A167457 -id:A167457 - cp's OEIS frontend

A167450 Smallest sequence which lists the position of digits "8" in the sequence.

2, 8, 9, 10, 11, 88, 880, 900, 901, 902, 903, 904, 905, 906, 907, 909, 910, 911, 912, 913, 914, 915, 916, 917, 919, 920, 921, 922, 923, 924, 925, 926, 8000, 9000, 9001, 9002, 9003, 9004, 9005, 9006, 9007, 9009, 9010, 9011, 9012, 9013, 9014, 9015, 9016, 9017

Offset: 1

M. F. Hasler, Nov 19 2009

The lexicographically earliest sequence such that a(1),a(2),a(3),... is the (increasing) list of the positions of digits "8" in the string obtained by concatenating all these terms, written in base 10.

			We cannot have a(1)=1 (since then there's no "8" in the first place), but a(1)=2 is possible.
This implies that a(2) must start with a digit "8", so a(2)=8 is the smallest possible choice.
This allows us to go on with a(3)=9, a(4)=10, a(5)=11, but then must be follow 4 digits "8" (the 8th through 11th digit of the sequence), so a(6)=88 and a(7)=880 are the smallest possible choices.
Then the reasoning continues in analogy with A167452-A167457.

Cf. A098645, A167519, A167520, A167451 - A167457.

concat([ [2,8,9,10,11,88,880], vector((88-11-1)\3,i,900-(i<=8)+i+(i>=18)), [8000], select(x->x%10-8 & x\10%10-8,vector((880-88)\4,i,9000-1+i)) ])

A167451 Smallest sequence which lists the position of digits "9" in the sequence.

Original entry on oeis.org

2, 9, 10, 11, 12, 99, 990, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1020, 1021, 1022, 1900, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2010, 2011, 2012, 2013, 2014, 2015, 2016

Offset: 1

M. F. Hasler, Nov 19 2009

The lexicographically earliest sequence such that a(1),a(2),a(3),... is the (increasing) list of the positions of digits "9" in the string obtained by concatenating all these terms, written in base 10.

			We cannot have a(1)=1 (since then there's no "9" in the first place), but a(1)=2 is possible.
This implies that a(2) must start with a digit "9", so a(2)=9 is the smallest possible choice.
This allows us to go on with a(3)=10, a(4)=11, a(5)=12, but then must be follow 4 digits "9" (the 9th through 12th digit of the sequence), so a(6)=99 and a(7)=990 are the smallest possible choices.
Then the reasoning continues in analogy with A167450-A167457.

Cf. A098645, A167519, A167520, A167450 - A167457.

concat([ [2,9,10,11,12,99,990], vector((99-11-1)\4,i,1000-(i<=9)+i+(i>=19)), [1900], select(x->x%10-9 & x\10%10-9,vector((990-99)\4,i,2000-1+i)) ])
/* The following code checks a sequence for consistency (i.e., the given digit occurs exactly at positions given by the terms), but it does not check the monotonicity neither the minimality.
In case of a contradiction, it returns [n,pos,d] where n is the index of the term, pos is the position in the concatenation, and d is the digit for which the contradiction occurred.
If d is different from the given digit, the term a(n) said that there should be that digit at position pos, but we found d instead.
If d equals the given digit, we found d at position pos, but the term a(n) said that the next d should occur elsewhere. */
check_self(a,d=9)={ my(t=Vecsmall(concat(concat([""],a))),c=0); d+=48;
for( i=1,#a, a[i]>#t & break; t[a[i]]==d || return([i,a[i],t[a[i]]-48]));
for( i=1,#t, t[i]==d & (a[c++ ]==I || return([c,i,d-48]))) /* no contradiction => empty result */}

cp's OEIS Frontend

A167450 Smallest sequence which lists the position of digits "8" in the sequence.

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