A290489 -id:A290489 - cp's OEIS frontend

A290488 Lower ends of record gaps between numbers that are either prime or twice a prime.

2, 7, 47, 181, 467, 526, 1069, 1262, 1327, 2477, 3274, 5119, 12854, 20809, 28229, 31397, 338038, 520462, 546391, 1050631, 1761206, 1964987, 3466943, 6084986, 27348829, 27915754, 111108769, 113652443, 230126378, 231902267, 327764081, 438981007, 581755318, 1837981547, 2489382697

Offset: 1

Bobby Jacobs, Aug 03 2017

nonn

The gap between 31397 and 31454 is due to the record prime gap between 31397 and 31469 being almost exactly twice the record prime gap between 15683 and 15727.

			a(3) = 47 because the next number that is a prime or twice a prime after 47 is 53, and that is a record gap of size 6.

Robert G. Wilson v, Table of n, a(n) for n = 1..37

Cf. A001751, A290489, A290541.

nxt[n_] := Block[{k = n +1}, While[ !PrimeQ[ k] && ! PrimeQ[ k/2], k++]; k]; p = 2; q = 3; mx = 0; lst = {}; While[p < 2500000000, If[q > mx + p, mx = q - p; AppendTo[ lst, p]]; p = q; q = nxt@ q]; lst (* Robert G. Wilson v, Aug 03 2017 *)

a(n) = A290489(n) - A290541(n).

a(11) onward from Robert G. Wilson v, Aug 03 2017

A290541 Record gaps between numbers that are either prime or twice a prime.

Original entry on oeis.org

1, 3, 6, 10, 11, 12, 13, 15, 19, 21, 25, 28, 35, 40, 48, 57, 60, 64, 70, 75, 83, 90, 95, 117, 120, 144, 148, 150, 153, 167, 168, 196, 205, 212, 214, 221, 234, 244, 254, 255

Offset: 1

Bobby Jacobs, Aug 05 2017

Records in A290496.

The gap of 57 between 31397 and 31454 is due to the record prime gap between 31397 and 31469 being almost exactly twice the record prime gap between 15683 and 15727.

			a(3) = 6 because the next number that is a prime or twice a prime after 47 is 53, and that is a record gap of size 6.

Cf. A001751, A290488, A290489, A290496.

With[{nn = 10^7}, Union@ FoldList[Max, Differences@ #] &@ Union@ Flatten@ {#, 2 TakeWhile[#, # < Prime[nn]/2 &]} &@ Prime@ Range@ nn] (* Michael De Vlieger, Aug 06 2017 *)

a(n) = A290489(n) - A290488(n).

a(36)-a(40) from Giovanni Resta, Aug 06 2017

A290496 First differences of A001751.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 2, 1, 3, 2, 3, 1, 3, 3, 2, 3, 3, 1, 3, 2, 3, 1, 6, 5, 1, 2, 1, 5, 4, 2, 1, 5, 3, 1, 3, 3, 5, 3, 4, 2, 3, 1, 2, 4, 5, 4, 5, 4, 3, 3, 2, 3, 4, 3, 2, 6, 1, 5, 3, 1, 6, 5, 1, 2, 10, 2, 1, 3, 2, 3, 4, 5, 3, 4, 5, 3, 1, 2, 4, 6, 2, 10, 3, 3, 5, 1

Offset: 1

Michel Marcus, Aug 04 2017

nonn

Cf. A001751, A290488, A290489, A290541.

With[{nn = 56}, Differences@ Union@ Flatten@ {#, 2 TakeWhile[#, # < Prime[nn]/2 &]} &@ Prime@ Range@ nn]

lista(nn) = {last = 2; for (n=3, nn, if (isprime(n) || (!(n%2) && isprime(n/2)), print1(n - last, ", "); last = n;););}

A290572 Least number that is the start of a gap of size n between numbers that are either prime or twice a prime (A001751).

Original entry on oeis.org

2, 11, 7, 67, 53, 47, 514, 401, 317, 181, 467, 526, 1069, 2819, 1262, 3142, 1382, 1913, 1327, 4178, 2477, 9697, 8123, 8329, 3274, 11213, 21031, 5119, 16382, 13063, 20446, 44417, 22193, 37747, 12854, 46957, 35617, 63863, 48679, 20809, 76166, 39251, 110359, 59282, 136898, 212923, 143006

Offset: 1

Bobby Jacobs and Robert G. Wilson v, Aug 06 2017

nonn

Numbers that are less than any later number are recorded in A290488.

			a(1) is  2 since  3 -  2 = 1;
a(2) is 11 since 13 - 11 = 2;
a(3) is  7 since 10 -  7 = 3;
a(4) is 67 since 71 - 67 = 4; etc.

Robert G. Wilson v, Table of n, a(n) for n = 1..261
Robert G. Wilson v, Table of n, a(n) for n = 0..290, or 0 if no such value is known.

Cf. A001751, A002386, A290488, A290489.

nxt[n_] := Block[{k = n +1}, While[ !PrimeQ[k] && !PrimeQ[k/2], k++]; k]; p = 2; q = 3; t[_] = 0; While[p < 215000, d = q - p; If[ t[d] == 0, t[d] = p]; p = q; q = nxt@ q]; t@# & /@ Range@ 47

cp's OEIS Frontend

A290488 Lower ends of record gaps between numbers that are either prime or twice a prime.

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A290541 Record gaps between numbers that are either prime or twice a prime.

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A290496 First differences of A001751.

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PARI

A290572 Least number that is the start of a gap of size n between numbers that are either prime or twice a prime (A001751).

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