A138755 -id:A138755 - cp's OEIS frontend

A138716 Primes p2 such that p1^2 + p2^3 is an average of twin primes and p1 < p2 are consecutive primes.

29, 107, 1481, 1613, 2393, 2879, 4421, 5021, 5519, 5573, 6269, 7817, 8447, 9629, 11489, 11981, 12011, 17159, 17573, 18461, 19961, 21713, 23021, 23291, 23747, 24917, 26339, 27947, 29021, 29201, 29663, 30893, 32063, 32717, 34217, 34589, 35159, 36527, 36899, 44753

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 15 2008

nonn

			29 is a term since 23 and 29 are consecutive primes, 23^2 + 29^3 = 24918, and (24917, 24919) are twin primes.

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

Cf. A001097, A001359, A006512, A014574.

Cf. A138755, A139777, A139780.

[NthPrime(k+1):k in [1..5000]| IsPrime(q-1) and IsPrime(q+1) where q is NthPrime(k)^2+ NthPrime(k+1)^3]; // Marius A. Burtea, Dec 22 2019

a={};Do[p1=Prime[n];p2=Prime[n+1];pp=p1^2+p2^3;If[PrimeQ[pp-1]&&PrimeQ[pp+1],AppendTo[a,p2]],{n,16^3}];Print[a];
Select[Partition[Prime[Range[5000]],2,1],AllTrue[#[[1]]^2+#[[2]]^3+{1,-1},PrimeQ]&] [[All,2]] (* Harvey P. Dale, Oct 29 2022 *)

More terms from Amiram Eldar, Dec 22 2019

A139777 Average of twin primes p4 = p1^3 + p2^2 such that p1 < p2 are consecutive primes and p3 = p1^2 + p2^3 is also an average of twin primes.

Original entry on oeis.org

13008, 9268057799643918, 1151303780719281840798, 1166398496059056623580, 1408815704665167877050, 1611023943160530038112, 1839284737645145603808, 1876391173984974899670, 2541672151459722294708, 3760269231809150191932, 13232137801909374644760, 19086525662779517405622

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 16 2008

nonn

			13008 = 23^3 + 29^2 is a term since 23 and 29 are consecutive primes, (13007, 13009) are twin primes, 23^2 + 29^3 = 24918, and (24917, 24919) are also twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A001097, A001359, A006512, A014574.

Cf. A138716, A138755, A139780.

a={};Do[p1=Prime[n];p2=Prime[n+1];p3=p1^2+p2^3;p4=p1^3+p2^2;If[PrimeQ[p3-1]&&PrimeQ[p3+1]&&PrimeQ[p4-1]&&PrimeQ[p4+1],AppendTo[a,p4]],{n,13^5}];Print[a];

More terms from Amiram Eldar, Dec 22 2019

A139780 Average of twin primes of the form p1^3 + p2^2, where p1 < p2 are twin primes.

Original entry on oeis.org

38318210940, 68484878220, 143165125680, 6353554336290, 75041090138100, 91851874324800, 116366750976990, 118525130067690, 368776631152800, 374826155288910, 431041855258500, 585102141663000, 767853933976740, 834602006112360, 845499208101600, 1560061877051100

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 16 2008

nonn

			38318210940 = 3371^3 + 3373^2 is a term since both (3371, 3373) and (38318210939, 38318210941) are twin primes.

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

Cf. A001097, A001359, A006512, A014574.

Cf. A138716, A138755, A139777.

[q:k in [1..10000]|IsPrime((NthPrime(k)+2)) and IsPrime(q-1)and IsPrime(q+1) where q is NthPrime(k)^3+ (NthPrime(k)+2)^2]; // Marius A. Burtea, Dec 22 2019

a={};Do[p1=Prime[n];p2=Prime[n+1];pp=p1^3+p2^2;If[(p2-p1)==2&&PrimeQ[pp-1]&&PrimeQ[pp+1],AppendTo[a,pp]],{n,10^4}];Print[a];
Select[#[[1]]^3+#[[2]]^2&/@Select[Partition[Prime[Range[15000]],2,1],#[[2]] - #[[1]]==2&],AllTrue[#+{1,-1},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 01 2021 *)

More terms from Amiram Eldar, Dec 22 2019

A171938 Record values in A138753 (a "prime" variation of the Collatz (3n+1) problem).

Original entry on oeis.org

1, 4, 5, 8, 21, 22, 24, 29, 60, 61, 72, 73, 97, 100, 184, 216, 239, 451, 469

Offset: 1

M. F. Hasler, Apr 01 2008

Cf. A124123, A138750-A138754, A138756, A006878 (analog for Collatz problem).

A138754[n_]:=A138754[n]=With[{p=Prime[n]},PrimePi[NextPrime[If[Mod[p,3]==2,p/2,2p]]]];
A138753[n_]:=Length[NestWhileList[A138754,n,UnsameQ,{1,4}]]-1;
A171938list[upto_]:=Module[{v,r=0},Table[If[(v=A138753[n])>r,r=v,Nothing],{n,upto}]];
A171938list[500] (* Paolo Xausa, Jul 29 2023 *)

m=0; for( i=1,#A138753, A138753[i] > m & print1( m=A138753[i],", "))

A171938(n) = A138753(A138756(n)).

A171938 = { A138753(m) | A138753(k) < A138753(m) for all k

Originally submitted as A138755, but mislaid by Editor-in-Chief; renumbered and added to OEIS, Oct 24 2010

a(15)-a(19) from Paolo Xausa, Jul 29 2023

A138757 a(n) = A007918(A138750(n)), that is, least prime > n/2 if n=2 (mod 3), > 2n otherwise.

Original entry on oeis.org

2, 2, 2, 7, 11, 3, 13, 17, 5, 19, 23, 7, 29, 29, 7, 31, 37, 11, 37, 41, 11, 43, 47, 13, 53, 53, 13, 59, 59, 17, 61, 67, 17, 67, 71, 19, 73, 79, 19, 79, 83, 23, 89, 89, 23, 97, 97, 29, 97, 101, 29, 103, 107, 29, 109, 113, 29, 127, 127, 31, 127, 127, 31, 127

Offset: 0

M. F. Hasler, Apr 04 2008

This can be considered as an analog of the Collatz (or 3n+1) map on the set of primes, see A138751 and A138754 for details.

Numbers 0,1,2 go immediately to the unique fixed point 2, all others end up in the cycle 7 -> 17 -> 11 -> 7, after a number of iterations given by A138753(A138757(n))-1 (= A138753(n)-2 if n is prime).

			a(7) = 17 since 7 = 1 (mod 3), thus A138750(7) = 2*7 = 14, nextprime(14) = 17.
a(11) = 7 since 11 = 2 (mod 3), thus A138750(11) = ceiling(11/2) = 6, nextprime(6) = 7.

Georges Brougnard, Definition of GB-sequences.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A124123, A138750, A138751, A138752, A138753, A138754, A138755, A138756, A007918.

np1[n_]:=Module[{x=Ceiling[n/2]},If[PrimeQ[x],x,NextPrime[x]]]; np2[n_]:= Module[{x=2n},If[PrimeQ[x],x,NextPrime[x]]]; Table[If[Mod[n,3]==2, np1[n], np2[n]],{n,0,70}] (* Harvey P. Dale, Jul 10 2013 *)

A138757(n)=nextprime(if(n%3==2,(n+1)\2,2*n))

a(n) = A007918(A138750(n)).

For p prime, a(p) = A138751(A000720(p))

Showing 1-5 of 5 results.

cp's OEIS Frontend

A138716 Primes p2 such that p1^2 + p2^3 is an average of twin primes and p1 < p2 are consecutive primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A139777 Average of twin primes p4 = p1^3 + p2^2 such that p1 < p2 are consecutive primes and p3 = p1^2 + p2^3 is also an average of twin primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A139780 Average of twin primes of the form p1^3 + p2^2, where p1 < p2 are twin primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A171938 Record values in A138753 (a "prime" variation of the Collatz (3n+1) problem).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A138757 a(n) = A007918(A138750(n)), that is, least prime > n/2 if n=2 (mod 3), > 2n otherwise.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula