A022885 -id:A022885 - cp's OEIS frontend

A235725 Values k(i) such that k(i) + k(i+3) = k(i+1) + k(i+2), where k(i) is A022885(i).

5, 353, 541, 853, 2341, 4217, 4229, 8219, 10663, 11047, 13591, 18593, 21577, 28387, 30181, 34457, 37853, 52021, 55333, 57203, 75389, 84431, 93229, 110603, 120811, 147451, 153499, 162907, 166357, 176797, 179581, 219953, 243671, 246203, 307253, 342037, 359701

Offset: 1

Vladimir Shevelev, Jan 15 2014

nonn

			Four consecutive Kimberling primes(A022885), beginning with 5 are 5,7,11,13. Since 5+13 = 7+11, then 5 is in the sequence; four consecutive Kimberling primes, beginning with 7 are 7,11,13,23. Since 7+23 is not equal to 11+13, then 7 is not in the sequence.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Cf. A022885.

Nest[Map[#[[1]]&,Select[Partition[#,4,1],#[[1]]+#[[4]]==#[[2]]+#[[3]]&]]&,Prime[Range[5000]],2]

isA022885(p) = {my(k = primepi(p)); (p == prime(k)) && ((prime(k) + prime(k+3)) == (prime(k+1) + prime(k+2)));}
lista(nn) = {prm = primes(nn); vkp = select(p->isA022885(p), prm); for(n=1, #vkp-3, if ((vkp[n] + vkp[n+3]) == (vkp[n+1] + vkp[n+2]), print1(vkp[n], ", ")););}  \\ Michel Marcus, Jan 15 2014

a(5)-a(37) from Giovanni Resta, Jan 15 2014

A064101 Primes p = prime(k) such that prime(k) + prime(k+5) = prime(k+1) + prime(k+4) = prime(k+2) + prime(k+3).

Original entry on oeis.org

5, 7, 19, 31, 97, 131, 151, 293, 587, 683, 811, 839, 857, 907, 1013, 1097, 1279, 2347, 2677, 2833, 3011, 3329, 4217, 4219, 5441, 5839, 5849, 6113, 8233, 8273, 8963, 9433, 10301, 10427, 10859, 11953, 13513, 13597, 13721, 13931, 14713, 15629, 16057

Offset: 1

Robert G. Wilson v, Sep 17 2001

			The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Take just the fourth through the ninth and rearrange them such that the first pairs with the sixth, the second with the fifth and the third with the fourth as follows: 7 and 23, 11 and 19 and 13 and 17. All three pairs sum to 30. Therefore a(2) = 7.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A022885, A359440.

A := {}: for n to 1000 do p1 := ithprime(n); p2 := ithprime(n+1); p3 := ithprime(n+2); p4 := ithprime(n+3); p5 := ithprime(n+4); p6 := ithprime(n+5); if `and`(p1+p6 = p2+p5, p2+p5 = p3+p4) then A := `union`(A, {p1}) end if end do; A := A;

a = {0, 0, 0, 0, 0, 0}; Do[ a = Delete[ a, 1 ]; a = Append[ a, Prime[ n ] ]; If[ a[ [ 1 ] ] + a[ [ 6 ] ] == a[ [ 2 ] ] + a[ [ 5 ] ] == a[ [ 3 ] ] + a[ [ 4 ] ], Print[ a[ [ 1 ] ] ] ], {n, 1, 20000} ] (* RGWv *)
Prime[Select[Range[100], Prime[#] + Prime[# + 5] == Prime[# + 1] + Prime[# + 4] && Prime[#] + Prime[# + 5] == Prime[# + 2] + Prime[# + 3] &]]
Select[Partition[Prime[Range[2000]],6,1],#[[1]]+#[[6]]==#[[2]]+#[[5]] == #[[3]]+ #[[4]]&][[All,1]] (* Harvey P. Dale, Jan 16 2022 *)

{ n=0; default(primelimit, 1500000); for (k=1, 10^9, p1=prime(k) + prime(k + 5); p2=prime(k + 1) + prime(k + 4); p3=prime(k + 2) + prime(k + 3); if (p1==p2 && p2==p3, write("b064101.txt", n++, " ", prime(k)); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 07 2009

Primes p = prime(k) = A000040(k) such that A359440(k+2) >= 2. - Peter Munn, Jan 09 2023

A022884 Numbers k such that prime(k) + prime(k+3) = prime(k+1) + prime(k+2).

Original entry on oeis.org

3, 4, 5, 6, 9, 12, 16, 21, 25, 26, 27, 29, 33, 37, 41, 43, 48, 54, 56, 63, 71, 74, 77, 80, 81, 84, 88, 92, 93, 100, 103, 105, 108, 124, 125, 126, 134, 140, 142, 147, 149, 151, 153, 156, 165, 171, 175, 176, 181, 185, 191, 200, 208, 211, 216, 224, 234, 235

Offset: 1

Clark Kimberling

nonn

			The ninth prime is 23. We verify that 23 + 37 = 60 = 29 + 31. Hence 9 is in the sequence.
The tenth prime is 29. We see that 29 + 41 = 70 but 31 + 37 = 68, so 10 is not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)

Cf. A000720, A022885

Cf. A261470. - Altug Alkan, Oct 28 2015

[n: n in [1..250] |(NthPrime(n)+NthPrime(n+3)) eq (NthPrime(n+1)+ NthPrime(n+2))]; // Vincenzo Librandi, Nov 04 2018

Select[Range@ 240, Prime[#] + Prime[# + 3] == Prime[# + 1] + Prime[# + 2] &] (* Michael De Vlieger, Oct 28 2015 *)

isok(k) = prime(k+3)+prime(k) == prime(k+1)+prime(k+2); \\ Michel Marcus, Aug 20 2015

is(n,p=prime(n))=my(q=nextprime(p+1),r=nextprime(q+1),s=nextprime(r+1)); p+s==q+r
n=0; forprime(p=2,1e5, if(is(n++,p), print1(n", "))) \\ Charles R Greathouse IV, Oct 28 2015

a(n) = A000720(A022885(n)). - Zak Seidov, Oct 23 2015

Name edited by Michel Marcus, Aug 20 2015

A105093 Numbers n such that n = prime(k) + prime(k+3) = prime(k+1) + prime(k+2) for some k.

Original entry on oeis.org

18, 24, 30, 36, 60, 84, 120, 162, 204, 210, 216, 240, 288, 330, 372, 390, 456, 520, 540, 624, 726, 762, 798, 840, 852, 882, 924, 978, 990, 1104, 1140, 1164, 1200, 1392, 1410, 1428, 1530, 1632, 1650, 1716, 1740, 1764, 1794, 1848, 1974, 2052, 2100, 2112, 2184

Offset: 1

Giovanni Teofilatto, Apr 06 2005

nonn

			a(1)=18 because prime(3)+prime(6)=prime(4)+prime(5)=5+13=7+11=18.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Goldbach Conjecture

Cf. A001172, A000954, first primes in A022885.

lst = {}; Do[ If[ Prime[n] + Prime[n + 3] == Prime[n + 1] + Prime[n + 2], AppendTo[lst, Prime[n] + Prime[n + 3]]], {n, 184}]; lst (* Robert G. Wilson v, Apr 07 2005 *)

from sympy import nextprime
A105093_list, plist = [], [2,3,5,7]
while len(A105093_list) < 10000:
    m = plist[0]+plist[3]
    if m == plist[1]+plist[2]:
        A105093_list.append(m)
    plist = plist[1:] + [nextprime(plist[-1])] # Chai Wah Wu, Mar 30 2020

Edited and extended by Robert G. Wilson v, Apr 07 2005

A263674 Double interprimes: a(n) = (q+r)/2 = (p+s)/2 with p

Original entry on oeis.org

9, 12, 15, 18, 30, 42, 60, 81, 102, 105, 108, 120, 144, 165, 186, 195, 228, 260, 270, 312, 363, 381, 399, 420, 426, 441, 462, 489, 495, 552, 570, 582, 600, 696, 705, 714, 765, 816, 825, 858, 870, 882, 897, 924, 987, 1026, 1050, 1056, 1092, 1113, 1167, 1230

Offset: 1

Antonio Roldán, Oct 23 2015

nonn

Values of p (lesser of consecutive primes) are in the sequence A022885.

			600 is in this sequence because 593, 599, 601 and 607 are consecutive primes, and 600 = (599+601)/2 = (593+607)/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Interprime

Cf. A022885, A075190, A075277, A075296, A078443, A130178, A263675, A263676.

(Prime@ # + Prime[# + 3])/2 & /@ Select[Range@ 240, (First@ # + Last@ #)/2 == (#[[2]] + #[[3]])/2 &@ Prime@ Range[#, # + 3] &] (* Michael De Vlieger, Nov 18 2015 *)
Mean/@Select[Partition[Prime[Range[300]],4,1],(#[[2]]+#[[3]])/2==(#[[1]]+#[[4]])/2&] (* Harvey P. Dale, Aug 18 2024 *)

{forprime(q=3,2000,p=precprime(q-1); r=nextprime(q+1); s=nextprime(r+1);m=(q+r)/2;if(m==(p+s)/2,print1(m,", ")))}

A260179 Primes pr(k) such that pr(k)+pr(k+1)+pr(k+6)+pr(k+7) = pr(k+2)+pr(k+3)+pr(k+4)+pr(k+5).

Original entry on oeis.org

17, 23, 71, 149, 173, 233, 331, 359, 389, 419, 431, 503, 677, 727, 839, 853, 937, 971, 1019, 1201, 1229, 1277, 1327, 2213, 2221, 2237, 2593, 2689, 2797, 2999, 3019, 3167, 3221, 3253, 3821, 3823, 4027, 4111, 4201, 4219, 4231, 4801, 5147, 5309, 5407, 5437

Offset: 1

Harvey P. Dale, Jul 17 2015

nonn

			Starting from 71, the eight consecutive primes are 71, 73, 79, 83, 89, 97, 101, 103; and they satisfy 71+73+101+103=79+83+89+97, so 71 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from R. J. Mathar)

Cf. A022885.

n := 1 ;
Lp := [[0,1,6,7],[2,3,4,5]] ;
for i from 1 do
    if add(ithprime(i+j),j=op(1,Lp)) = add(ithprime(i+j),j=op(2,Lp)) then
        printf("%d %d\n",n,ithprime(i)) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 06 2015

pr8Q[lst_]:=With[{tadr=TakeDrop[lst,{3,6}]},Total[tadr[[1]]] == Total[ tadr[[2]]]]; Transpose[Select[Partition[Prime[ Range[ 1000]],8,1], pr8Q]][[1]] (* The program uses the TakeDrop function from Mathematica version 10.2 *)

A261470 a(n) = prime(n+3) - prime(n+2) - prime(n+1) + prime(n).

Original entry on oeis.org

1, 2, 0, 0, 0, 0, 4, -2, 0, 2, -4, 0, 4, 2, -4, 0, 2, -4, 2, 2, 0, 4, -2, -6, 0, 0, 0, 12, 0, -8, -2, 4, 0, -4, 4, -2, 0, 2, -4, 4, 0, -6, 0, 8, 10, -8, -10, 0, 4, -2, 4, 4, -4, 0, -4, 0, 2, -4, 6, 12, -6, -12, 0, 12, 2, -4, -4, -6, 4, 4, 0, -2, -2, 0, 4, -2, 0

Offset: 1

Altug Alkan, Aug 20 2015

			a(5) = 19 - 17 - 13 + 11 = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A022884, A022885, A093667.

Cf. A001223.

a261470 n = a261470_list !! (n-1)
a261470_list = zipWith (-) (drop 2 a001223_list) a001223_list
-- Reinhard Zumkeller, Aug 22 2015

Table[Prime[i+3] - Prime[i+2] - Prime[i+1] + Prime[i], {i,100}] (* G. C. Greubel, Aug 20 2015 *)

first(m)=vector(m,i,prime(i+3)+prime(i)-prime(i+1)-prime(i+2)) \\ Anders Hellström, Aug 20 2015

a(n) = A001223(n+2) - A001223(n). - Reinhard Zumkeller, Aug 22 2015

A064102 Primes p = prime(k) such that prime(k) + prime(k+7) = prime(k+1) + prime(k+6) = prime(k+2) + prime(k+5) = prime(k+3) + prime(k+4).

Original entry on oeis.org

17, 149, 677, 853, 1277, 5437, 6101, 13499, 13921, 19853, 22073, 41863, 49667, 51307, 51797, 55799, 61637, 66337, 83227, 91121, 100957, 103963, 109111, 113147, 128747, 136309, 137933, 148157, 158849, 163117, 167249, 179033, 205171, 208927

Offset: 1

Robert G. Wilson v, Sep 17 2001

			17 + 43 = 19 + 41 = 23 + 37 = 29 + 31.

Harry J. Smith, Table of n, a(n) for n = 1..400

Cf. A000040, A022885, A064101, A359440.

a = {0, 0, 0, 0, 0, 0, 0, 0}; Do[ a = Delete[ a, 1 ]; a = Append[ a, Prime[ n ] ]; If[ a[ [ 1 ] ] + a[ [ 8 ] ] == a[ [ 2 ] ] + a[ [ 7 ] ] == a[ [ 3 ] ] + a[ [ 6 ] ] == a[ [ 4 ] ] + a[ [ 5 ] ], Print[ a[ [ 1 ] ] ] ], {n, 1, 10^4} ]
Select[Partition[Prime[Range[20000]],8,1],#[[1]]+#[[8]]==#[[2]]+#[[7]]==#[[3]]+#[[6]]==#[[4]]+#[[5]]&][[;;,1]] (* Harvey P. Dale, Jul 03 2025 *)

{ n=0; default(primelimit, 8300000); for (k=1, 10^9, p1=prime(k) + prime(k + 7); p2=prime(k + 1) + prime(k + 6); p3=prime(k + 2) + prime(k + 5); p4=prime(k + 3) + prime(k + 4); if (p1==p2 && p2==p3 && p3==p4, write("b064102.txt", n++, " ", prime(k)); if (n==400, break)) ) } \\ Harry J. Smith, Sep 07 2009

Primes p = prime(k) = A000040(k) such that A359440(k+3) >= 3. - Peter Munn, Jan 09 2023

A235743 Primes p(k) such that p(k) + p(k+3) = p(k+1) + p(k+2) + 4.

Original entry on oeis.org

17, 41, 79, 131, 149, 173, 227, 233, 239, 347, 349, 379, 439, 463, 521, 599, 641, 673, 677, 983, 1013, 1091, 1231, 1277, 1427, 1429, 1453, 1487, 1549, 1607, 1811, 1949, 2099, 2203, 2309, 2579, 2609, 2687, 2689, 2833, 2857, 2903, 2909, 2917, 3083, 3167, 3299

Offset: 1

Vladimir Shevelev, Jan 15 2014

nonn

If p(k) is in the sequence, then the four consecutive primes p(k), p(k+1), p(k+2), p(k+3) possess a property of quadruplet of consecutive squares: n^2 + (n+3)^2 = (n+1)^2 + (n+2)^2 + 4.

Cf. A022885, where such quadruplets possess a linear property: n + (n+3) = (n+1) + (n+2).

			17 is in the sequence since 17, 19, 23, and 29 are four consecutive primes and it holds 17 + 29 = 19 + 23 + 4.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A022885.

f[{a_,b_,c_,d_}]:= a-b-c+d; First /@ Select[Partition[Prime@ Range@ 500, 4, 1], f@# == 4 &] (* Giovanni Resta, Jan 16 2014 *)
Transpose[Select[Partition[Prime[Range[5000]], 4, 1], First[#] + Last[#]==#[[2]] + #[[3]] + 4&]][[1]] (* Vincenzo Librandi, Feb 02 2014 *)

isok(p) = { my(k = primepi(p)); (p == prime(k)) && ((prime(k) + prime(k+3)) == (prime(k+1) + prime(k+2) + 4));} \\ Michel Marcus, Jan 15 2014

More terms from Michel Marcus, Jan 15 2014

A260265 Numbers n such that prime(n) + prime(n+4) + prime(n+5) + prime(n+7) + prime(n+8) = prime(n+1) + prime(n+2) + prime(n+3) + prime(n+6) + prime(n+9).

Original entry on oeis.org

180, 205, 223, 258, 297, 326, 350, 424, 502, 514, 521, 548, 666, 757, 765, 796, 804, 861, 909, 1006, 1098, 1150, 1264, 1317, 1348, 1349, 1400, 1416, 1592, 1613, 1663, 1757, 1821, 1914, 1939, 2034, 2048, 2073, 2085, 2109, 2164, 2276, 2316, 2360, 2380, 2398, 2655

Offset: 1

K. D. Bajpai, Jul 21 2015

nonn

			180 is in the list because 1069 + 1097 + 1103 + 1117 + 1123 = 1087 + 1091 + 1093 + 1109 + 1129.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A022885, A260179.

[n: n in [1..3000] | NthPrime(n) + NthPrime(n+4)+ NthPrime(n+5)+ NthPrime(n+7)+ NthPrime(n+8) eq NthPrime(n+1)+ NthPrime(n+2)+ NthPrime(n+3)+ NthPrime(n+6)+ NthPrime(n+9)];

Select[Range[5000], Prime[#] + Prime[# + 4] + Prime[# + 5] + Prime[# + 7] + Prime[# + 8] == Prime[# + 1] + Prime[# + 2] + Prime[# + 3] + Prime[# + 6] + Prime[# + 9] &]

for(n=1,5000, if(prime(n) + prime(n+4) + prime(n+5) + prime(n+7) + prime(n+8)  == prime(n+1) + prime(n+2) + prime(n+3) + prime(n+6) + prime(n+9), print1(n,", ")));

cp's OEIS Frontend

A235725 Values k(i) such that k(i) + k(i+3) = k(i+1) + k(i+2), where k(i) is A022885(i).

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A064101 Primes p = prime(k) such that prime(k) + prime(k+5) = prime(k+1) + prime(k+4) = prime(k+2) + prime(k+3).

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A022884 Numbers k such that prime(k) + prime(k+3) = prime(k+1) + prime(k+2).

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A105093 Numbers n such that n = prime(k) + prime(k+3) = prime(k+1) + prime(k+2) for some k.

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A263674 Double interprimes: a(n) = (q+r)/2 = (p+s)/2 with p

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A260179 Primes pr(k) such that pr(k)+pr(k+1)+pr(k+6)+pr(k+7) = pr(k+2)+pr(k+3)+pr(k+4)+pr(k+5).

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A261470 a(n) = prime(n+3) - prime(n+2) - prime(n+1) + prime(n).

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