67, 70, 76, 77, 82, 91, 94, 104, 107, 109, 110, 115, 121, 122, 124, 130, 136, 142, 149, 151, 154, 157, 160, 161, 164, 170, 172, 181, 184, 185, 187, 196, 202, 205, 206, 214, 220, 226, 227, 229

Offset: 1

Ray Chandler, Jul 10 2004

nonn

a(n)*4 + 3 = A096787(n).

Cf. A060254, A096784, A096785, A096786, A096787, A096788.

A060333 Primes which are the sum of eight consecutive composite numbers.

Original entry on oeis.org

193, 277, 353, 433, 443, 613, 643, 653, 673, 683, 739, 881, 1109, 1129, 1237, 1511, 1531, 1609, 1619, 1697, 1873, 1999, 2017, 2027, 2113, 2207, 2239, 2281, 2371, 2447, 2621, 2657, 2677, 2687, 2749, 2801, 2833, 2843, 2909, 2927, 3023, 3083, 3121, 3167

Offset: 1

Robert G. Wilson v, Mar 30 2001

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A060254, A060328, A060329, A060330, A060331, A060332, A151738, A151739.

comps:= remove(isprime, [$4..1000]):
S:= add(comps[i+1..i-8],i=0..7):
select(isprime,S); # Robert Israel, Dec 12 2019

composite[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1 != n, k++ ]; k); a = {}; Do[ p = Sum[ composite[ n + k ], {k, 0, 7} ]; If[ PrimeQ[ p ], a = Append[ a, p ] ], {n, 1, 600} ]; a
Select[Total /@ Partition[ Select[ Range@ 500, CompositeQ], 8, 1], PrimeQ] (* Giovanni Resta, Dec 13 2019 *)

A135170 Primes equal to a sum c1+c2 of two consecutive composite numbers such that lpf(c1)-spf(c1)+lpf(c2)-spf(c2) from their largest and smallest prime factors is prime.

Original entry on oeis.org

19, 29, 31, 41, 43, 53, 67, 71, 79, 89, 101, 109, 131, 149, 151, 173, 197, 199, 233, 239, 241, 251, 269, 271, 283, 307, 311, 317, 331, 337, 349, 367, 401, 419, 439, 449, 461, 487, 491, 499, 509, 521, 593, 599, 617, 641, 647, 683, 691, 727, 739, 751, 769, 809

Offset: 1

Giovanni Teofilatto, Feb 14 2008

nonn

Cf. A111426.

A002808 := proc(n) option remember ; local a ; if n = 1 then 4; else for a from A002808(n-1)+1 do if not isprime(a) then RETURN(a) ; fi ; od: fi ; end:
isA060254 := proc(n) local i,sComp ; if isprime(n) then for i from 1 do sComp := A002808(i)+A002808(i+1) ; if sComp = n then RETURN(i); elif sComp > n then RETURN(-1) ; fi ; od: else -1 ; fi ; end:
A046665 := proc(n) local a,ifs ; a := 0 ; ifs := seq(op(1, i),i=ifactors(n)[2]) ; max(ifs)-min(ifs) ; end:
A111426 := proc(n) A046665(A002808(n)) ; end:
isA135170 := proc(p) local i ; i := isA060254(p) ; if i > 0 then A111426(i) + A111426(i+1) ; isprime(%) ; else false ; fi ; end:
for n from 1 to 300 do p := ithprime(n) ; if isA135170(p) then printf("%d,",p) ; fi ; od: # R. J. Mathar, Feb 19 2008

{A060254(j): A002808(i)+A002808(i+1)=A060254(j) and A111426(i)+A111426(i+1) in A000040}. Subsequence of A060254. - R. J. Mathar, Feb 19 2008

Corrected and extended by R. J. Mathar, Feb 19 2008

More precise definition by R. J. Mathar, Sep 17 2009

A151744 Primes which are the sum of two, three, four and five consecutive composite numbers.

Original entry on oeis.org

17783, 25057, 47303, 48383, 49297, 76343, 89783, 205703, 412343, 516457, 704183, 754417, 790703, 938183, 1105343, 1110743, 1279583, 1563503, 1632817, 1744583, 1890743, 1903103, 2062943, 2276303, 2714617, 2802383, 2812897, 2932703

Offset: 1

Claudio Meller, Jun 15 2009

nonn

is in the list because: 17783 = 8891 + 8892 (sum of two consecutive composite numbers)
= 5926 + 5928 + 5929 (sum of three consecutive composite numbers)
= 4444 + 4445 + 4446 + 4448 (sum of four consecutive composite numbers)
= 3554 + 3555 + 3556 + 3558 + 3560 (sum of five consecutive composite numbers)

Harvey P. Dale, Table of n, a(n) for n = 1..100

CompositeNext[n_]:=Module[{k=n+1},While[PrimeQ[k],k++ ];k]; q=9!; lst2={};Do[If[ !PrimeQ[n],c=CompositeNext[n];a2=n+c;If[PrimeQ[a2],AppendTo[lst2,a2]]],{n,q}];lst2; lst3={};Do[If[ !PrimeQ[n],c1=CompositeNext[n];c2=CompositeNext[c1];a3=n+c1+c2;If[PrimeQ[a3],AppendTo[lst3,a3]]],{n,q}];lst3; lst4={};Do[If[ !PrimeQ[n],c1=CompositeNext[n];c2=CompositeNext[c1];c3=CompositeNext[c2];a4=n+c1+c2+c3;If[PrimeQ[a4],AppendTo[lst4,a4]]],{n,q}];lst4; lst5={};Do[If[ !PrimeQ[n],c1=CompositeNext[n];c2=CompositeNext[c1];c3=CompositeNext[c2];c4=CompositeNext[c3];a5=n+c1+c2+c3+c4;If[PrimeQ[a5],AppendTo[lst5,a5]]],{n,q}];lst5; Intersection[lst2,lst3,lst4,lst5] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2009 *)
Module[{comps=Select[Range[2*10^6],CompositeQ]},Intersection@@ Table[ Select[ Total/@ Partition[comps,n,1],PrimeQ],{n,2,5}]] (* Harvey P. Dale, Apr 16 2015 *)

Intersection of A060254, A060328, A060329 and A060330. - R. J. Mathar, Jun 17 2009

Extended beyond a(7) by Klaus Brockhaus, Jun 16 2009

A167915 Primes which are the sums of two consecutive nonprimes (A141468).

Original entry on oeis.org

5, 17, 19, 29, 31, 41, 43, 53, 67, 71, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 163, 173, 181, 191, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 281, 283, 293, 307, 311, 317, 331, 337, 349, 353, 367, 373, 379, 389, 401, 409

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2009

nonn

Five together with primes that are the sum of two consecutive composite numbers.

			a(1)=1+4=5, a(2)=8+9=17.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A060254, A141468, A176902.

[2*n+1: n in [1..300] | (not IsPrime(n) eq not IsPrime(n+1)) and IsPrime(2*n+1)]; // G. C. Greubel, Nov 10 2023

2*Select[Range[300], !PrimeQ[#] == !PrimeQ[#+1] && PrimeQ[2*#+1] &] + 1 (* G. C. Greubel, Jul 01 2016; Nov 10 2023 *)

[2*n+1 for n in (1..300) if  (not is_prime(n)) - (not is_prime(n+1)) == 0 and is_prime(2*n+1)] # G. C. Greubel, Nov 10 2023

a(n+1) = A060254(n) = A176902(n+1). - Juri-Stepan Gerasimov, Apr 28 2010

Typo corrected and terms checked by D. S. McNeil, Nov 17 2010

A096675 a(n) = A096786(n)/2.

Original entry on oeis.org

4, 7, 10, 13, 22, 24, 25, 27, 28, 34, 37, 43, 45, 49, 57, 58, 60, 64, 67, 70, 73, 79, 84, 87, 88, 93, 97, 100, 102, 108, 112, 115, 127, 130, 139, 142, 144, 148, 150, 154, 160, 163, 169, 175, 177, 190, 192, 193, 199, 202, 205, 207, 213, 214, 220, 232, 234, 235, 238

Offset: 1

Ray Chandler, Jul 10 2004

nonn

a(n)*4 + 1 = A096785(n).

Cf. A060254, A096784, A096785, A096786, A096787, A096788.

A176902 Primes p such that p-1 and p+1 are both non-semiprime.

Original entry on oeis.org

2, 17, 19, 29, 31, 41, 43, 53, 67, 71, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 163, 173, 181, 191, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 281, 283, 293, 307, 311, 317, 331, 337, 349, 353, 367, 373, 379, 389, 401, 409

Offset: 1

Juri-Stepan Gerasimov, Apr 28 2010

nonn

2 together with A060254.

			a(1)=2 because 2-1=1=non-semiprime and 2+1=3=non-semiprime.

a(n+1)=A060254(n)=A167915(n+1).

Corrected (233, 239 inserted, 279 and 289 replaced by 379 and 389) by R. J. Mathar, Aug 12 2010

A096678 A096785 indexed by A000040.

Original entry on oeis.org

7, 10, 13, 16, 24, 25, 26, 29, 30, 33, 35, 40, 42, 45, 50, 51, 53, 55, 57, 60, 62, 66, 68, 70, 71, 74, 77, 79, 80, 84, 87, 89, 97, 98, 102, 104, 106, 108, 110, 113, 116, 119, 123, 126, 127, 135, 136, 137, 139, 140, 142, 145, 147, 148, 152, 158, 159, 160, 162, 165

Offset: 1

Ray Chandler, Jul 10 2004

nonn

Cf. A060254, A096784, A096785, A096786, A096787, A096788.

a(n) = k such that A000040(k) = A096785(n).

cp's OEIS Frontend

A109998 Non-Cunningham primes: primes isolated from any Cunningham chain under any iteration of 2p+-1 or (p+-1)/2.

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A037174 Primes which are not the sum of consecutive composite numbers.

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A096676 a(n) = (A096788(n)-1)/2.

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A060333 Primes which are the sum of eight consecutive composite numbers.

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A135170 Primes equal to a sum c1+c2 of two consecutive composite numbers such that lpf(c1)-spf(c1)+lpf(c2)-spf(c2) from their largest and smallest prime factors is prime.

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A151744 Primes which are the sum of two, three, four and five consecutive composite numbers.

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A167915 Primes which are the sums of two consecutive nonprimes (A141468).

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A096675 a(n) = A096786(n)/2.

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A176902 Primes p such that p-1 and p+1 are both non-semiprime.

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