A192476 -id:A192476 - cp's OEIS frontend

A192527 Difference sequence of A192476.

1, 3, 3, 18, 3, 21, 15, 3, 21, 39, 549, 3, 21, 39, 102, 3, 21, 39, 447, 165, 165, 819, 3, 21, 39, 612, 165, 885, 3, 21, 39, 336, 3, 21, 39, 213, 99, 66, 234, 165, 1260, 399, 798, 3, 21, 39, 465, 147, 165, 87, 399, 1173, 1725, 399, 3297, 543, 3, 21, 39, 612, 165

Offset: 1

Clark Kimberling, Jul 04 2011

nonn

See A192476

			A192476=(1,2,5,8,26,29,...), so that
a(1)=2-1, a(2)=5-2, a(3)=8-5,...

A192476.

Mathematica
```
(See A192476.)
```

a(n)=A192476(n)-A192476(n-1), n>=1.

A192580 Monotonic ordering of set S generated by these rules: if x and y are in S and xy+1 is a prime, then xy+1 is in S, and 2 is in S.

Original entry on oeis.org

2, 5, 11, 23, 47

Offset: 1

Clark Kimberling, Jul 04 2011

Following the discussion at A192476, the present sequence introduces a restriction: that the generated terms must be prime. A192580 is the first of an ascending chain of finite sequences, determined by the initial set called "start":
A192580:  f(x,y)=xy+1 and start={2}
A192581:  f(x,y)=xy+1 and start={2,4}
A192582:  f(x,y)=xy+1 and start={2,4,6}
A192583:  f(x,y)=xy+1 and start={2,4,6,8}
A192584:  f(x,y)=xy+1 and start={2,4,6,8,10}
For other choices of the function f(x,y) and start, see A192585-A192598.
A192580 consists of only 5 terms, A192581 of 7 terms, and A192582 of 28,...; what can be said about the sequence (5,7,28,...)?
2, 5, 11, 23, 47 is the complete Cunningham chain that begins with 2. Each term except the last is a Sophie Germain prime A005384. - Jonathan Sondow, Oct 28 2015

			2 is in the sequence by decree.
The generated numbers are 5=2*2+1, 11=2*5+1, 23=2*11+1, 47=2*23+1.

Wikipedia, Cunningham chain

Cf. A005384, A192476, A192581, A192582, A192583, A192584.

start = {2}; primes = Table[Prime[n], {n, 1, 10000}];
f[x_, y_] := If[MemberQ[primes, x*y + 1], x*y + 1]
b[x_] := Block[{w = x}, Select[Union[Flatten[AppendTo[w, Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # < 50000 &]];
t = FixedPoint[b, start]  (* A192580 *)

A009293 If a, b in sequence, so is a*b+1.

Original entry on oeis.org

2, 5, 11, 23, 26, 47, 53, 56, 95, 107, 113, 116, 122, 131, 191, 215, 227, 233, 236, 245, 254, 263, 266, 281, 287, 383, 431, 455, 467, 473, 476, 491, 509, 518, 527, 530, 533, 536, 563, 566, 575, 581, 584, 599, 611, 617, 656, 677, 767, 863, 911, 935, 947, 953, 956, 983

Offset: 1

David W. Wilson

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from Reinhard Zumkeller)

Cf. A009388, A009299, A192476.

import Data.Set (singleton, deleteFindMin, insert)
a009293 n = a009293_list !! (n-1)
a009293_list = f [2] (singleton 2) where
   f xs s = m : f xs' (foldl (flip insert) s' (map ((+ 1) . (* m)) xs'))
     where xs' = m : xs
           (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 15 2011

f[l_] := Block[{k = l}, Select[ Union[ Flatten[ AppendTo[k, Table[ k[[i]]*k[[j]] + 1, {i, 1, Length[k]}, {j, 1, i}]]]], # < 1000 &]]; NestList[f, {2}, 7][[ -1]] (* Robert G. Wilson v, May 23 2004 *)

A192646 First differences of A192645.

Original entry on oeis.org

1, 1, 2, 3, 8, 5, 3, 15, 16, 5, 3, 72, 50, 7, 39, 16, 5, 3, 65, 49, 8, 39, 16, 5, 3, 72, 39, 16, 5, 3, 369, 1, 135, 185, 192, 39, 8, 8, 5, 3, 1, 368, 190, 369, 1, 46, 89, 47, 8, 130, 192, 39, 16, 5, 3, 17, 118, 185, 49, 48, 87, 8, 39, 16, 5, 3, 114, 192, 39, 16, 5, 3, 48

Offset: 1

Clark Kimberling, Jul 06 2011

nonn

Does this sequence include 1 infinitely many times?

			A192645 = (1, 2, 3, 5, 8, 16, 21, ...) gives (2-1, 3-2, 5-3, 8-5, 16-8, 21-16, ...)

Cf. A192476, A192645.

start = {1, 2};
f[x_, y_] := If[MemberQ[Range[1, 5000], x^2 - y^2], x^2 - y^2]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # <
      5000 &]];
t = FixedPoint[b, start]  (* A192645 *)
Differences[t] (* A192646 *)

A008318 Smallest number strictly greater than previous one which is the sum of squares of two previous distinct terms (a(1)=1, a(2)=2).

Original entry on oeis.org

1, 2, 5, 26, 29, 677, 680, 701, 842, 845, 866, 1517, 458330, 458333, 458354, 459005, 459170, 462401, 462404, 462425, 463076, 463241, 491402, 491405, 491426, 492077, 492242, 708965, 708968, 708989, 709640, 709805, 714026, 714029, 714050, 714701

Offset: 1

R. Muller

A003095 is a subsequence apart from the initial term. - Reinhard Zumkeller, Jan 17 2008

The subsequence of primes begins: 2, 5, 29, 677, 701, 458333, 462401, 492077, 708989, 714029, ... - Jonathan Vos Post, Nov 21 2012

F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House 2000
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

T. D. Noe, Table of n, a(n) for n=1..1000
Mihaly Bencze [Beneze], Smarandache Recurrence Type Sequences, in Bull. Pure Appl. Sciences, Vol. 16E, No. 2, 231-236, 1997.
F. Smarandache, Definitions, Solved and Unsolved Problems, Conjectures, ...
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences.

Cf. A192476.

import Data.Set (singleton, deleteFindMin, insert)
a008318 n = a008318_list !! (n-1)
a008318_list = f [1] (singleton 1) where
   f xs s =
     m : f (m:xs) (foldl (flip insert) s' (map (+ m^2) (map (^ 2) xs)))
     where (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 15 2011

a[1]=1; a[2]=2; a[n_] := a[n] = First[ Select[ Sort[ Flatten[ Table[a[j]^2 + a[k]^2, {j, 1, n-1}, {k, j+1, n-1}]]], # > a[n-1] & , 1]]; Table[a[n], {n, 1, 36}](* Jean-François Alcover, Nov 10 2011 *)

More terms from David W. Wilson

A192645 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2 - y^2 > 0 then x^2 - y^2 is in S, and 1 and 2 are in S.

Original entry on oeis.org

1, 2, 3, 5, 8, 16, 21, 24, 39, 55, 60, 63, 135, 185, 192, 231, 247, 252, 255, 320, 369, 377, 416, 432, 437, 440, 512, 551, 567, 572, 575, 944, 945, 1080, 1265, 1457, 1496, 1504, 1512, 1517, 1520, 1521, 1889, 2079, 2448, 2449, 2495, 2584, 2631, 2639

Offset: 1

Clark Kimberling, Jul 06 2011

nonn

See A192476 for a general discussion.  Related sequences:
A192645:  f(x,y) = x^2 - y^2 > 0, start={1,2};
A192647:  f(x,y) = x^2 - y^2 > 0, start={1,3};
A192648:  f(x,y) = x^2 - y^2 > 0, start={2,3};
A192649:  f(x,y) = x^2 - y^2 > 0, start={1,2,4}.

			2^2 - 1^2 = 3;
3^2 - 2^2 = 5, 3^2 - 1^2 = 8;
5^2 - 3^2 = 16, 5^2 - 2^2 = 21, 5^2 - 1^2 = 24.
Taking the generating procedure in the order just indicated results in the monotonic ordering of the sequence and also suggests a triangular format for the generated terms:
    3;
    5,   8;
   16,  21,  24;
   39,  55,  60,  63;
  135, 185, 192, 231, 247;
  ...

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A192476, A192646 (first differences).

start = {1, 2};
f[x_, y_] := If[MemberQ[Range[1, 5000], x^2 - y^2], x^2 - y^2]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # <
      5000 &]];
t = FixedPoint[b, start]  (* A192645 *)
Differences[t] (* A192646 *)

A009299 If a, b in sequence, so is a*b+2.

Original entry on oeis.org

2, 6, 14, 30, 38, 62, 78, 86, 126, 158, 174, 182, 198, 230, 254, 318, 350, 366, 374, 398, 422, 462, 470, 510, 518, 534, 638, 702, 734, 750, 758, 798, 846, 870, 902, 926, 942, 950, 1022, 1038, 1046, 1070, 1094, 1142, 1190, 1206, 1278, 1382, 1406, 1446, 1470, 1502

Offset: 1

David W. Wilson

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A009293, A192476.

import Data.Set (singleton, deleteFindMin, insert)
a009299 n = a009299_list !! (n-1)
a009299_list = f [2] (singleton 2) where
   f xs s = m : f xs' (foldl (flip insert) s' (map ((+ 2) . (* m)) xs'))
     where xs' = m : xs
           (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 15 2011

A009388 If a, b in sequence, so is a*b-1.

Original entry on oeis.org

2, 3, 5, 8, 9, 14, 15, 17, 23, 24, 26, 27, 29, 33, 39, 41, 44, 45, 47, 50, 51, 53, 57, 63, 65, 68, 69, 71, 74, 77, 80, 81, 84, 86, 87, 89, 93, 98, 99, 101, 105, 111, 113, 114, 116, 119, 122, 125, 129, 131, 134, 135, 137, 140, 141, 144, 147, 149, 152, 153, 158, 159, 161, 164

Offset: 1

David W. Wilson

nonn

All terms are congruent to 0 or 2 mod 3. It follows that no three consecutive integers are in the sequence. - Franklin T. Adams-Watters, Aug 31 2016, conjectured by David W. Wilson.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A009293. This is superset of A005659 - 1.

Cf. A009299, A192476.

import Data.Set (singleton, deleteFindMin, insert)
a009388 n = a009388_list !! (n-1)
a009388_list = f [2] (singleton 2) where
   f xs s = m : f xs' (foldl (flip insert) s' (map (pred . (* m)) xs'))
     where xs' = m : xs
           (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 15 2011

f[l_] := Block[{k = l}, Select[ Union[ Flatten[ AppendTo[k, Table[ k[[i]]*k[[j]] - 1, {i, 1, Length[k]}, {j, 1, i}]]]], # < 170 &]]; NestList[f, {2}, 6][[ -1]] (* Robert G. Wilson v, May 23 2004 *)

A192583 Monotonic ordering of set S generated by these rules: if x and y are in S and xy+1 is a prime, then xy+1 is in S, and 2, 4, 6, and 8 are in S.

Original entry on oeis.org

2, 4, 5, 6, 8, 11, 13, 17, 23, 31, 37, 41, 47, 53, 67, 79, 83, 89, 103, 107, 137, 139, 149, 167, 179, 223, 269, 283, 317, 359, 499, 557, 619, 643, 719, 823, 857, 1097, 1193, 1433, 1439, 1699, 1997, 2153, 2477, 2879, 3343, 4457, 6857, 7159, 8599, 12919, 41143

Offset: 1

Clark Kimberling, Jul 04 2011

See the discussion at A192580.

Cf. A192476.

start = {2, 4, 6, 8}; primes = Table[Prime[n], {n, 1, 10000}];
f[x_, y_] := If[MemberQ[primes, x*y + 1], x*y + 1]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # <
      10000000 &]];
t = FixedPoint[b, start]  (* A192583 *)
PrimePi[t] (* A192530 Nonprimes 4,6,8 are represented by "next prime down". *)

A192523 Monotonic ordering of set S generated by these rules: if x and y are in S then 2xy-x-y is in S, and 2 is in S.

Original entry on oeis.org

2, 4, 10, 24, 28, 66, 70, 82, 164, 180, 192, 196, 208, 244, 446, 458, 486, 490, 522, 538, 570, 574, 586, 622, 730, 1104, 1144, 1244, 1256, 1292, 1320, 1336, 1340, 1368, 1372, 1452, 1456, 1468, 1512, 1548, 1564, 1612, 1704, 1708, 1720, 1756, 1864

Offset: 1

Clark Kimberling, Jul 03 2011

nonn

Cf. A192476.

start = {2}; f[x_, y_] := 2 x*y - x - y
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # <
      5000 &]];
t = NestList[b, start, 10][[-1]] (* A192523 *)
t/2 (* A192524 *)
Table[t[[i]] - t[[i - 1]], {i, 2, Length[t]}]  (* differences *)

cp's OEIS Frontend

A192527 Difference sequence of A192476.

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A192580 Monotonic ordering of set S generated by these rules: if x and y are in S and xy+1 is a prime, then xy+1 is in S, and 2 is in S.

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A009293 If a, b in sequence, so is a*b+1.

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A192646 First differences of A192645.

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A008318 Smallest number strictly greater than previous one which is the sum of squares of two previous distinct terms (a(1)=1, a(2)=2).

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A192645 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2 - y^2 > 0 then x^2 - y^2 is in S, and 1 and 2 are in S.

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A009299 If a, b in sequence, so is a*b+2.

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A009388 If a, b in sequence, so is a*b-1.

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A192583 Monotonic ordering of set S generated by these rules: if x and y are in S and xy+1 is a prime, then xy+1 is in S, and 2, 4, 6, and 8 are in S.