A257762 -id:A257762 - cp's OEIS frontend

A062234 From Bertrand's postulate: a(n) = 2*prime(n) - prime(n+1).

1, 1, 3, 3, 9, 9, 15, 15, 17, 27, 25, 33, 39, 39, 41, 47, 57, 55, 63, 69, 67, 75, 77, 81, 93, 99, 99, 105, 105, 99, 123, 125, 135, 129, 147, 145, 151, 159, 161, 167, 177, 171, 189, 189, 195, 187, 199, 219, 225, 225, 227, 237, 231, 245, 251, 257, 267, 265, 273, 279

Offset: 1

Reinhard Zumkeller, Jun 29 2001

The theorem that a(n) > 0 for all n is known as "Bertrand's Postulate", and was proved by Tchebycheff in 1852.

The analog for Ramanujan primes is Paksoy's theorem that 2*R(n) - R(n+1) > 0 for n > 1. See A233822. - Jonathan Sondow, Dec 16 2013

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (first 1000 terms from Harry J. Smith)

Cf. A000040, A001223, A215808 (prime terms), A233822.
Cf. A098764, A100484, A258383 (run lengths), A258432, A258469, A257762, A258449, A257892, A257951.
When negated, forms the left edge of irregular triangle A252750, and also the leftmost column of square array A372562.

a062234 n = a062234_list !! (n-1)
a062234_list = zipWith (-) (map (* 2) a000040_list) (tail a000040_list)
-- Reinhard Zumkeller, May 31 2015

a:= n-> (p-> 2*p(n)-p(n+1))(ithprime):
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 09 2022

Table[2*Prime[n]-Prime[n+1],{n,60}] (* James C. McMahon, Apr 27 2024 *)
2#[[1]]-#[[2]]&/@Partition[Prime[Range[70]],2,1] (* Harvey P. Dale, Jul 29 2024 *)
ListConvolve[{-1, 2}, Prime[Range[100]]] (* Paolo Xausa, Nov 02 2024 *)

a(n) = 2*prime(n) - prime(n + 1); \\ Harry J. Smith, Aug 03 2009

a(n) = A000040(n) - A001223(n). - Zak Seidov, Sep 07 2012
a(n) = 2*A000040(n) - A000040(n+1). - Zak Seidov, May 12 2020
a(n) = A098764(n) - A000040(n). - Anthony S. Wright, Feb 19 2024

Edited by N. J. A. Sloane, Feb 24 2023

A257951 Numbers n with property that A062234(n)=A062234(n+1)=A062234(n+2)=A062234(n+3)=A062234(n+4).

Original entry on oeis.org

465460, 672832, 829363, 891802, 919088, 1703659, 2656715, 2669971, 3035410, 3223041, 3585960, 3608292, 3636024, 4047253, 4058989, 4232549, 4591286, 4785400, 4797700, 5054313, 5120280, 5599321, 5872369, 6089675, 6541163, 6963642, 7957852, 8234393, 9069087, 9082140, 9312431

Offset: 1

Zak Seidov, May 14 2015

nonn

a(n) = A258432(m), where m such that A258383(m) = 5. - Reinhard Zumkeller, May 31 2015

			For k=465460..465464, 2*prime(k)-prime(k+1)=6824895.
a(1) = A258437(A258432(5)) = 465460.

Subsequence of A257892, A257762 and A062234.

Cf. A258383, A258432, A258437, A258469, A258449, A257951.

a257951 n = a257951_list !! (n-1)
a257951_list = map a258432 $ filter ((== 5) . a258383) [1..]
-- Reinhard Zumkeller, May 31 2015

More terms from Zak Seidov, Jul 29 2015

A257892 Numbers n with property that A062234(n) = A062234(n+1) = A062234(n+2) = A062234(n+3).

Original entry on oeis.org

332, 878, 1999, 3949, 4524, 5953, 6576, 8676, 10068, 11840, 17107, 17208, 19034, 19525, 46771, 46828, 52767, 54567, 54927, 56879, 58695, 61748, 65926, 77168, 77676, 79722, 92775, 92823, 96099, 101607, 111007, 136141, 160095, 160418, 173404

Offset: 1

Zak Seidov, May 14 2015

nonn

a(n) = A258432(m), where m such that A258383(m) = 4. - Reinhard Zumkeller, May 31 2015

			a(1) = A258437(A258432(4)) = 332. - _Reinhard Zumkeller_, May 31 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..110

Subsequence of A257762 and A062234.

Cf. A258383, A258432, A258437, A258469, A258449, A257892, A257951.

a257892 n = a257892_list !! (n-1)
a257892_list = map a258432 $ filter ((== 4) . a258383) [1..]
-- Reinhard Zumkeller, May 31 2015

d(k) = 2*prime(k)-prime(k+1);
isok(n) = (d(n) == d(n+1)) && (d(n+1) == d(n+2)) && (d(n+2) == d(n+3)); \\ Michel Marcus, May 29 2015

use Math::Prime::Util::PrimeArray; tie my @prime,"Math::Prime::Util::PrimeArray"; sub d { 2*$prime[$[0]-1] - $prime[$[0]] } for (1..1e6) { my $dk=d($); say if d($+1)==$dk && d($+2)==$dk && d($+3)==$dk } # Dana Jacobsen, May 31 2015

use ntheory ":all"; { my @prime; sub d { 2*$prime[$[0]-1] - $prime[$[0]] } sub list { my $n=shift; @prime=@{primes(nth_prime($n+5))}; my @d=map{d($)}1..4; my @list; for (1..$n) { push @list,$ if $d[0]==$d[1] && $d[0]==$d[2] && $d[0]==$d[3]; @d=(@d[1..3],d($+4)) } @list; } } say join ", ",list(1e6); # _Dana Jacobsen, May 31 2015

A258437 Smallest number m such that A062234(m) = A062234(m-1+k) for k = 1..n.

Original entry on oeis.org

9, 1, 302, 332, 465460, 67928439

Offset: 1

Reinhard Zumkeller, May 31 2015

From Michel Marcus, Feb 09 2022: (Start)
Previous name: "Smallest number m such that A258383(m) = n" was not ok. For instance, for a(1) the smallest m such that A258383(m)=1 is 5, then we have to sum up the first 5 terms 2+2+2+2+1 to get 9, as shown in the example table (whose 2nd and 3rd column names I edited too).
Note that prime([302, 332, 465460]) = [1997, 2237, 6824897] which is a subsequence of A090807. Then one can verify that primepi(1356705137 = A090807(7)) = 67928439 and primepi(3637803390827 = A090807(8)) = 130463972798 are good candidates for a(6) and a(7). a(6) has been confirmed by program. (End)

			   n |   f(n) | a(n) = A258432(f(n)) |     Run in A062234
  ---+--------+----------------------+--------------------------
   1 |      5 |       9 = A258469(1) | [17]
   2 |      1 |       1 = A257762(1) | [1, 1]
   3 |    265 |     302 = A258449(1) | [1995, 1995, 1995]
   4 |    290 |     332 = A257892(1) | [2235, 2235, 2235, 2235]
   5 | 440676 |  465460 = A257951(1) | [ ___ 5 x 6824895 ___ ]

Cf. A062234, A258383, A258432, A258469, A257762, A258449, A257892, A257951.

Cf. A090807.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258437 = (+ 1) . fromJust . (`elemIndex` a258383_list)

f(n) = 2*prime(n) - prime(n+1); \\ A062234
lista(nn) = {my(vp=primes(nn)); my(v=vector(nn-1, k, 2*vp[k] - vp[k+1]), last=v[1], nb=1, list=List()); kill(vp); for (n=2, nn-1, if (v[n]==last, nb++, listput(list, nb); last=v[n]; nb=1);); Vec(list);} \\ A258383
find(k, v) = {my(i=1); while (v[i] != k, i++); i;}
listr(nn) = {my(v=lista(nn)); for (k=1, 6, my(pos = find(k, v)); print1(sum(i=1, pos, v[i])- k + 1, ", "););}
listr(9*10^7) \\ Michel Marcus, Feb 09 2022

A258383(a(n)) = n and A258383(m) != n for m < a(n);

let m = A258432(a(n)): A062234(m) = A062234(m-1+k) for k = 1..n.

New name and a(6) from Michel Marcus, Feb 09 2022

A258449 Numbers n such that A062234(n) = A062234(n+1) = A062234(n+2).

Original entry on oeis.org

302, 336, 384, 805, 1016, 1043, 1963, 2201, 2364, 2398, 2495, 2506, 2528, 2574, 2683, 2734, 3208, 4267, 4561, 4659, 5234, 5415, 5525, 5620, 5759, 5903, 6044, 6258, 6543, 7737, 7928, 8019, 8039, 8115, 8521, 8717, 8833, 9056, 9165, 9379, 9730, 10302, 10495

Offset: 1

Reinhard Zumkeller, May 31 2015

nonn

a(n) = A258432(m), where m such that A258383(m) = 3.

			a(1) = A258437(A258432(3)) = 302.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A062234, A258383, A258432, A258437, A258469, A257762, A257892, A257951.

a258449 n = a258449_list !! (n-1)
a258449_list = map a258432 $ filter ((== 3) . a258383) [1..]

A258469 Numbers m such that A062234(m) != A062234(m+1).

Original entry on oeis.org

9, 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 80, 81, 82, 83, 84, 85, 86, 87, 88, 91

Offset: 1

Reinhard Zumkeller, May 31 2015

nonn

a(n) = A258432(m), where m such that A258383(m) = 1.

			a(1) = A258437(A258432(1)) = 9.

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

Cf. A062234, A258383, A258432, A258437, A257762, A258449, A257892, A257951.

a258469 n = a258469_list !! (n-1)
a258469_list = map a258432 $ filter ((== 1) . a258383) [1..]

A303112 Primes p such that (r-q)/(q-p) = 2 or 1/2, and p < q < r are three consecutive primes.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 37, 41, 67, 89, 97, 101, 103, 107, 191, 193, 223, 227, 277, 307, 311, 347, 389, 397, 449, 457, 461, 479, 487, 491, 503, 613, 641, 739, 757, 761, 821, 823, 853, 857, 877, 881, 907, 929, 991, 1087, 1091, 1231, 1277, 1297, 1301, 1423, 1427, 1439, 1447, 1453

Offset: 1

Andres Cicuttin, Apr 18 2018

nonn

Conjecture: The two most frequent ratios between consecutive prime gaps are 2 and 1/2, and both ratios occur with about the same frequency.

			The first three consecutive primes are 2, 3 and 5, and (5-3)/(3-2)=2, so the first term is a(1)=2, that is, the first prime of (2,3,5).
The next three consecutive primes are 3, 5 and 7, and (7-5)/(5-3)=1, so the first prime of (3,5,7) is not in the list.
The next three consecutive primes are 5, 7 and 11, and (11-7)/(7-5)=2, so the second term is a(2)=5, that is, the first prime of (5,7,11).
The prime 13 is also in the list because (19-17)/(17-13)=1/2.

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

Cf. A001223, A274263, A276309, A022885.

Cf. A257762 (indices of primes with above ratio = 2).

b={};
Do[If[Abs[Log[2,(Prime[j+2]-Prime[j+1])/(Prime[j+1]-Prime[j])]]==1,AppendTo[b,Prime[j]]],{j,1,200}];
Print@b
Select[Partition[Prime[Range[250]],3,1],(#[[3]]-#[[2]])/(#[[2]]-#[[1]]) == 2||(#[[3]]-#[[2]])/(#[[2]]-#[[1]])==1/2&][[All,1]] (* Harvey P. Dale, Mar 14 2022 *)

isok(p) = my(q = nextprime(p+1), r = nextprime(q+1), f = (r-q)/(q-p)); (f == 2) || (f == 1/2);
forprime(p=2, 1000, if (isok(p), print1(p, ", "))); \\ Michel Marcus, Apr 23 2018

Conjecture: lim_{n->inf} n/primepi(a(n)) > k > 0 for some k.

cp's OEIS Frontend

A062234 From Bertrand's postulate: a(n) = 2*prime(n) - prime(n+1).

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A257951 Numbers n with property that A062234(n)=A062234(n+1)=A062234(n+2)=A062234(n+3)=A062234(n+4).

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A257892 Numbers n with property that A062234(n) = A062234(n+1) = A062234(n+2) = A062234(n+3).

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A258437 Smallest number m such that A062234(m) = A062234(m-1+k) for k = 1..n.

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A258449 Numbers n such that A062234(n) = A062234(n+1) = A062234(n+2).

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A258469 Numbers m such that A062234(m) != A062234(m+1).

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A303112 Primes p such that (r-q)/(q-p) = 2 or 1/2, and p < q < r are three consecutive primes.

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