A062234 -id:A062234 - cp's OEIS frontend

A258383 Run lengths of consecutive identical terms in A062234.

2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, May 31 2015

nonn

a(A258437(n)) = n and a(m) != n for m < A258437(n). [Because of recent changes that were made to A258437, this comment may now need to be changed. - N. J. A. Sloane, Oct 31 2024]

			See A258432.

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

Cf. A062234, A258432, A258437.

import Data.List (group)
a258383 n = a258383_list !! (n-1)
a258383_list = map length $ group a062234_list

Map[Length, Most[Split[ListConvolve[{-1, 2}, Prime[Range[200]]]]]] (* Paolo Xausa, Oct 30 2024 *)

A258432 Smallest number m such that A062234(m) = A062234(m-1+k) for k = 1..A258383(n).

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Reinhard Zumkeller, May 31 2015

nonn

			.       n | 1 2   3 4   5 6    7  8    9   10   11   12   13 14   15
. A062234 | 1 1   3 3   9 9   15 15   17   27   25   33   39 39   41
. --------+-----+-----+-----+-------+----+----+----+----+-------+----+-
. A258383 | 2   | 2   | 2   | 2     | 1  |  1 |  1 |  1 |  2    |  1 |
. A258432 | 1   | 3   | 5   | 7     | 9  | 10 | 11 | 12 | 13    | 15 |
. run #   | 1   | 2   | 3   | 4     | 5  |  6 |  7 |  8 |  9    | 10 |

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

Cf. A062234, A258383.

import Data.List (groupBy); import Data.Function (on))
a258432 n = a258432_list !! (n-1)
a258432_list = map (snd . head) $
                    groupBy ((==) `on` fst) $ zip a062234_list [1..]

A257762 Numbers n with property that A062234(n) = A062234(n+1).

Original entry on oeis.org

1, 3, 5, 7, 13, 26, 28, 43, 49, 64, 69, 78, 89, 93, 96, 116, 131, 134, 142, 148, 152, 155, 167, 182, 202, 206, 212, 225, 231, 234, 236, 238, 247, 253, 258, 281, 286, 302, 303, 311, 313, 330, 332, 333, 334, 336, 337, 356, 362, 384, 385, 390, 435, 438, 455, 458, 484, 492, 512, 516

Offset: 1

Zak Seidov, May 07 2015

nonn

Numbers n with property that 2*prime(n)-prime(n+1) = 2*prime(n+1)-prime(n+2), or 2*prime(n)+prime(n+2) = 3*prime(n+1).
Numbers n with property that 2*A001223(n) = A001223(n+1). - Gionata Neri, May 22 2015
a(n) = A258432(m), where m such that A258383(m) = 2. - Reinhard Zumkeller, May 31 2015

			a(1) = A258437(A258432(2)) = 1.

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

Cf. A258383, A258432, A258437, A258469, A257762, A258449, A257892, A257951.

a257762 n = a257762_list !! (n-1)
a257762_list = map a258432 $ filter ((== 2) . a258383) [1..]
-- Reinhard Zumkeller, May 31 2015

[n: n in [0..600] | 2*NthPrime(n)-NthPrime(n+1) eq 2*NthPrime(n+1)-NthPrime(n+2)]; // Vincenzo Librandi, May 12 2015

Primes:= select(isprime,[2,(2*i+1 $ i=1..10^4)]):
Gaps:= Primes[2..-1] - Primes[1..-2]:
G2:= Gaps[2..-1] - 2*Gaps[1..-2]:
ListTools:-SearchAll(0,G2); # Robert Israel, May 22 2015

Select[Range@ 600, 2 Prime[#] - Prime[# + 1] == 2 Prime[# + 1] - Prime[# + 2] &] (* Michael De Vlieger, May 11 2015 *)

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..]

A111005 Terms in A062234 that are powers > 1.

Original entry on oeis.org

9, 9, 27, 25, 81, 125, 225, 225, 361, 441, 1089, 1089, 1225, 1225, 2025, 2025, 2601, 3249, 3249, 3375, 3721, 5041, 6859, 7569, 7921, 12321, 13689, 13689, 15129, 18225, 21609, 21609, 30625, 31329, 38809, 42025, 47961, 53361, 59049, 65025, 65025

Offset: 1

Walter Kehowski, Oct 02 2005

nonn

			a(20) = 2*prime(478) - prime(479) = 2*3391 - 3407 = 3375 = 15^3.

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

Cf. A062234.

with(numtheory); egcd := proc(n) local L; L:=map(proc(z) z[2] end, ifactors(n)[2]); igcd(op(L)) end: PW:=[]: cnt:=0: for z to 1 do for j from 1 to 50000 do x:=2*ithprime(j)-ithprime(j+1); if egcd(x)>1 then cnt:=cnt+1; PW:=[op(PW),[cnt,j,x]] fi od od; PW; map(proc(z) z[3] end, PW);

fn[n_]:=2*Prime[n]-Prime[n+1];Select[fn[Range[3,7000]],ResourceFunction["PerfectPowerQ"][#]&] (* James C. McMahon, Apr 27 2024 *)

lista(pmax) = {my(prev = 2, k); forprime(p = 3, pmax, k = 2*prev - p; if(ispower(k), print1(k, ", ")); prev = p);} \\ Amiram Eldar, Aug 02 2024

Title modified by James C. McMahon, Apr 27 2024

A256274 Numbers k such that all values of A062234(k+i) are equal for i = 0..5, where A062234(n) = 2*prime(n) - prime(n+1).

Original entry on oeis.org

67928439, 266947136, 282959564, 290650720, 411522246, 412395369, 442605172, 469294445, 474377829, 484820862, 516574437, 540188314, 566750674, 611253120, 656844834, 717566734, 835153917, 930156158, 1195351247, 1463919418, 1524111225, 1684626377, 1744328226, 1785356368, 2025764074

Offset: 1

Zak Seidov, Jun 02 2015

nonn

Subsequence of A257951: a(1) = 67928439 = A257951(166).

Cf. A062234, A257951.

a(12)-a(25) from Michel Marcus, Feb 14 2022

cp's OEIS Frontend

A258383 Run lengths of consecutive identical terms in A062234.

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A258432 Smallest number m such that A062234(m) = A062234(m-1+k) for k = 1..A258383(n).

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A257762 Numbers n with property that A062234(n) = A062234(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).