A167493 -id:A167493

Search: a167493 -id:a167493

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A167495

Records in A167494.

+0
8

2, 3, 5, 13, 31, 61, 139, 283, 571, 1153, 2311, 4651, 9343, 19141, 38569, 77419, 154873, 310231, 621631, 1243483, 2486971, 4974721

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Conjecture: each term > 3 of the sequence is the greater member of a twin prime pair (A006512).

Indices of the records are 1, 2, 4, 6, 9, 10, 15, 18, 21, 25, 28, 30, 38, 72, 90, ... [R. J. Mathar, Nov 05 2009]

One can formulate a similar conjecture without verification of the primality of the terms (see Conjecture 4 in my paper). [Vladimir Shevelev, Nov 13 2009]

LINKS

Table of n, a(n) for n=1..22.

E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol.11 (2008), Article 08.2.8.

E. S. Rowland, A natural prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.

V. Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.

V. Shevelev, Three theorems on twin primes, arXiv:0911.5478 [math.NT], 2009-2010. [Vladimir Shevelev, Dec 03 2009]

MATHEMATICA

nxt[{n_, a_}] := {n + 1, If[EvenQ[n], a + GCD[n+1, a], a + GCD[n-1, a]]};

A167494 = DeleteCases[Differences[Transpose[NestList[nxt, {1, 2}, 10^7]][[2]]], 1];

Tally[A167494][[All, 1]] //. {a1___, a2_, a3___, a4_, a5___} /; a4 <= a2 :> {a1, a2, a3, a5} (* Jean-François Alcover, Oct 29 2018, using Harvey P. Dale's code for A167494 *)

CROSSREFS

Cf. A167494, A167493, A167197, A167195, A167170, A167168, A106108, A132199, A167054, A167053, A166944, A166945, A116533, A163961, A163963, A084662, A084663, A134162, A135506, A135508, A118679, A120293.

KEYWORD

nonn,more

AUTHOR

Vladimir Shevelev, Nov 05 2009

EXTENSIONS

Simplified the definition to include all records; one term added by R. J. Mathar, Nov 05 2009

a(16) to a(21) from R. J. Mathar, Nov 19 2009

a(22) from Jean-François Alcover, Oct 29 2018

STATUS

approved

A167494

List of first differences of A167493 that are different from 1.

+0
5

2, 3, 3, 5, 3, 13, 5, 3, 31, 61, 7, 5, 3, 7, 139, 5, 3, 283, 5, 3, 571, 7, 5, 3, 1153, 5, 3, 2311, 31, 4651, 17, 5, 13, 3, 3, 5, 3, 9343, 5, 3, 11, 3, 59, 3, 29, 3, 19, 7, 5, 3, 7, 19, 5, 3, 17, 3, 113

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Conjecture. All terms of the sequence are primes.

The conjecture is false: a(144)=27, a(146)=25, a(158)=45, etc., which are composite numbers. - Harvey P. Dale, Dec 05 2015

LINKS

Michel Marcus, Table of n, a(n) for n = 1..211

E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8.

V. Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.

V. Shevelev, Three theorems on twin primes, arXiv:0911.5478 [math.NT], 2009-2010. [From Vladimir Shevelev, Dec 03 2009]

MATHEMATICA

nxt[{n_, a_}]:={n+1, If[EvenQ[n], a+GCD[n+1, a], a+GCD[n-1, a]]}; DeleteCases[ Differences[ Transpose[NestList[nxt, {1, 2}, 20000]][[2]]], 1] (* Harvey P. Dale, Dec 05 2015 *)

PROG

(PARI) lista(nn) = {my(va = vector(nn)); va[1] = 2; for (n=2, nn, va[n] = if (n%2, va[n-1] + gcd(n, va[n-1]), va[n-1] + gcd(n-2, va[n-1])); ); select(x->(x!=1), vector(nn-1, n, va[n+1] - va[n])); } \\ Michel Marcus, Dec 13 2018

CROSSREFS

Cf. A167493, A167197, A167195, A167170, A167168, A106108, A132199, A167054, A167053, A166944, A166945, A116533, A163961, A163963, A084662, A084663, A134162, A135506, A135508, A118679, A120293.

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Nov 05 2009

STATUS

approved

A168144

First differences of A168143 which are different from 1, incremented by 14.

+0
0

19, 23, 31, 47, 79

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

All terms of the sequence are primes greater than 17.

Are there more than 5 terms?

LINKS

Table of n, a(n) for n=1..5.

E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, 11 (2008), Article 08.2.8.

V. Shevelev, An infinite set of generators of primes based on the Rowland idea and conjectures concerning twin primes, arXiv:0910.4676 [math.NT], 2009.

V. Shevelev, Generalizations of the Rowland theorem, arXiv:0911.3491 [math.NT], 2009-2010.

MATHEMATICA

A168143[17] = 37;

A168143[n_] := A168143[n] = If[GCD[n, A168143[n - 1]] > 1 && FactorInteger[n][[1, 1]] > 17, 3 n - 14, A168143[n - 1] + 1]

DeleteCases[Differences[A168143 /@ Range[17, 100]], 1] + 14 (* Eric Rowland, Jan 27 2019 *)

CROSSREFS

Cf. A168143, A167495, A167494, A167493, A167197, A167195, A167170, A167168, A106108, A132199, A167054, A167053, A166944, A166945, A116533, A163961, A163963, A084662, A084663, A134162, A135506, A135508, A118679, A120293.

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Nov 19 2009

EXTENSIONS

Corrected and edited by Eric Rowland, Jan 27 2019

STATUS

approved

A168143

a(17)=37; for n>=17, a(n)=3n-14 if gcd(n,a(n-1))>1 and all prime divisors of n more than 17; a(n)=a(n-1)+1, otherwise

+0
2

37, 38, 43, 44, 45, 46, 55, 56, 57, 58, 59, 60, 61, 62, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157

(list; graph; refs; listen; history; text; internal format)

OFFSET

17,1

COMMENTS

a(n+1)-a(n)+14 is either 15 or a prime > 17. For a generalization, see the second Shevelev link. - Edited by Robert Israel, Aug 21 2017

LINKS

Robert Israel, Table of n, a(n) for n = 17..10000

E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8.

V. Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.

V. Shevelev, Generalizations of the Rowland theorem, arXiv:0911.3491 [math.NT], 2009-2010.

MAPLE

A[17]:= 37:

q:= convert(select(isprime, [$2..17]), `*`);

for n from 18 to 100 do

if igcd(n, A[n-1]) > 1 and igcd(n, q) = 1 then A[n]:= 3*n-14

else A[n]:= A[n-1]+1 fi

od:

seq(A[i], i=17..100); # Robert Israel, Aug 21 2017

MATHEMATICA

nxt[{n_, a_}]:={n+1, If[GCD[n+1, a]>1&&FactorInteger[n+1][[1, 1]]>17, 3(n+1)-14, a+1]}; NestList[nxt, {17, 37}, 60][[All, 2]] (* Harvey P. Dale, Aug 15 2017 *)

CROSSREFS

Cf. A167495, A167494, A167493, A167197, A167195, A167170, A167168, A106108, A132199, A167054, A167053, A166944, A166945, A116533, A163961, A163963, A084662, A084663, A134162, A135506, A135508, A118679, A120293.

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Nov 19 2009

EXTENSIONS

Corrected by Harvey P. Dale, Aug 15 2017

STATUS

approved

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