cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A006512 Greater of twin primes.

Original entry on oeis.org

5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Also primes that are the sum of two primes (which is possible only if 2 is one of the primes). - Cino Hilliard, Jul 02 2004, edited by M. F. Hasler, Nov 14 2019
The set of greater of twin primes larger than five is a proper subset of the set of primes of the form 3n + 1 (A002476). - Paul Muljadi, Jun 05 2008
Smallest prime > n-th isolated composite. - Juri-Stepan Gerasimov, Nov 07 2009
Subsequence of A175075. Union of a(n) and sequence A175080 is A175075. - Jaroslav Krizek, Jan 30 2010
A164292(a(n))=1; A010051(a(n)+2)=0 for n > 1. - Reinhard Zumkeller, Mar 29 2010
Omega(n) = Omega(n-2); d(n) = d(n-2). - Juri-Stepan Gerasimov, Sep 19 2010
Aside from the first term, all subsequent terms have digital root 1, 4, or 7. - J. W. Helkenberg, Jul 24 2013
Also primes p with property that the sum of the successive gaps between primes <= p is a prime number. - Robert G. Wilson v, Dec 19 2014
The phrase "x is an element of the {primes, positive integers} and there {exist no, exist} elements a,b of {1 and primes, primes}: a+b=x" determines A133410, A067829, A025584, A006512, A166081, A014092, A014091 and A038609 for the first few hundred terms with only de-duplication or omitting/including 3, 4 and 6 in the case of A166081/A014091 and one case of omitting/including 3 given 1 isn't prime. - Harry G. Coin, Nov 25 2015
The yet unproved Twin Prime Conjecture states that this sequence is infinite. - M. F. Hasler, Nov 14 2019

References

  • See A001359 for further references and links.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Subsequence of A139690.
Bisection of A077800.
Cf. A001097, A001359, A014574, A067829, A002476.

Programs

  • Haskell
    a006512 = (+ 2) . a001359 -- Reinhard Zumkeller, Feb 10 2015
  • Magma
    [n: n in PrimesUpTo(1610)|IsPrime(n-2)]; // Bruno Berselli, Feb 28 2011
  • Maple
    for i from 1 to 253 do if ithprime(i+1) = ithprime(i) + 2 then print({ithprime(i+1)}); fi; od; # Zerinvary Lajos, Mar 19 2007
P := select(isprime,[$1..1609]): select(p->member(p-2,P),P); # Peter Luschny, Mar 03 2011
A006512 := proc(n)
    2+A001359(n) ;
end proc: # R. J. Mathar, Nov 26 2014
  • Mathematica
    Select[Prime[Range[254]], PrimeQ[# - 2] &] (* Robert G. Wilson v, Jun 09 2005 *)
Transpose[Select[Partition[Prime[Range[300]], 2, 1], Last[#] - First[#] == 2 &]][[2]] (* Harvey P. Dale, Nov 02 2011 *)
Cases[Prime[Range[500]] + 2, ?PrimeQ] (* _Fred Patrick Doty, Aug 23 2017 *)
  • PARI
    select(p->isprime(p-2),primes(1000))
  • PARI
    a(n)=p=3; while(p+2 < (p=nextprime(p+1)) || n-->0, ); p
vector(100, n, a(n)) \\ Altug Alkan, Dec 04 2015
  • Python
    from sympy import primerange, isprime
print([n for n in primerange(1, 2001) if isprime(n - 2)]) # Indranil Ghosh, Jul 20 2017

A175075 Primes q with result 2 under iterations of {r mod (max prime p < r)} starting at r = q.

Original entry on oeis.org

2, 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489
Offset: 1

Views

Author

Jaroslav Krizek, Jan 23 2010

Keywords

Comments

a(1) = 2, a(n) = A006512(n-1) for 2 <= n <= 82, a(83) = 2999. Sequence is the union of A006512 and A175080. Subsequence of A175072. Primes q with some results of {2, 28, 36, 52, 58, 66, ... } under first step of iteration of {r mod (max prime p < r)} starting at r = q, i.e. number 2 and primes q such that difference q and previous prime is equal to some of the values 2, 28, 36, 52, 58, 66, ...
Not the same as A094743: contains 2999, 3299, 5147, 5981, 8999, 9587, 10037, 10427, 10559, 10937, 11579, 12889, ... that are absent from that sequence. Up to 10^9, there are 3247366 terms in this sequence that are not in A094743, though every term from that sequence appears here. Is A094743 a subsequence of this sequence? - Charles R Greathouse IV, Jan 12 2022
It suffices to stop after the iterations yield a number less than 5 and check if the result is 2. Under this procedure, 2 takes 0 iterations, 5 is the first prime to take 1 iteration, 29 is the first to take 2 iterations, 2999 is the first to take 3 iterations, and 401429925999155063 is the first to take 4 iterations. - Charles R Greathouse IV, Jan 14 2022

Examples

			Iteration procedure for a(5) = 19: 19 mod 17 = 2. Iteration procedure for a(83) = 2999: 2999 mod 2971 = 28, 28 mod 23 = 5, 5 mod 3 = 2.

Links

Programs

  • Mathematica
    fQ[p_] := Block[{r = p}, While[r > 2, r = Mod[r, NextPrime[r, -1]]]; r == 2]; Select[ Prime@ Range@ 253, fQ] (* Robert G. Wilson v, Aug 09 2010 *)
  • PARI
    is(n)=if(!isprime(n), return(0)); while(n>4, n-=precprime(n-1)); n==2 \\ Charles R Greathouse IV, Jan 12 2022
  • PARI
    has(n)=while(n>4, n-=precprime(n-1)); n==2
list(lim)=my(v=List([2]),p=3); forprime(q=5,lim, if(has(q-p), listput(v,q)); p=q); Vec(v) \\ Charles R Greathouse IV, Jan 12 2022

Formula

A175072 \ A175076. [A-number corrected by R. J. Mathar, Sep 25 2010] - Jaroslav Krizek, Jan 30 2010

Extensions

More terms from Robert G. Wilson v, Aug 09 2010
A175080 inserted in comment - R. J. Mathar, Sep 25 2010

A175073 Primes q with result 1 under iterations of {r mod (max prime p < r)} starting at r = q.

Original entry on oeis.org

3, 11, 17, 23, 29, 37, 41, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 107, 113, 127, 131, 137, 149, 157, 163, 167, 173, 179, 191, 197, 211, 223, 227, 233, 239, 251, 257, 263, 269, 277, 281, 293, 307
Offset: 1

Views

Author

Jaroslav Krizek, Jan 23 2010

Keywords

Comments

Subsequence of A175071.
Union of a(n) and A175074 is A175071. - Jaroslav Krizek, Jan 30 2010
The terms in A025584 but not in here are 2, 2999, 3299, 5147, 5981, 8999, 9587, ... , apparently those listed in A175080. - R. J. Mathar, Feb 01 2010
a(n-1)=A156828(n) in the range n=3..348, but afterwards the sequences differ because numbers like 2999 and 3229 are in A156828 but not in here. - R. J. Mathar, Mar 01 2010
Conjecture: under this iteration procedure, all primes eventually will yield either a 2 or a 1. If a 2 results, all subsequent terms are zeros; if a 1 results, all subsequent terms are -1s. The conjecture is true for the first 2 million primes. - Harvey P. Dale, Jan 17 2014

Examples

			Iteration procedure for a(2) = 11: 11 mod 7 = 4, 4 mod 3 = 1.

Links

Crossrefs

Note that all three of A025584, A156828, A175073 are different sequences. - N. J. A. Sloane, Apr 10 2011

Programs

  • Maple
    isA175073 := proc(p)
    local r,rold;
    if not isprime(p) then
        return false;
    end if;
    r := p ;
    while true do
        rold :=r ;
        if r = 2 then
            return false ;
        end if;
        r := modp(r,prevprime(r)) ;
        if r = 1 then
            return true;
        elif r= rold then
            return false ;
        end if;
    end do:
end proc:
A175073 := proc(n)
    option remember ;
    if n= 1 then
        3;
    else
        for p from procname(n-1)+2 by 2 do
            if isA175073(p) then
                return p;
            end if;
        end do:
    end if;
end proc:
seq(A175073(n),n=1..40) ; # R. J. Mathar, Mar 25 2024
  • Mathematica
    r1Q[n_] := FixedPoint[Mod[#, NextPrime[#, -1]] &, n] == -1; Select[Prime[ Range[70]],r1Q] (* This program relies upon the conjecture described in the comments above *) (* Harvey P. Dale, Jan 17 2014 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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