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.

User: Edward Bernstein

Edward Bernstein's wiki page.

Edward Bernstein has authored 4 sequences.

A316792 a(n) is the least prime p such that the second forward difference of three consecutive primes p, q and r is n = -(p - 2q + r)/2.

Original entry on oeis.org

3, 7, 23, 1531, 139, 113, 523, 1069, 887, 6397, 1129, 3137, 5351, 2971, 1327, 14107, 9973, 19333, 84871, 16141, 15683, 73189, 31907, 28229, 35617, 35677, 44293, 43331, 107377, 34061, 221327, 134513, 31397, 480209, 173359, 332317, 933073, 736279, 265621, 843911, 404851, 155921
Offset: 0

Views

Author

Edward Bernstein and Robert G. Wilson v, Jul 14 2018

Keywords

Comments

Inspired by A295973.
Except for the first three primes {2, 3, 5}, all sfds are even.
The only other sfd which is not covered by this sequence is when the primes are {2, 3, 5} which results in an sfd of 1.
Except for an sfd of 0 or 1, all values of sfd occur infinitely often.
As an example, sfd = -2 for p = 7, 13, 31, 37, 61, 67, 73, 97, 103, 157, 193, 223, 271, 277, 307, ..., .

Examples

			a(0) = 3 since the three consecutive primes {3, 5, 7} have an sfd of 0;
a(1) = 7 since the three consecutive primes {7, 11, 13} have a sfd of -2;
a(2) = 23 since the three consecutive primes {23, 29, 31} have a sfd of -4;
a(3) = 1531 since the three consecutive primes {1531, 1543, 1549} have an sfd of -6;
a(4) =  since the three consecutive primes {} have an sfd of -8; etc.

Links

Crossrefs

Cf. A000040, A000230, A036263, A137501, A295746, A295973, A316791.

Programs

  • Mathematica
    p = 2; q = 3; r = 5; t[_] := 0; While[p < 1000000, d = p - 2q + r; If[ t[d] == 0, t[d] = p]; p = q; q = r; r = NextPrime@ r]; Array[ t[-2#] &, 42, 0]
  • PARI
    a(n) = my(p=2, q=3); while ((p - 2*q + nextprime(q+1))/2 != -n, p=q; q=nextprime(q+1)); p; \\ Michel Marcus, Mar 08 2023

A316791 a(n) is the least prime p such that the second forward difference of three consecutive primes p, q and r is n = (p - 2q + r)/2.

Original entry on oeis.org

3, 5, 29, 503, 137, 109, 1063, 1931, 521, 7951, 1949, 1667, 5743, 2969, 1321, 15817, 9547, 28349, 45433, 20807, 15679, 113837, 43793, 19603, 40283, 25469, 40637, 156151, 79697, 34057, 282487, 134507, 552401, 770663, 31393, 188021, 480203, 461707, 281429, 1078241, 265619, 637937
Offset: 0

Views

Author

Edward Bernstein and Robert G. Wilson v, Jul 14 2018

Keywords

Comments

Inspired by A295973.
Except for the first three primes {2, 3, 5}, all sfds are even.
The only other sfd which is not covered by this sequence is when the primes are {2, 3, 5} which results in an sfd of 1.
Except for an sfd of 0 or 1, all values of sfd occur infinitely often.
As an example, sfd=2 for p = 5, 11, 17, 19, 41, 43, 79, 83, 101, 107, 127, 163, ..., .

Examples

			a(0) = 3 since the three consecutive primes {3, 5, 7} have an sfd of 0;
a(1) = 5 since the three consecutive primes {5, 7, 11} have an sfd of 2;
a(2) = 29 since the three consecutive primes {29, 31, 37} have an sfd of 4;
a(3) = 503 since the three consecutive primes {503, 509, 521} have an sfd of 6;
a(4) = 137 since the three consecutive primes {137, 139, 149} have an sfd of 8; etc.

Links

Crossrefs

Cf. A000040, A000230, A036263, A137501, A295746, A295973, A316792.

Programs

  • Mathematica
    p = 2; q = 3; r = 5; t[_] := 0; While[p < 1100000, d = p - 2q + r; If[ t[d] == 0, t[d] = p]; p = q; q = r; r = NextPrime@ r]; Array[ t[2#] &, 42, 0]
  • PARI
    a(n) = my(p=2, q=3); while ((p - 2*q + nextprime(q+1))/2 != n, p=q; q=nextprime(q+1)); p; \\ Michel Marcus, Mar 08 2023

A295973 Primes introducing new second differences in A036263.

Original entry on oeis.org

3, 5, 7, 11, 29, 31, 113, 127, 139, 149, 509, 523, 541, 907, 1069, 1087, 1151, 1327, 1361, 1543, 1669, 1933, 1951, 2971, 2999, 3163, 5381, 5749, 6421, 7963, 9551, 10007, 14143, 15683, 15727, 15823, 16183, 19373, 19609, 20809, 25471, 28277, 28351, 31397, 31469, 31957, 34061, 34123, 35671
Offset: 1

Views

Author

Edward Bernstein, Nov 30 2017

Keywords

Comments

This list consists of those primes corresponding to new second differences in A036263. There are 97 new second differences introduced up to the 100000th prime.

Examples

			The new values in A036263 are 1, 0, 2, -2, -4, 4, 10, -10, 8, -8, ... at indices 1, 2, 3, 4, 9, 10, 29, 30, ... and the middle primes of the prime triple starting at these indices are 3, 5, 7, 11, 29, ...

Links

Crossrefs

Cf. A036263, A295746.

Programs

  • Maple
    A036263s := proc(maxn)
    s := {} ;
    for n from 1 to maxn do
        s := s union {A036263(n)} ;
    end do:
    s ;
end proc:
A295973a := proc(n)
    if n = 1 then
        return 2;
    end if;
    p := nextprime(procname(n-1)) ;
    pidx := numtheory[pi](p) ;
    while true do
        candD := A036263(pidx) ;
        if not candD in A036263s(pidx-1) then
            return ithprime(pidx) ;
        end if ;
        pidx := pidx+1 ;
    end do:
end proc:
A295973 := proc(n)
    nextprime(A295973a(n)) ;
end proc:
seq(A295973(n),n=1..40) ; # R. J. Mathar, Jan 06 2018

A295746 Distinct second differences in the sequence of primes in order of appearance.

Original entry on oeis.org

1, 0, 2, -2, -4, 4, 10, -10, 8, -8, 6, 16, -12, -16, 12, -14, -20, 28, -28, -6, 22, 14, 20, 26, -26, -22, -24, 24, -18, 18, 32, -32, -30, 40, -40, 30, -38, -34, 46, 38, 50, -46, 34, 68, -64, -44, 58, -58, -48, -50, 48, 52, -54, 44, -52, 36, -42, 56
Offset: 1

Views

Author

Edward Bernstein, Nov 29 2017

Keywords

Comments

A036263 excluding repeated terms. - Iain Fox, Nov 30 2017

Examples

			From _Jon E. Schoenfield_, Jan 15 2018: (Start)
The first several primes and their 1st and 2nd differences are as follows:
.
   k prime(k)  1st difference    2nd difference
  -- --------  --------------  -------------------
   1     2
                 3 -  2 = 1
   2     3                     2 - 1 =  1 (new)
                 5 -  3 = 2
   3     5                     2 - 2 =  0 (new)
                 7 -  5 = 2
   4     7                     4 - 2 =  2 (new)
                11 -  7 = 4
   5    11                     2 - 4 = -2 (new)
                13 - 11 = 2
   6    13                     4 - 2 =  2 (repeat)
                17 - 13 = 4
   7    17                     2 - 4 = -2 (repeat)
                19 - 17 = 2
   8    19                     4 - 2 =  2 (repeat)
                23 - 19 = 4
   9    23                     6 - 4 =  2 (repeat)
                29 - 23 = 6
  10    29                     2 - 6 = -4 (new)
                31 - 29 = 2
  11    31
.
and the 2nd differences that are not repeats of 2nd differences encountered earlier are, in order of appearance, 1, 0, 2, -2, -4, ..., i.e., the terms of this sequence. (End)

Links

Crossrefs

Cf. A036263. A295973 are the primes associated with the new second differences.

Programs

  • Maple
    P:= select(isprime, [2,seq(i,i=3..10^6,2)]):
DP:= P[2..-1]-P[1..-2]:
DDP:= DP[2..-1]-DP[1..-2]:
ListTools:-MakeUnique(DDP); # Robert Israel, Jan 15 2018
  • Mathematica
    DeleteDuplicates@ Differences[Prime@ Range[10^4], 2] (* Michael De Vlieger, Dec 09 2017 *)
  • PARI
    first(n) = { my(res = vector(n), i=3, j=3); res[1]=1; res[2]=0; while(i<=n, my(d=prime(j+2)+prime(j)-2*prime(j+1)); if(!setsearch(Set(res), d), res[i]=d; i++); j++); res; } \\ Iain Fox, Nov 30 2017

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