Rodolfo Ruiz-Huidobro - cp's OEIS frontend

A377465 Exponents of Mersenne primes that are emirps.

13, 17, 31, 107, 1279, 9941, 3021377, 13466917

Offset: 1

Rodolfo Ruiz-Huidobro, Oct 29 2024

The just discovered Mersenne exponent 136279841 is also an emirp. However, it cannot be included as the next member of the sequence as there are Mersenne exponents over 70000000 that have not been double checked yet.

			107 is a term because it is in A000043 and in A006567.

Eric Weisstein's World of Mathematics, Emirp.
Wikipedia, Emirp.

Intersection of A000043 and A006567.

Select[MersennePrimeExponent[Range[48]], PrimeQ[p=FromDigits[Reverse[IntegerDigits[#]]]] && p!=# &] (* Stefano Spezia, Nov 02 2024 *)

A330414 Cyclops primes that become a cube when the middle "0" is removed.

Original entry on oeis.org

68059, 1170649, 4560533, 7530571, 136501919, 158103251, 173703979, 212503933, 226605187, 356101289, 362604691, 382702753, 439806977, 518905117, 811802737, 954403993, 19484041249, 19956016979, 22635071297, 24658046551, 27263097773, 34635012697, 35326042667, 37166072149, 39668022287, 41499095543, 44839062449

Offset: 1

Rodolfo Ruiz-Huidobro, Dec 14 2019

			a(1) = 68059 because 6859 = 19^3 is the first cube that results from the removal of the 0 digit from a cyclops prime.
136501919 is a term because 13651919 is 239^3.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 60 terms from Rodolfo Ruiz-Huidobro)

Cf. A329737, A134809, A016755.

count:= 0: Res:= NULL:
for d from 2 to 6 do
  for n from ceil(10^((2*d-1)/3)) to floor((10^(2*d)-1)^(1/3)) do
    L:=convert(n^3,base,10);
    if member(0,L) then next fi;
    a:= n^3 mod 10^d;
    p:= 10*(n^3-a)+a;
    if isprime(p) then
      count:= count+1; Res:= Res, p;
    fi
od od:
Res; # Robert Israel, Dec 24 2019

seq(n)={my(i=0, L=List()); while(#Lt==0,v), my(m=fromdigits(concat([v[1..k], 0, v[k+1..#v]]))); if(isprime(m), listput(L,m)))); Vec(L)} \\ Andrew Howroyd, Dec 20 2019

A329737 Cyclops primes that remain prime after being "blinded".

Original entry on oeis.org

101, 103, 107, 109, 307, 401, 503, 509, 601, 607, 701, 709, 809, 907, 11071, 11087, 11093, 12037, 12049, 12097, 13099, 14029, 14033, 14051, 14071, 14081, 14083, 14087, 15031, 15053, 15083, 16057, 16063, 16067, 16069, 16097, 17021, 17033, 17041, 17047, 17053

Offset: 1

Rodolfo Ruiz-Huidobro, Nov 20 2019

There are 14 of these primes with 3 digits and 302 with 5 digits.

			The first term, a(1), is 101 because if you remove the "cyclops' eye" it remains a prime (11) and because 101 is the 1st cyclops prime.
307 is a term because when you remove the "0" it remains a prime: 37.

Rodolfo Ruiz-Huidobro, Table of n, a(n) for n = 1..3194

Intersection of A256186 and A134809.

a:=[]; f:=func; g:=func; for n in [1..20000] do if f(n) and IsPrime(g(n)) then Append(~a,n); end if; end for; a; // Marius A. Burtea, Nov 20 2019

A307176 Number of Sophie Germain primes of the form 4k + 1 less than 10^n.

Original entry on oeis.org

1, 5, 17, 89, 589, 3833, 27940, 211439, 1653257, 13283194, 109058142, 911411528, 7731354496

Offset: 1

Rodolfo Ruiz-Huidobro, Mar 27 2019

Sophie Germain primes can alternatively be Lucasian primes, primes of the form 4k + 1, or, the individual prime 2.

			There are five Sophie Germain Primes of the form 4k + 1 below 10^2: {5, 29, 41, 53, 89}, therefore a(2) = 5.

Cf. A091098, A092816, A002515, A307121, A002144, A005384, A103579.

nonLucSophies = Select[4Range[2500000] + 1, PrimeQ[#] && PrimeQ[2# + 1] &]; Table[Length[Select[nonLucSophies, # < 10^n &]], {n, 0, 7}]

a(n) < A092816(n).
a(n) <= A091098(n) (with equality for n = 1).
a(n) = A092816(n) - A307121(n) - 1.

A307121 Number of Lucasian primes less than 10^n.

Original entry on oeis.org

1, 4, 19, 100, 581, 3912, 28091, 211700, 1655601, 13286320, 109058381, 911436949, 7731247492

Offset: 1

Rodolfo Ruiz-Huidobro, Mar 26 2019

The Lucasian primes are those Sophie Germain primes of the form 4k + 3. They are interesting because if they are infinite in number, then the sequence of Mersenne numbers with prime exponents contains an infinite number of composite integers.

Conjecture: about half of all Sophie Germain primes are Lucasian primes, and the rest are either 2 or a prime of the form 4k + 1.

			There are 4 Lucasian primes below 10^2: {3,11,23,83}, therefore a(2) = 4.

Simon Davis, Arithmetical sequences for the exponents of composite Mersenne numbers, Notes on Number Theory and Discrete Mathematics 20, no. 1 (2014): 19-26.

Cf. A092816, A002515, A307176.

c = 0; r = 10; s = {}; Do[If[p > r, AppendTo[s, c]; r *= 10]; If[PrimeQ[p] && PrimeQ[2p + 1], c++], {p, 3, 1000003, 4}]; s (* Amiram Eldar, Mar 27 2019 *)
lucSophies = Select[4Range[2500000] - 1, PrimeQ[#] && PrimeQ[2# + 1] &]; Table[Length[Select[lucSophies, # < 10^n &]], {n, 0, 7}]

a(n) = { my(t=0); forprime(p=2,10^n,p%4==3 && ispseudoprime(1+(2*p)) && t++);t } \\ Dana Jacobsen, Mar 28 2019

use ntheory ":all"; sub a { my($n,$t)=(shift,0); forprimes { $t++ if ($&3) == 3 && is_prime(1+($<<1)) } 10**$n; $t; } # Dana Jacobsen, Mar 28 2019

a(9)-a(11) from Amiram Eldar, Mar 27 2019
a(12) from Amiram Eldar, Mar 31 2019
a(13) from Dana Jacobsen, Apr 02 2019

A316917 Let g(n) be the n-th maximal prime gap; a(n) = 1 if g(n) has record merit, 0 if it does not.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Rodolfo Ruiz-Huidobro, Jul 16 2018

a(n) = 1 if A002386(n) is in A111870, a(n) = 0 if A002386(n) is not in A111870.
The merit M of a prime gap of measure g following the prime p_1 is defined as M=g/ln(p_1). It is the ratio of the measure of the gap to the "average" measure of gaps near that point. As an example, the merit of the sixth maximal gap, of size 14, after prime 113 is 2.96.
a(81) = 0 because there are previous maximal gaps with higher merits. - Rodolfo Ruiz-Huidobro, Jan 23 2024
a(82) = 1 as the merit of the gap is 1572/log(18571673432051830099)=1572/44.37=35.43 (which is a record merit). - Rodolfo Ruiz-Huidobro, May 10 2024
a(83) =1 as the merit for the gap is 1676/log(20733746510561444539) =1676/44.48=37.681 (which is a record merit). - Rodolfo Ruiz-Huidobro, Dec 20 2024

			The 5th record prime gap from 89 to 97 does not have record merit, so a(5) = 0.
The 10th record prime gap from 1327 to 1361 has record merit, so a(10) = 1.

Prime Gap List Community, Record prime gaps, 2021.
Index entries for characteristic functions

Cf. A000101, A002386, A111870, A111871, A005250.

Block[{nn = 10^6, s, t, u, v}, s = Prime@ Range[nn]; t = Differences@ s; u = Map[(#2 - #1)/Log[#1] & @@ # &, Partition[Prime@ Range[nn], 2, 1]]; v = Map[Prime@ FirstPosition[u, #][[1]] &, Union@ FoldList[Max, u]]; Boole[! FreeQ[v, s[[FirstPosition[t, #][[1]] ]] ] ] & /@ Union@ FoldList[Max, t]] (* Michael De Vlieger, Jul 19 2018 *)

a(81) from Rodolfo Ruiz-Huidobro, Jan 23 2024
a(82) from Rodolfo Ruiz-Huidobro, May 10 2024
a(83) from Rodolfo Ruiz-Huidobro, Dec 09 2024

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User: Rodolfo Ruiz-Huidobro

A377465 Exponents of Mersenne primes that are emirps.

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A330414 Cyclops primes that become a cube when the middle "0" is removed.

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A329737 Cyclops primes that remain prime after being "blinded".

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A307176 Number of Sophie Germain primes of the form 4k + 1 less than 10^n.

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A307121 Number of Lucasian primes less than 10^n.

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A316917 Let g(n) be the n-th maximal prime gap; a(n) = 1 if g(n) has record merit, 0 if it does not.

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