A009293 -id:A009293 - cp's OEIS frontend

A192476 Monotonic ordering of set S generated by these rules: if x and y are in S then x^2 + y^2 is in S, and 1 is in S.

1, 2, 5, 8, 26, 29, 50, 65, 68, 89, 128, 677, 680, 701, 740, 842, 845, 866, 905, 1352, 1517, 1682, 2501, 2504, 2525, 2564, 3176, 3341, 4226, 4229, 4250, 4289, 4625, 4628, 4649, 4688, 4901, 5000, 5066, 5300, 5465, 6725, 7124, 7922, 7925, 7946, 7985

Offset: 1

Clark Kimberling, Jul 01 2011

nonn

Let N denote the positive integers, and suppose that f(x,y): N x N->N. Let "start" denote a subset of N, and let S be the set of numbers defined by these rules: if x and y are in S, then f(x,y) is in S, and "start" is a subset of S. The monotonic increasing ordering of S is a sequence:
A192476:  f(x,y)=x^2+y^2, start={1}
A003586:  f(x,y)=x*y, start={1,2,3}
A051037:  f(x,y)=x*y, start={1,2,3,5}
A002473:  f(x,y)=x*y, start={1,2,3,5,7}
A003592:  f(x,y)=x*y, start={2,5}
A009293:  f(x,y)=x*y+1, start={2}
A009388:  f(x,y)=x*y-1, start={2}
A009299:  f(x,y)=x*y+2, start={3}
A192518:  f(x,y)=(x+1)(y+1), start={2}
A192519:  f(x,y)=floor(x*y/2), start={3}
A192520:  f(x,y)=floor(x*y/2), start={5}
A192521:  f(x,y)=floor((x+1)(y+1)/2), start={2}
A192522:  f(x,y)=floor((x-1)(y-1)/2), start={5}
A192523:  f(x,y)=2x*y-x-y, start={2}
A192525:  f(x,y)=2x*y-x-y, start={3}
A192524:  f(x,y)=4x*y-x-y, start={1}
A192528:  f(x,y)=5x*y-x-y, start={1}
A192529:  f(x,y)=3x*y-x-y, start={2}
A192531:  f(x,y)=3x*y-2x-2y, start={2}
A192533:  f(x,y)=x^2+y^2-x*y, start={2}
A192535:  f(x,y)=x^2+y^2+x*y, start={1}
A192536:  f(x,y)=x^2+y^2-floor(x*y/2), start={1}
A192537:  f(x,y)=x^2+y^2-x*y/2, start={2}
A192539:  f(x,y)=2x*y+floor(x*y/2), start={1}

			1^2+1^2=2, 1^2+2^2=5, 2^2+2^2=8, 1^2+5^2=26.

Reinhard Zumkeller, Table of n, a(n) for n = 1..6171

Cf. A009293, A008318.

import Data.Set (singleton, deleteFindMin, insert)
a192476 n = a192476_list !! (n-1)
a192476_list = f [1] (singleton 1) where
   f xs s =
     m : f xs' (foldl (flip insert) s' (map (+ m^2) (map (^ 2) xs')))
     where xs' = m : xs
           (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 15 2011

start = {1}; f[x_, y_] :=  x^2 + y^2  (* start is a subset of t, and if x,y are in t then f(x,y) is in t. *)
b[z_] :=  Block[{w = z}, Select[Union[Flatten[AppendTo[w, Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # < 30000 &]];
t = FixedPoint[b, start] (* A192476 *)
Differences[t]
(* based on program by Robert G. Wilson v at A009293 *)

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 *)

A180337 Numbers which cannot be expressed as a sum 1 + p1 + p1p2 + p1p2*p3 + ... for some collection of primes {p1, p2, p3, ...}.

Original entry on oeis.org

2, 5, 11, 23, 26, 47, 56, 95, 116, 122, 236, 254, 518, 530, 1082, 2210

Offset: 1

Jack W Grahl, Aug 28 2010

nonn

I conjecture, but have not been able to prove, that this sequence is finite with only the terms given above. In that case it can be constructed by taking a1=2, and adjoining all numbers aj*ak + 1, where aj and ak are two prime members of the sequence.

Any number which can be expressed as p*q + 1, where p is prime and q does not belong to the sequence, does not belong to the sequence either.

			8 is not a member of the sequence since it is equal to 1 + 7.
9 is not a member of the sequence since it can be written 1 + 2 + 2*3.
10 is not a member of the sequence since it is equal to 1 + 3 + 3*2.
11 is a member of the sequence. If 11 could be written in this form, then p1 must divide 10. We would have 11 = 1 + p1(1 + p2 + ...), which would imply that 5 is not a member of the sequence if p1 = 2, or vice versa. Since both 2 nor 5 are members, so is 11.

All terms given above belong to A009293.

Cf. A317240, A317242.

q:= proc(n) option remember; is(n=1 or ormap(p->
      q((n-1)/p), numtheory[factorset](n-1)))
    end:
remove(q, [$1..3000])[];  # Alois P. Heinz, Jul 24 2018

q[1] = True; q[2] = False;
q[n_] := q[n] = AnyTrue[FactorInteger[n-1][[All, 1]], q[(n-1)/#]&];
Select[Range[3000], !q[#]&] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)

#!/usr/bin/perl $max = 10; if (defined($ARGV[0])) { $max = $ARGV[0]; } $primes{1} = 0; $list{1} = 1; $list{2} = 0; print "2, "; foreach $k (2..$max){ $p = 1; $l = 0; foreach $j (1..$k) { if ($primes{$j}){ if (($k % $j) == 0){ $p = 0; if ($list{$k / $j}){ $l = 1; } } } } $primes{$k} = $p; $list{$k + 1} = $l || $p; if (!$list{$k + 1}){ $t = $k + 1; print "$t, " } }

A317240(a(n)) = 0. - Alois P. Heinz, Jul 24 2018

A093906 Start with {2} and close under the operations XY and XY+1; sequence gives complete list of numbers that do not appear.

Original entry on oeis.org

1, 3, 6, 7, 12, 13, 14, 15, 24, 27, 28, 29, 30, 31, 48, 49, 54, 57, 58, 59, 60, 61, 62, 63, 97, 98, 108, 109, 114, 117, 118, 119, 120, 123, 124, 127, 194, 197, 217, 218, 219, 228, 229, 237, 238, 239, 240, 241, 246, 247, 248, 249, 389, 394, 434, 439, 457, 458, 479, 482, 492, 493, 497, 498, 499, 788, 789, 878, 879, 917, 959, 984, 985, 986, 994, 997, 998, 1579, 1757, 1758, 1759, 1968, 1971, 1988, 1994, 1997, 3514, 3517, 3518, 3989, 7028, 7034, 7037, 7978, 14074, 28148

Offset: 1

Franklin T. Adams-Watters, May 25 2004

If we start with {2} and close under the operations XY-1 and XY+1, we get the complement of A014574 (midpoints of twin primes), except for the absence of 1.

Cf. A009388 and A009293.

Corrected by Don Reble, May 25 2004

A180432 Start with 3. If a, b in sequence, so is ab+1.

Original entry on oeis.org

3, 10, 31, 94, 101, 283, 304, 311, 850, 913, 934, 941, 962, 1011, 2551, 2740, 2803, 2824, 2831, 2887, 2915, 3034, 3041, 3111, 3132, 7654, 8221, 8410, 8473, 8494, 8501, 8662, 8746, 8774, 8837, 9103, 9124, 9131, 9334, 9341, 9397, 9411, 9425, 9495, 9621

Offset: 1

Robert G. Wilson v, Sep 04 2010

nonn

The difference between any two terms is == 0 (mod 7).

Cf. A009293.

f[s_List] := Select[ Union@ Join[s, Union[1 + Times @@@ Subsets[ Join[s, s], {2}]]], # < 10000 &]; Nest[f, {3}, 8]

cp's OEIS Frontend

A192476 Monotonic ordering of set S generated by these rules: if x and y are in S then x^2 + y^2 is in S, and 1 is in S.

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

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Haskell

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

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Mathematica

A180337 Numbers which cannot be expressed as a sum 1 + p1 + p1*p2 + p1*p2*p3 + ... for some collection of primes {p1, p2, p3, ...}.

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Perl

Formula

A093906 Start with {2} and close under the operations XY and XY+1; sequence gives complete list of numbers that do not appear.

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Extensions

A180432 Start with 3. If a, b in sequence, so is ab+1.

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Mathematica

A180337 Numbers which cannot be expressed as a sum 1 + p1 + p1p2 + p1p2*p3 + ... for some collection of primes {p1, p2, p3, ...}.