A299952 The sum a(n) + a(n+1) is a substring of the concatenation of all terms up to a(n+1). Lexicographic first sequence of positive integers without duplicate terms having this property.

1, 10, 99, 11, 80, 19, 61, 30, 31, 49, 12, 2, 4, 5, 3, 6, 7, 15, 9, 13, 17, 14, 8, 16, 20, 25, 23, 22, 26, 27, 18, 34, 28, 21, 24, 29, 32, 35, 36, 44, 37, 43, 38, 33, 41, 39, 42, 40, 51, 45, 46, 47, 52, 57, 53, 56, 54, 55, 63, 59, 50, 60, 58, 64, 66, 65, 74, 48, 62, 68, 71, 77, 72, 67, 78, 70

Offset: 1

Author

Eric Angelini and Lars Blomberg, Feb 22 2018

Keywords

• nonn
• base

Comments

The sequence starts with a(1) = 1 and is always extended with the smallest integer not yet present that does not lead to a contradiction.

This is probably a permutation of the natural numbers (after 10000 terms, the smallest integer not yet present is 9990).

Examples

			a(1) + a(2) = 1 + 10 = 11 and “11” is visible in [1,10]
a(2) + a(3) = 10 + 99 = 109 and “109” is visible in [10,99]
a(3) + a(4) = 99 + 11 = 110 and “110” is visible in [1,10]
a(4) + a(5) = 11 + 80 = 91 and “91” is visible in [99,11]
a(5) + a(6) = 80 + 19 = 99 and “99” is visible in [99]
a(6) + a(7) = 19 + 61 = 80 and “80” is visible in [80]
...
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A300000.

For a different constraint on a(n)+a(n+1) (must have a digit '1'), see A299957 and A299970, A299982, ..., A299988, A299969 (nonnegative analog with digit 0, 2, ..., 9), A299971, A299972, ..., A299979 (positive analog with digit 0, 2, ..., 9).

Cf. A299980, A299981, A299402, A299403, A298974, A298975, A299996, A299997, A298978, A298979 for the analog using multiplication: a(n)*a(n+1) has a digit 0, resp. 1, ..., resp. 9.

Programs

Mathematica

Nest[Function[a, Append[a, Block[{k = 1, d}, While[Nand[FreeQ[a, k], SequenceCount[Flatten@ IntegerDigits[Append[a, k]], IntegerDigits[a[[-1]] + k]] > 0], k++]; k]]], {1}, 75] (* Michael De Vlieger, Feb 22 2018 *)

PARI

a(n,show=1,a=1,s=a,u=[a],t,m)={for(n=2,n, show&&print1(a","); for(k=u[1]+1,oo, setsearch(u,k)&&next;m=Mod(a+k,10^#Str(a+k));t=s*10^#Str(k)+k; until(k>=t\=10,t==m&&(a=k)&&break(2)));s=s*10^#Str(a)+a;u=setunion(u,[a]); u[2]==u[1]+1&&u=u[^1]);a} \\ M. F. Hasler, Feb 22 2018