This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(3) = 659426 because d(659426) = d(659426+1) = d(659426+2) = d(6594286+3) or 9843019, 9843049, 9843079, 9843109, 9843139 are five consecutive primes with same difference and prime(659426) = 9843019 is the smallest prime number with this property. Also a(4) = 6904737 because d(6904737) = d(6904737+1) = ... = d(6904737+4) or 121174811, 121174841, 121174871, 121174901, 121174931, 121174961 are six consecutive primes with same difference and prime(6904737) = 121174811 is the smallest prime number with this property.
The antiruns of odd primes (differing by > 2) begin: 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 with lengths: 1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, ... with runs: 1 1 2 2 3 3 4 3 6 2 5 2 6 2 2 4 3 5 3 4 with lengths a(n).
Length/@Split[Length/@Split[Select[Range[3,1000],PrimeQ],#2-#1>2&]//Most]//Most
The first 10 weakly increasing prime quartets:
2 3 5 7
3 5 7 11
17 19 23 29
41 43 47 53
43 47 53 59
79 83 89 97
107 109 113 127
149 151 157 163
163 167 173 179
197 199 211 223
For example, 43 is the 14th prime, and the primes (43,47,53,59) have differences (4,6,6), which are weakly increasing, so 14 is in the sequence.
The first 10 unequal prime quartets: 17 19 23 29 19 23 29 31 29 31 37 41 31 37 41 43 41 43 47 53 59 61 67 71 61 67 71 73 67 71 73 79 71 73 79 83 79 83 89 97 For example, 83 is the 23rd prime, and the primes (83,89,97,101) have differences (6,8,4), which are all distinct, so 23 is in the sequence.
The first 10 partially unequal prime quartets: 5 7 11 13 7 11 13 17 11 13 17 19 13 17 19 23 17 19 23 29 19 23 29 31 23 29 31 37 29 31 37 41 31 37 41 43 37 41 43 47
ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x!=z-y&&z-y!=t-z:>PrimePi[x]]
PrimePi[#]&/@(Select[Partition[Prime[Range[90]],4,1],#[[2]]-#[[1]]!=#[[3]]-#[[2]]&&#[[3]]-#[[2]]!=#[[4]]-#[[3]]&][[;;,1]]) (* Harvey P. Dale, Aug 05 2025 *)
The first 10 weakly decreasing prime quartets: 31 37 41 43 47 53 59 61 61 67 71 73 89 97 101 103 151 157 163 167 167 173 179 181 199 211 223 227 211 223 227 229 241 251 257 263 251 257 263 269 For example, 241 is the 53rd prime, and the primes (241,251,257,263) have differences (10,6,6), which are weakly decreasing, so 53 is in the sequence.
The runs of odd primes differing by 2 begin: 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 with lengths: 3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, ... which have runs beginning: 3 2 2 1 2 1 2 1 1 2 1 2 1 1 1 1 2 2 1 1 1 with lengths: 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 2, 4, 3, ... with positions of first appearances a(n).
t=Length/@Split[Length/@Split[Select[Range[3,10000], PrimeQ],#1+2==#2&]//Most]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]
prime(4) = 7, 6!+1 = 721 gives residue 1 when divided by prime(4)+2 = 9.
pnmQ[n_]:=Module[{p=Prime[n]},Mod[(p-1)!+1,p+2]==1]; Select[Range[ 100],pnmQ] (* Harvey P. Dale, Jun 24 2017 *)
isok(n) = (((prime(n)-1)! + 1) % (prime(n)+2)) == 1; \\ Michel Marcus, Dec 31 2013
p=c=g=P=0;forprime(q=1,, p+g==(p+=g=q-p)|| next; q==P+2*g&& c++|| c=3; c>5&& print1(P-3*g,","); P=q-g) \\ M. F. Hasler, Oct 26 2018
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