A001223 -id:A001223 - cp's OEIS frontend

A086980 Late occurring prime gaps in the prime gap sequence A001223.

12, 16, 32, 38, 46, 56, 66, 70, 74, 80, 88, 94, 102, 108, 116, 124, 134, 144, 150, 158, 166, 186, 194, 200, 228, 256, 264, 278, 294, 298, 316, 328, 334, 362, 370, 388, 422, 436, 442, 452, 466, 472, 482, 488, 510, 520, 536, 568, 576, 580, 590, 608, 628, 632

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is the gap g = p_k+1 - p_k between consecutive primes with all even gaps smaller than g occurring at a smaller prime and the next even gap g+2 also occurring earlier.

			16 is in this list because the first time a prime gap of 16 occurs is between consecutive primes 1831 and 1847. All even prime gaps less than 16 occur for a smaller prime. The next even prime gap of 18 also occurs earlier.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 144.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

Cf. A000230, A001223, A001632, A038664, A086977, A086978, A086979, A002386.

A119595 Primes for which the weight as defined in A117078 is 15 and the gap as defined in A001223 is 8.

Original entry on oeis.org

743, 1193, 1523, 1733, 2003, 2243, 2273, 3623, 4583, 4943, 5573, 5693, 6143, 6203, 6473, 7673, 8573, 8933, 9803, 10103, 11243, 11813, 12413, 12503, 13163, 14423, 14843, 15053, 15233, 15383, 16103

Offset: 1

Rémi Eismann, Jun 01 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (30i-7) with i=(level(n)+1)/2, level(n) defined in A117563.

			a(1)=743 because of 751=743+mod(743;15) and g(n)=751-743=8
30*((49+1)/2)-7=743
a(2)=1193 because of 1201=1193+mod(1193;15) and g(n)=1201-1193=8
30*((79+1)/2)-7=1193

Remi Eismann, Table of n, a(n) for n=1..10000

Cf. A117078, A001223, A117563, A001359, A118359, A074822, A118922, A118924, A119504.

A119596 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 10.

Original entry on oeis.org

241, 1627, 2089, 4201, 4663, 4861, 5323, 6247, 6379, 6709, 8821, 9283, 9679, 10141, 12253, 12517, 12781, 13441, 15091, 15289, 15619, 17599, 17929, 19249, 19447, 19843, 21757, 23539, 26839, 28687, 33703, 34429, 34693, 35089, 35353, 36343

Offset: 1

Rémi Eismann, Jun 01 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (22i-1) with i=(level(n)+1)/2, level(n) defined in A117563.

			a(1)=241 because of 251=241+mod(241;11) and 251-241=10.
22*((21+1)/2)-1=241, level=21
a(2)=1627 because of 1637=1627+mod(1627;11) and 1637-1627=10
22*((147+1)/2)-1=1627, level=147

Remi Eismann, Table of n, a(n) for n=1..10000

Cf. A117078, A001223, A117563, A001359, A118359, A074822, A118922, A118924, A119504.

A119597 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 6.

Original entry on oeis.org

61, 677, 941, 1117, 1601, 2063, 2371, 3691, 3911, 4021, 5297, 5407, 6067, 6353, 6991, 7541, 7717, 8311, 8641, 8663, 9103, 9851, 10973, 11897, 12491, 12953, 13591, 13613, 13723, 14537, 15131, 15263, 15307, 15461, 15901, 16363

Offset: 1

Rémi Eismann, Jun 01 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (22i-5) with i=(level(n)+1)/2, level(n) defined in A117563.

			a(1)=61 because of 67=61+mod(61;11) and 67-61=6.
22*((5+1)/2)-5=61, level=5
a(2)=677 because of 683=677+mod(677;11) and 683-677=6
22*((61+1)/2)-5=677, level=5

Remi Eismann, Table of n, a(n) for n=1..10000

Cf. A117078, A001223, A117563, A001359, A118359, A074822, A118922, A118924, A119504.

A130642 Numbers n such that 1 + Sum{k=1..n/2} A001223(2k-1)*(-1)^k = 0.

Original entry on oeis.org

2, 6, 14, 190, 194, 200, 306, 462, 468, 474, 478, 490, 560, 1208, 1890, 1938, 23716, 23850, 25226, 25834, 25968, 26642, 26650, 26998, 48316, 311888, 311922, 313946, 331540, 331762, 331782, 377078, 377518, 377666, 377674, 377748, 378422, 378428

Offset: 1

Manuel Valdivia, Jun 20 2007

Sequence has 170 terms < 10^8.

Being prime(n) = 1 + Sum{k=1..n-1}A000040(k)*(-1)^Floor(k/2), for n/2 odd and, prime(n) = (1 + Sum{k=1..n- 1}A000040(k)*(-1)^Floor(k/2))*(-1), for n/2 even.

			1 + ( -A001223(1)) = 1+(-1) = 0, hence 2 is a term.
1 + ( -A001223(1) + A001223(3) - A001223(5)) = 1+(-1+2-2) = 0, hence 6 is a term.

Cf. A127596, A128039, A001223, A000101, A002386.

S=0; a=0; Do[S=S+(Prime[2*k]-Prime[2*k-1])*(-1)^k; If[1+S==0, a++; Print[a, " ", 2*k]], {k, 1, 10^8, 1}]

A130643 Numbers n such that 1 - Sum{k=1..n/2} A001223(2k-1)*(-1)^k = 0.

Original entry on oeis.org

4, 8, 12, 22, 38, 302, 308, 464, 472, 476, 1186, 1884, 2006, 2026, 2106, 23636, 23656, 23698, 25984, 25990, 26706, 26924, 27000, 311914, 311938, 313866, 313880, 331676, 332002, 332676, 377102, 377634, 377670, 379026, 379090, 379108, 387618, 389076

Offset: 1

Manuel Valdivia, Jun 20 2007

Sequence has 177 terms < 10^8.

Being prime(n) = 1 - Sum{k=1..n-1}A000040(k)*(-1)^Floor(k/2), for n/2 even and, prime(n) = (1 - Sum{k=1..n- 1}A000040(k)*(-1)^Floor(k/2))*(-1), for n/2 odd.

			1 - ( -A001223(1) + A001223(3)) = 1-(-1+2) = 0, hence 4 is a term.
1 - ( -A001223(1) + A001223(3) - A001223(5) + A001223(7)) = 1-(-1+2-2+2) = 0, hence 8 is a term.

Cf. A127596, A128039, A001223, A000101, A002386.

S=0; a=0; Do[S=S+(Prime[2*k]-Prime[2*k-1])*(-1)^k; If[1-S==0, a++; Print[a, " ", 2*k]], {k, 1, 10^8, 1}]

A162188 Numbers k such that A001223(k) > A000005(k).

Original entry on oeis.org

4, 9, 11, 15, 16, 19, 21, 23, 25, 29, 30, 31, 34, 37, 39, 42, 46, 47, 51, 53, 55, 58, 59, 61, 62, 66, 67, 68, 71, 73, 74, 77, 79, 82, 86, 87, 91, 92, 94, 97, 99, 101, 103, 106, 107, 111, 114, 115, 118, 119, 121, 123, 124, 125, 127, 129, 131, 133, 137, 138, 139, 141, 145

Offset: 1

Omar E. Pol, Jul 04 2009

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..5000
Omar E. Pol, Illustration: Divisors and pi(x)

Cf. A000005, A001223, A068526, A161911, A161912, A161913, A162189.

for(n=1,200,if(prime(n+1)-prime(n)>numdiv(n),print1(n","))) \\ Franklin T. Adams-Watters, May 26 2010

More terms from Franklin T. Adams-Watters, May 26 2010

A162189 Numbers k such that A001223(k) < A000005(k).

Original entry on oeis.org

10, 12, 20, 26, 28, 33, 35, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 57, 60, 63, 64, 69, 70, 72, 75, 78, 81, 84, 88, 90, 96, 98, 100, 102, 104, 105, 108, 110, 112, 116, 117, 120, 126, 130, 132, 136, 140, 142, 144, 147, 148, 152, 153, 156, 160, 165

Offset: 1

Omar E. Pol, Jul 04 2009

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Omar E. Pol, Illustration: Divisors and pi(x)

Cf. A000005, A001223, A068526, A161911, A161912, A161913, A162188.

p2:=2:for n from 1 to 200 do p1:=p2:p2:=ithprime(n+1):if(p2-p1Nathaniel Johnston, Apr 15 2011

a(30)-a(56) from Nathaniel Johnston, Apr 15 2011

A196175 Positions of local minima in A001223.

Original entry on oeis.org

5, 7, 10, 13, 17, 20, 22, 26, 28, 31, 33, 35, 38, 41, 43, 45, 49, 52, 57, 60, 64, 67, 69, 75, 78, 81, 83, 85, 89, 93, 95, 98, 100, 104, 109, 113, 116, 120, 122, 126, 131, 134, 136, 138, 140, 142, 144, 148, 152, 155, 159, 163, 167, 169

Offset: 1

Zak Seidov, Oct 27 2011

nonn

Or, numbers n such that A001223(n-1)>A001223(n)<A001223(n+1).
We start with A001223:
S1= 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4.
Local minima are shown in brackets:
S2= 1, 2, 2, {4,2,4}, {4,2,4}, {6,2,6}, {4,2,4}, 6, {6,2,6}, {4,2,6}, {6,4,6}, 8, {4,2,4}, {4,2,4}, {14,4,6}, {6,2,10};
values of local minima are 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, and positions of local minima in A001223 give this sequence. Note that in the first and second brackets we take A001223(6)=4 twice. Also note that all 2's starting with A001223(5) and so on are local minima but there are many other local minima.

			n=5 A001223(4)=4, A001223(5)=2, A001223(6)=4, and A001223(5) is the local minimum;
n=38: A001223(38)=4 is the local minimum because A001223(37)=6 and A001223(39)=6 both > A001223(38).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Zak Seidov, Crests and troughs in A001223

Cf. A001223 (differences between consecutive primes).

Cf. A036263 (second differences of primes).

a196175 n = a196175_list !! (n-1)
a196175_list = map (+ 2) $ elemIndices True $
   zipWith (\x y -> x < 0 && y > 0) a036263_list $ tail a036263_list
-- Reinhard Zumkeller, Oct 29 2011

nn = 1001; t = Differences[Prime[Range[nn]]]; t2 = {}; Do[If[t[[n - 1]] > t[[n]] && t[[n]] < t[[n + 1]], AppendTo[t2, {n, t[[n]]}]], {n, 2, nn - 2}]; Transpose[t2][[1]] (* T. D. Noe, Dec 27 2011 *)

A347101 Fully multiplicative with a(prime(k)) = A001223(k), where A001223 gives the distance from the k-th prime to the (k+1)-th prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 1, 4, 2, 2, 2, 4, 4, 4, 1, 2, 4, 4, 2, 8, 2, 6, 2, 4, 4, 8, 4, 2, 4, 6, 1, 4, 2, 8, 4, 4, 4, 8, 2, 2, 8, 4, 2, 8, 6, 6, 2, 16, 4, 4, 4, 6, 8, 4, 4, 8, 2, 2, 4, 6, 6, 16, 1, 8, 4, 4, 2, 12, 8, 2, 4, 6, 4, 8, 4, 8, 8, 4, 2, 16, 2, 6, 8, 4, 4, 4, 2, 8, 8, 16, 6, 12, 6, 8, 2, 4, 16, 8, 4, 2, 4, 4, 4, 16

Offset: 1

Antti Karttunen, Aug 19 2021

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for primes, gaps between

Cf. A000079 (positions of 1's), A001223, A347102, A347123.

A347101(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = (nextprime(f[i, 1]+1)-f[i,1])); factorback(f); };

For all n >= 0, a(2^n) = 1.

cp's OEIS Frontend

A086980 Late occurring prime gaps in the prime gap sequence A001223.

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A119595 Primes for which the weight as defined in A117078 is 15 and the gap as defined in A001223 is 8.

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A119596 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 10.

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A119597 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 6.

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A130642 Numbers n such that 1 + Sum{k=1..n/2} A001223(2k-1)*(-1)^k = 0.

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Mathematica

A130643 Numbers n such that 1 - Sum{k=1..n/2} A001223(2k-1)*(-1)^k = 0.

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Mathematica

A162188 Numbers k such that A001223(k) > A000005(k).

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PARI

Extensions

A162189 Numbers k such that A001223(k) < A000005(k).

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Maple

Extensions

A196175 Positions of local minima in A001223.

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Haskell