A029707 -id:A029707 - cp's OEIS frontend

A356222 Array read by antidiagonals upwards where A(n,k) is the position of the k-th appearance of 2n in the sequence of prime gaps A001223. If A001223 does not contain 2n at least k times, set A(n,k) = -1.

2, 4, 3, 9, 6, 5, 24, 11, 8, 7, 34, 72, 15, 12, 10, 46, 42, 77, 16, 14, 13, 30, 47, 53, 79, 18, 19, 17, 282, 62, 91, 61, 87, 21, 22, 20, 99, 295, 66, 97, 68, 92, 23, 25, 26, 154, 180, 319, 137, 114, 80, 94, 32, 27, 28, 189, 259, 205, 331, 146, 121, 82, 124, 36, 29, 33

Offset: 1

Gus Wiseman, Aug 04 2022

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

This is a permutation of the positive integers > 1.

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9
  n=1:   2   3   5   7  10  13  17  20  26
  n=2:   4   6   8  12  14  19  22  25  27
  n=3:   9  11  15  16  18  21  23  32  36
  n=4:  24  72  77  79  87  92  94 124 126
  n=5:  34  42  53  61  68  80  82 101 106
  n=6:  46  47  91  97 114 121 139 168 197
  n=7:  30  62  66 137 146 150 162 223 250
  n=8: 282 295 319 331 335 378 409 445 476
  n=9:  99 180 205 221 274 293 326 368 416
For example, the positions in A001223 of appearances of 2*3 begin: 9, 11, 15, 16, 18, 21, 23, ..., which is row n = 3 (A320701).

The row containing n is A028334(n).
Row n = 1 is A029707.
Row n = 2 is A029709.
Column k = 1 is A038664.
The column containing n is A274121(n).
Column k = 2 is A356221.
The diagonal A(n,n) is A356223.
A001223 lists the prime gaps.
A073491 lists numbers with gapless prime indices.
A356224 counts even divisors with gapless prime indices, complement A356225.
Cf. A066205, A119313, A193829, A287170, A328457, A356226, A356232.

gapa=Differences[Array[Prime,10000]];
Table[Position[gapa,2*(k-n+1)][[n,1]],{k,6},{n,k}]

A378620 Lesser prime index of twin primes with nonsquarefree mean.

Original entry on oeis.org

2, 5, 7, 17, 20, 28, 35, 41, 43, 45, 49, 52, 57, 64, 69, 81, 83, 98, 109, 120, 140, 144, 152, 171, 173, 176, 178, 182, 190, 206, 215, 225, 230, 236, 253, 256, 262, 277, 286, 294, 296, 302, 307, 315, 318, 323, 336, 346, 373, 377, 390, 395, 405, 428, 430, 444

Offset: 1

Gus Wiseman, Dec 10 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

This is a subset of A029707 (twin prime indices). The other twin primes are A068361, so A029707 is the disjoint union of A068361 and A378620.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

The lesser of twin primes is A001359, index A029707 (complement A049579).
The greater of twin primes is A006512, index A107770 (complement appears to be A168543).
A subset of A029707 (twin prime lesser indices).
Prime indices of the primes listed by A061368.
Indices of twin primes with squarefree mean are A068361.
A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
A005117 lists the squarefree numbers, differences A076259.
A006562 finds balanced primes.
A013929 lists the nonsquarefree numbers, differences A078147.
A014574 is the intersection of A006093 and A008864.
A038664 finds the first position of a prime gap of 2n.
A046933 counts composite numbers between primes.
A120327 gives the least nonsquarefree number >= n.
Cf. A007674, A072284, A068360.
Cf. A057627, A061398, A061399, A067535, A070321, A071403, A112925.
Cf. A000720, A025584, A067774, A122535, A155752, A179067.

Select[Range[100],Prime[#]+2==Prime[#+1]&&!SquareFreeQ[Prime[#]+1]&]
PrimePi/@Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&&!SquareFreeQ[Mean[#]]&][[;;,1]] (* Harvey P. Dale, Jul 13 2025 *)

prime(a(n)) = A061368(n).

A074326 Numbers n such that difference between (1+2^n)-th and (2^n)-th primes is 2.

Original entry on oeis.org

1, 6, 8, 9, 17, 23, 27, 39, 48

Offset: 1

Labos Elemer, Aug 21 2002

			n=39: 2^39=549755813888, prime(549755813889) = 16149760533343, prime(549755813888) = 16149760533341, difference=2, just twin primes.

Cf. A029707, A051439, A033844, A073798, A000079.

s=0; Do[s=Prime[1+2^n]-Prime[2^n]; If[s==2, Print[{n, Prime[2^n]}]], {n, 1, 40}]
diffQ[n_]:=Module[{prn=Prime[2^n]},NextPrime[prn]-prn==2]; Select[ Range[ 40],diffQ] (* Harvey P. Dale, Aug 21 2014 *)

Solutions to A051439(x) - A033844(x) = 2,

a(9) from Chai Wah Wu, Feb 28 2019

A074328 Numbers m such that prime(m^2+1)-prime(m^2)=2, where prime(j) is the j-th prime.

Original entry on oeis.org

7, 8, 9, 12, 15, 16, 22, 25, 27, 34, 53, 83, 85, 88, 95, 107, 108, 144, 149, 187, 196, 223, 234, 238, 249, 255, 268, 274, 315, 324, 350, 355, 358, 367, 386, 410, 411, 416, 424, 436, 440, 445, 450, 462, 469, 471, 481, 494, 501, 509, 511, 517, 522, 549, 554, 564

Offset: 1

Labos Elemer, Aug 21 2002

nonn

Square roots of squares in A029707. - Michel Marcus, Oct 20 2022

			25 is a term because the 626th and 625th primes are twin primes: 4639 - 4637 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A029707, A001359, A006512, A011757.

t=Table[0, {250}]; t1=Table[0, {250}]; s=0; k=0; Do[s=Prime[1+n^2]-Prime[n^2]; If[s==2, k=k+1; t[[k]]=n; t1[[k]]=Prime[n^2]; Print[{k, n, Prime[n^2]}]], {n, 1, 2500}] t t1

isok(m) = my(p=prime(m^2)); nextprime(p+1) - p == 2; \\ Michel Marcus, Oct 20 2022

list(lim) = {my(k = 1, prv = 2); forprime(p = 3, lim, if(p - prv == 2 && issquare(k), print1(sqrtint(k), ", ")); k++; prv = p);} \\ Amiram Eldar, Mar 20 2025

A173400 n-th difference between consecutive primes=n-th difference between consecutive nonnegative nonprimes.

Original entry on oeis.org

1, 3, 7, 20, 26, 33, 43, 49, 52, 81, 116, 140, 176, 265, 288, 313, 320, 323, 373, 377, 395, 398, 405, 408, 486, 492, 530, 555, 567, 592, 671, 681, 772, 805, 849, 874, 884, 931, 936, 1016, 1030, 1149, 1204, 1324, 1347, 1406, 1464, 1550, 1621, 1639, 1707, 1712

Offset: 1

Juri-Stepan Gerasimov, Feb 17 2010

Numbers n such that A001223(n)=A054546(n).

Cf. A001223, A075526, A173390, A173401, A029707.

A001223(a(n))=A054546(a(n)).

Extended by Charles R Greathouse IV, Mar 25 2010

A305318 Numbers k such that A071866(k)=3.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 17, 18, 20, 21, 24, 25, 26, 28, 29, 30, 33, 35, 36, 37, 41, 42, 43, 44, 45, 49, 50, 52, 53, 57, 58, 59, 60, 64, 65, 67, 69, 70, 73, 74, 78, 79, 81, 82, 83, 84, 87, 88, 89, 98, 99, 100, 104, 105, 109, 110, 111, 112, 113, 115, 116, 120, 121, 122, 125, 129, 130, 133

Offset: 1

Robert Israel, May 29 2018

nonn

All terms are in A066940. The first member of A066940 not in this sequence is 61.

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

Cf. A000720, A066940, A071866, A275697. Contains A029707.

select(n -> nops(convert(ithprime(n+1)/ithprime(n),confrac))=3, [$1..1000]);

isok(n) = length(contfrac(prime(n+1)/prime(n))) == 3; \\ Michel Marcus, May 31 2018

a(n) = A000720(A275697(n+1)). - Robert Israel, May 31 2018

A320709 Indices of primes followed by a gap (distance to next larger prime) of 22.

Original entry on oeis.org

189, 297, 344, 375, 457, 487, 522, 549, 557, 721, 836, 914, 1010, 1158, 1170, 1197, 1233, 1242, 1272, 1290, 1370, 1390, 1444, 1471, 1625, 1633, 1672, 1683, 1757, 1858, 1975, 1983, 2039, 2074, 2158, 2243, 2248, 2250, 2327, 2370, 2388, 2614, 2638, 2703, 2725, 2838, 2842, 2872

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A061779.

Vincenzo Librandi, Table of n, a(n) for n = 1..10040
Index entries for primes, gaps between.

Equals A000720 o A061779.
Row 11 of A174349.
Indices of 22's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..3000] | NthPrime(n+1) - NthPrime(n) eq 22]; // Vincenzo Librandi, Mar 22 2019

Select[Range[3000], Prime[#] + 22 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 22 2019 *)

A(N=100,g=22,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A061779(n)).

A320709 = { i > 0 | prime(i+1) = prime(i) + 22 }.

A320710 Indices of primes followed by a gap (distance to next larger prime) of 24.

Original entry on oeis.org

263, 327, 574, 615, 641, 697, 804, 834, 869, 909, 938, 987, 1022, 1045, 1127, 1336, 1399, 1421, 1446, 1452, 1551, 1577, 1865, 1883, 1908, 1938, 1939, 1968, 2032, 2064, 2128, 2130, 2176, 2214, 2313, 2508, 2555, 2592, 2612, 2652, 2736, 2762, 2991, 3162, 3243, 3285, 3483, 3489

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A098974.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between.

Equals A000720 o A098974.
Row 12 of A174349.
Indices of 24's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

P:= [seq(ithprime(i),i=1..4001)]:
B:= P[2..-1]-P[1..-2]:
select(t -> B[t]=24, [$1..4000]); # Robert Israel, May 03 2019

A(N=100,g=24,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A098974(n)).

A320710 = { i > 0 | prime(i+1) = prime(i) + 24 }.

A320711 Indices of primes followed by a gap (distance to next larger prime) of 26.

Original entry on oeis.org

367, 446, 732, 1357, 1440, 1475, 1746, 1864, 1912, 1933, 2293, 2714, 2888, 2912, 3159, 3204, 3362, 3523, 3715, 3786, 3801, 3840, 3870, 3920, 3931, 4107, 4164, 4235, 4240, 4502, 4643, 4809, 4957, 4990, 5110, 5371, 5440, 5451, 5581, 5712, 5736, 5743, 5870, 5882, 5906, 5923, 5933, 6018, 6277

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A124594.

Index entries for primes, gaps between.

Equals A000720 o A124594.
Row 13 of A174349.
Indices of 26's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=26,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A124594(n)).

A320711 = { i > 0 | prime(i+1) = prime(i) + 26 }.

A320712 Indices of primes followed by a gap (distance to next larger prime) of 28.

Original entry on oeis.org

429, 462, 685, 781, 1116, 1231, 1274, 1288, 1327, 1392, 1585, 1708, 1710, 1891, 1944, 2065, 2154, 2367, 2417, 2606, 2663, 2729, 2980, 3012, 3069, 3227, 3519, 3653, 3990, 4018, 4168, 4196, 4595, 4603, 4618, 4797, 4856, 4867, 5123, 5191, 5294, 5375, 5432, 5476, 5498, 5593, 5627, 5688, 5703

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A124595.

Index entries for primes, gaps between.

Equals A000720 o A124595.
Indices of 28's in A001223.
Row 14 of A174349.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=28,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A124595(n)).

A320712 = { i > 0 | prime(i+1) = prime(i) + 28 }.

cp's OEIS Frontend

A356222 Array read by antidiagonals upwards where A(n,k) is the position of the k-th appearance of 2n in the sequence of prime gaps A001223. If A001223 does not contain 2n at least k times, set A(n,k) = -1.

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A378620 Lesser prime index of twin primes with nonsquarefree mean.

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A074326 Numbers n such that difference between (1+2^n)-th and (2^n)-th primes is 2.

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A074328 Numbers m such that prime(m^2+1)-prime(m^2)=2, where prime(j) is the j-th prime.

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A173400 n-th difference between consecutive primes=n-th difference between consecutive nonnegative nonprimes.

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A305318 Numbers k such that A071866(k)=3.

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Maple

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A320709 Indices of primes followed by a gap (distance to next larger prime) of 22.

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Magma

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A320710 Indices of primes followed by a gap (distance to next larger prime) of 24.

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