A034881 -id:A034881 - cp's OEIS frontend

A093483 a(1) = 2; for n>1, a(n) = smallest integer > a(n-1) such that a(n) + a(i) + 1 is prime for all 1 <= i <= n-1.

2, 4, 8, 14, 38, 98, 344, 22268, 79808, 187124, 347978, 2171618, 4219797674, 98059918334, 22518029924768, 54420534706118, 252534792143648

Offset: 1

Amarnath Murthy, Apr 14 2004

a(i) == 2 mod 6 for i > 2. - Walter Kehowski, Jun 03 2006
a(i) == either 8 or 14 (mod 30) for i > 2. - Robert G. Wilson v, Oct 16 2012
The Hardy-Littlewood k-tuple conjecture would imply that this sequence is infinite. Note that, for n > 2, a(n)+3 and a(n)+5 are both primes, so a proof that this sequence is infinite would also show that there are infinitely many twin primes. - N. J. A. Sloane, Apr 21 2007
No more terms less than 7*10^12. - David Wasserman, Apr 03 2007

			a(5) = 38 because 38+2+1, 38+4+1, 38+8+1 and 38+14+1 are all prime.

G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
Carlos Rivera, Puzzle 37. Set of even numbers { ai } such that every ai + aj + 1 is prime ( i & j are different ), The Prime Puzzles and Problems Connection.

Cf. A034881, A117480, A121404, A103828.

Cf. A010051.

a093483 n = a093483_list !! (n-1)
a093483_list = f ([2..7] ++ [8,14..]) [] where
   f (x:xs) ys = if all (== 1) $ map (a010051 . (+ x)) ys
                    then x : f xs ((x+1):ys) else f xs ys
-- Reinhard Zumkeller, Dec 11 2011

EP:=[2,4]: P:=[]: for w to 1 do for n from 1 to 800*10^6 do s:=6*n+2; Q:=map(z-> z+s+1); if andmap(isprime,Q) then EP:=[op(EP),s]; P:=[op(P),op(Q)] fi; od od; EP; P: # Walter Kehowski, Jun 03 2006

f[1] = 2; f[2] = 4; f[3] = 8; f[n_] := f[n] = Block[{lst = Array[f, n - 1], k = f[n - 1] + 7}, While[ Union[ PrimeQ[k + lst]] != {True}, k += 6]; k-1]; Array[f, 13] (* Robert G. Wilson v, Oct 16 2012 *)

a(7) from Jonathan Vos Post, Mar 22 2006
More terms from Joshua Zucker, Jul 24 2006
Edited and extended to a(14) by David Wasserman, Apr 03 2007
a(15)-a(17) from Don Reble, added by N. J. A. Sloane, Sep 18 2012

A219761 a(1) = 1; for n>1, a(n) = smallest integer > a(n-1) such that a(n)*a(n-i)+1 is prime for all 0 <= i <= n-1.

Original entry on oeis.org

1, 2, 6, 156, 4260, 117306, 160650, 13937550, 32742516, 3306719796, 7746764190

Offset: 1

N. J. A. Sloane, Dec 01 2012

			After a(1)=1, a(2)=2, a(3)=6, we want the smallest m>6 such that 1+m, 1+2m, 1+6m and 1+m^2 are all prime: this is m = 156 = a(4).

Rainer Rosenthal, Posting to Sequence Fans Mailing List, Nov 30 2012.

Cf. A034881, A081942, A093483.

f[a_List] := Block[{m = a, k = a[[-1]] + 6}, While[ Union@ PrimeQ[k*Join[m, {k}] + 1] != {True}, k += 6]; k]; s = {1, 2, 6}; Do[ Print[{n, a = f[s]}]; s = Append[s, a], {n, 4, 9}] (* Robert G. Wilson v, Dec 03 2012 *)

a(8) and a(9) from Robert G. Wilson v, Dec 03 2012

a(10) and a(11) from Robert G. Wilson v, Dec 04 2012

A219953 a(1) = 1; for n > 1, a(n) = smallest integer > a(n-1) such that a(n)*a(i)+1 is semiprime for all 1 <= i <= n-1.

Original entry on oeis.org

1, 3, 8, 38, 86, 318, 504, 3600, 8132, 83160, 116850, 202272, 399126, 6190086, 8756916, 25253676, 309709400, 1112878446, 1478724036, 11062089360, 97331025386

Offset: 1

Jonathan Vos Post, Dec 01 2012

This is to A034881 as semiprimes A001358 are to primes A000040.

a(20) > 6*10^9. - Giovanni Resta, Jul 26 2015

			a(1) = 1 by definition.
a(2) = 3: 3 > 1, and 1*3 + 1 = 4 = 2^2 is semiprime.
a(3) = 8: 8 > 3, and 1*8 + 1 = 9 = 3^2 is semiprime, and 3*8 + 1 = 25 = 5^2 is semiprime.
a(4) = 38: 38 > 8, and 1*38 + 1 = 39 = 3*13 is semiprime, and 3*38 + 1 = 115 = 5*23 is semiprime, and 8*38 + 1 = 305 = 5*61 is semiprime.
From _Michel Marcus_, Jul 26 2015: (Start)
The resulting semiprimes are:
    4;
    9,  25;
   39, 115,  305;
   87, 259,  689,  3269;
  319, 955, 2545, 12085, 27349;
  ...
(End)

Cf. A030063, A001358, A034881, A219761.

A219953 := proc(n)
    option remember;
    if n= 1 then
        1;
    else
        for a from procname(n-1)+1 do
            issp := true ;
            for i from 1 to n-1 do
                if numtheory[bigomega]( a*procname(n-i)+1) = 2 then
                    ;
                else
                    issp := false;
                    break ;
                end if;
            end do:
            if issp then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Dec 15 2012

a = {1}; Do[k = a[[n - 1]] + 1; While[! AllTrue[(k a[[n - #]] + 1) & /@ Range@ (n - 1), Total[Last /@ FactorInteger@ #] == 2 &], k++]; AppendTo[a, k], {n, 2, 13}]; a (* Michael De Vlieger, Jul 26 2015, Version 10 *)

ok(v, n, k) = {v[n] = k; for (j=1, n-1, if (bigomega(1+v[n]*v[j]) != 2, return (0));); return (1);}
lista(nn) = {print1(k=1, ", "); v = [k]; for (n=2, nn, k = v[n-1]+1; v = concat(v, k); while (! ok(v, n, k), k++); v[n] = k; print1(k, ", "););} \\ Michel Marcus, Jul 26 2015

a(14)-a(17) from Luke March, Jul 26 2015
a(18)-a(19) from Giovanni Resta, Jul 26 2015
a(20)-a(21) from Tyler Busby, Jan 31 2023

A127903 Primes arising in A093483.

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 41, 43, 47, 53, 101, 103, 107, 113, 137, 347, 349, 353, 359, 383, 443, 22271, 22273, 22277, 22283, 22307, 22367, 22613, 79811, 79813, 79817, 79823, 79847, 79907, 80153, 102077, 187127, 187129, 187133, 187139, 187163, 187223

Offset: 1

Jonathan Vos Post, Apr 05 2007

nonn

Cf. A093483, A034881, A103828.

Data corrected by Giovanni Resta, Jun 19 2016

A274694 Variation on Fermat's Diophantine m-tuple: 1 + the product of any two distinct terms is a prime power.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 211050, 3848880, 20333040, 125038830, 2978699430

Offset: 1

Robert C. Lyons, Jul 02 2016

a(1) = 1; for n>1, a(n) = smallest integer > a(n-1) such that a(n)*a(i)+1 is a prime power for all 1 <= i <= n-1.

			After a(1)=1, a(2)=2, a(3)=3, we want m, the smallest number > 3 such that m+1, 2m+1 and 3m+1 are all prime powers: this is m = 4 = a(4).

Cf. A030063, A034881, A246655.

seq = [1]
prev_element = 1
max_n = 8
for n in range(2, max_n+1):
    next_element = prev_element + 1
    while True:
        all_match = True
        for element in seq:
            x = element * next_element + 1
            if not x.is_prime_power():
                all_match = False
                break
        if all_match:
            seq.append( next_element )
            break
        next_element += 1
    prev_element = next_element
print(seq)

cp's OEIS Frontend

A093483 a(1) = 2; for n>1, a(n) = smallest integer > a(n-1) such that a(n) + a(i) + 1 is prime for all 1 <= i <= n-1.

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A219761 a(1) = 1; for n>1, a(n) = smallest integer > a(n-1) such that a(n)*a(n-i)+1 is prime for all 0 <= i <= n-1.

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A219953 a(1) = 1; for n > 1, a(n) = smallest integer > a(n-1) such that a(n)*a(i)+1 is semiprime for all 1 <= i <= n-1.

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A127903 Primes arising in A093483.

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A274694 Variation on Fermat's Diophantine m-tuple: 1 + the product of any two distinct terms is a prime power.

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