A167493 -id:A167493 - cp's OEIS frontend

A167494 List of first differences of A167493 that are different from 1.

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

Offset: 1

Vladimir Shevelev, Nov 05 2009

nonn

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

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]

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

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 *)

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

A167495 Records in A167494.

Original entry on oeis.org

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

Offset: 1

Vladimir Shevelev, Nov 05 2009

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]

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]

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

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 *)

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

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.

Original entry on oeis.org

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

Offset: 17

Vladimir Shevelev, Nov 19 2009

nonn

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

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.

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

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

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 *)

Corrected by Harvey P. Dale, Aug 15 2017

A168144 First differences of A168143 which are different from 1, incremented by 14.

Original entry on oeis.org

19, 23, 31, 47, 79

Offset: 1

Vladimir Shevelev, Nov 19 2009

nonn

All terms of the sequence are primes greater than 17.

Are there more than 5 terms?

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.

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.

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 *)

Corrected and edited by Eric Rowland, Jan 27 2019

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A167494 List of first differences of A167493 that are different from 1.

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A167495 Records in A167494.

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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.

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A168144 First differences of A168143 which are different from 1, incremented by 14.

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