A167519 -id:A167519 - cp's OEIS frontend

A210414 List the positions of all digits 0 in the concatenation of all terms, not necessarily in order. This is the lexicographically earliest such sequence.

3, 10, 6, 20, 9, 100, 14, 30, 18, 40, 50, 24, 60, 28, 70, 80, 34, 90, 38, 1000, 39, 46, 101, 110, 54, 102, 59, 200, 64, 103, 69, 300, 74, 104, 79, 400, 84, 105, 89, 500, 94, 106, 99, 100000000, 1010, 108, 112, 121, 201, 127, 202, 133, 203, 139, 204, 145, 205, 151

Offset: 1

Paolo P. Lava, Mar 26 2012

See A210415 for comments, links, and code.

			The sequence cannot start with 0 (offset starting from 1), 1 (in the first position we have 1, not 0) or 2 (the second entry cannot start with 0). So the sequence starts with 3. The next term is 10, which is the minimum number with 0 as its second digit. And so on.

Danny Rorabaugh, Table of n, a(n) for n = 1..1000

Cf. A210415-A210423. See A167519 for another version.

Name edited and a(22)-a(58) corrected by Danny Rorabaugh, Nov 30 2015

A210415 List the positions of all digits 1 in the concatenation of all terms, not necessarily in order. This is the lexicographically earliest such sequence.

Original entry on oeis.org

1, 3, 10, 6, 11, 7, 21, 13, 15, 17, 19, 101, 24, 100, 29, 102, 34, 103, 39, 104, 44, 105, 49, 106, 54, 107, 59, 108, 64, 109, 69, 110, 70, 76, 111, 77, 78, 85, 112, 86, 91, 94, 113, 95, 211, 1111, 11111, 1110, 115, 116, 118, 119, 121, 122, 124, 125, 127, 129

Offset: 1

Paolo P. Lava, Mar 26 2012

Original name: "A self-describing sequence: The a(n) say the positions of the digits 1 inside the sequence when it is read as a string of digits."
For each n, the digit in position a(n) is equal to 1. At each step, choose the minimum integer not yet present in the sequence and not leading to a contradiction.
This sequence had been mentioned by Wasserman in 2008, cf. A098645, a variant of this sequence with additional restriction a(n+1) > a(n). - M. F. Hasler, Oct 08 2013

			The sequence starts with 1: the first digit is equal to 1. In the second position we cannot write 2 because the second digit would not be 1 but 2. Then we write 3. The third digit must be 1 and the minimum number starting with 1 is 10. And so on.

Danny Rorabaugh, Table of n, a(n) for n = 1..1000
Index to the OEIS: Entries related to self-referencing sequences.

This construction for other decimal digits is A210414, A210416, A210417, A210418, A210419, A210420, A210421, A210422, A210423.

Variants of this sequence include A098645, A098670, A114134, A167519.

#Returns the first n terms of this sequence for digit d
def dig_loc(d,n):
  L, S = [], ""
  while len(L)lenS or S[new-1]==str(d) ) and ( (new-lenS-1 not in range(ext)) or a[new-lenS-1]==str(d) ) and ( d!=0 or lenS+ext+1!=new ):
          L.append(new)
          S += str(new)
          break
        else: new = 0
  return L
dig_loc(1,58) # Danny Rorabaugh, Nov 27 2015

Edited by M. F. Hasler, Oct 10 2013

A167452 Smallest sequence which lists the position of digits "2" in the sequence.

Original entry on oeis.org

3, 4, 22, 30, 31, 33, 34, 35, 36, 37, 38, 42, 43, 44, 45, 52, 202, 222, 223, 302, 2220, 3000, 3200, 3300, 3301, 3303, 3304, 3305, 3306, 3307, 3308, 3309, 3310, 3311, 3313, 3314, 3315, 3316, 3317, 3318, 3319, 3330, 3331, 3333, 3334, 3335, 3336, 3337, 3338

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 "1" in the string obtained by concatenating all these terms, written in base 10.

			We cannot have a(1)=1 (since then there's no 2 in the first place), nor a(1)=2 (since then the first occurrence of a "2" would be at position 1).
But a(1)=3 is possible, "predicting" that the first occurrence of a digit "2" will be in the 3rd digit.
Then a(2)=4 is the smallest possible choice for a(2).
The next two digits (= the 3rd and 4th digit of the sequence) must be a "2", in view of a(1) and a(2). Thus a(3)=22 is the smallest possible choice.
This means that the next digit "2" will occur as the 22nd digit of the sequence, so the following terms are the least possible numbers without digit "2": 30,31,33,...,38. These make up digits 5 to 20 of the sequence.
The following number must have a "2" as second digit, the smallest possibility is 42.

Cf. A098645, A167519, A167520.

concat([ [3,4,22], vector((22-4)/2-1,i,i+30-(i<=2)), vector(4,i,42+i-1), [52,202,222,223,302,2220,3000,3200], select(x -> x%10-2 & x\10%10-2 & x\100%10-2, vector((202-52)\4+13,i,3300+i-1)) ])

A167453 Smallest sequence which lists the position of digits "3" in the sequence.

Original entry on oeis.org

2, 3, 30, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 63, 330, 333, 3333, 33333, 33400, 40300, 40400, 40401, 40402, 40404, 40405, 40406, 40407, 40408, 40409, 40410, 40411, 40412, 40414, 40415, 40416, 40417, 40418, 40419, 40420

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 "3" in the string obtained by concatenating all these terms, written in base 10.

			We cannot have a(1)=1 (since then there's no "3" in the first place), but a(1)=2 is possible.
Then a(2)=3 is a possible choice and certainly the smallest.
This "predicts" that a(3) starts with a digit "3", so a(3)=30 is the smallest possible choice.
The next digit "3" must not appear until the 30th digit of the sequence, so we fill in terms 40,41,42,44,45... (omitting 43 which has a digit "3").
Now it happens that the term 53 would correspond to digits # 29 and 30=a(3) of the sequence, so we can simply continue with this and 4 more terms, up to 57.
The next term must have its second digit (digit # 40=a(4) of the sequence) equal to 3, so 63 is the smallest choice.
The terms a(5)=41, a(6)=42 leave 330 as the smallest possible choice for the next term.
The terms 44,45,46 and 47,48,49,50 and 51,52,53,54,55 lead to the subsequent terms 333, 3333, 33333.

Cf. A098645, A167519, A167520, A167452.

concat([[2,3,30],vector((40-4)/2-1,i,40-(i<=3)+i), [63, 330, 333, 3333, 33333, 33400,40300], select(x->x%10-3 & x\10%10-3,vector((330-63)\5+10,i,40400+i-1)) ])

A167457 Smallest sequence which lists the position of digits "7" in the sequence.

Original entry on oeis.org

2, 7, 8, 9, 10, 77, 770, 800, 801, 802, 803, 804, 805, 806, 808, 809, 810, 811, 812, 813, 814, 815, 816, 818, 819, 820, 821, 822, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 838, 839, 840, 841, 842, 843, 844, 845, 846, 848, 849, 850, 851, 852, 853, 854

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 "7" in the string obtained by concatenating all these terms, written in base 10.

			We cannot have a(1)=1 (since then there's no "7" in the first place), but a(1)=2 is possible.
Then a(2) must start with a digit "7", so a(2)=7 is the smallest possible choice.
This allows us to go on with a(3)=8, a(4)=9, a(5)=10, but then must be follow 4 digits "6" (the 7th through 10th digit of the sequence), so a(6)=77 and a(7)=770 are the smallest possible choices.
Then the reasoning continues in analogy with A167452-A167456.

Cf. A098645, A167519, A167520, A167452 - A167456.

concat([ [2,7,8,9,10,77,770], vector((77-10)\3-1,i,800-(i<=7)+i+(i>=17)), [827], select(x->x%10-7 & x\10%10-7,vector((770-77)\3+20,i,827+i)) ])

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

Original entry on oeis.org

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 */}

A167454 Smallest sequence which lists the position of digits "4" in the sequence.

Original entry on oeis.org

2, 4, 5, 44, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 400, 500, 4444, 5444, 44444, 45444, 444000, 500000, 500001, 500002, 500003, 500005, 500006, 500007, 500008, 500009, 500010, 500011, 500012, 500013, 500015, 500016

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 "4" in the string obtained by concatenating all these terms, written in base 10.

			We cannot have a(1)=1 (since then there's no "4" in the first place), but a(1)=2 is possible.
Then a(2)=4 is the smallest possible choice.
This allows us to take a(3)=5, but this must be followed by two digits "4" (the 4th and 5th of the sequence), thus a(4)=44. Terms a(5) through a(5+(44-6)/2) are now to be filled with 50,51,..., omitting terms with a digit "4".
The last term of this sequence is 70, which must be followed by 400 (whose first digit is the 44th digit of the sequence), 500, and then 4444 (digits 50-53), 5444 (digits 54-57), 44444 (digits 58-62), 45444 (digits 63-67), 444000 (digits 68-73). This "predicts" that a(3) starts with a digit "3", so a(3)=30 is the smallest possible choice.
The next digit "3" must not appear until the 30th digit of the sequence, so we fill in terms 40,41,42,44,45... (omitting 43 which has a digit "3").
Now it happens that the term 53 would correspond to digits # 29 and 30=a(3) of the sequence, so we can simply continue with this and 4 more terms, up to 57.
The next term must have its second digit (digit # 40=a(4) of the sequence) equal to 3, so 63 is the smallest choice.
The terms a(5)=41, a(6)=42 leave 330 as the smallest possible choice for the next term.
The terms 44,45,46 and 47,48,49,50 and 51,52,53,54,55 lead to the subsequent terms 333, 3333, 33333.

Cf. A098645, A167519, A167520, A167452, A167453.

concat([[2,4,5,44],vector((44-6)/2,i,50-(i<=4)+i+(i>=14)),[400,500,4444,5444,44 444,45 444, 444000], select(x->x%10-4 & x\10%10-4,vector((400-70)\6+10,i,500 000+i-1)) ])

A167455 Smallest sequence which lists the position of digits "5" in the sequence.

Original entry on oeis.org

2, 5, 6, 7, 55, 56, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 550, 605, 5555, 6555, 55555, 56555, 555555, 600000, 600001, 600002, 600003, 600004, 600006, 600007, 600008, 600009, 600010, 600011, 600012, 600013

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 "5" in the string obtained by concatenating all these terms, written in base 10.

			We cannot have a(1)=1 (since then there's no "5" in the first place), but a(1)=2 is possible.
Then a(2) must start with a digit "5", so a(2)=5 is the smallest possible choice.
This allows us to go on with a(3)=6, a(4)=6, but then must be follow 3 digits "5" (the 5th, 6th and 7th digit of the sequence), so a(5)=55 and a(6)=56 are the smallest possible choice.
The reasoning continues in analogy with A167452-A167454.

Cf. A098645, A167519, A167520, A167452, A167453, A167454.

concat([ [2,5,6,7,55,56], vector((55-8)\2,i,60-(i<=5)+i+(i>=15)), [550, 605, 5555, 6555, 55 555, 56 555, 555 555], select(x->x%10-5 & x\10%10-5,vector((550-84)\6+10,i,600 000+i-1)) ])

A167456 Smallest sequence which lists the position of digits "6" in the sequence.

Original entry on oeis.org

2, 6, 7, 8, 9, 66, 660, 700, 701, 702, 703, 704, 705, 707, 708, 709, 710, 711, 712, 713, 714, 715, 717, 718, 719, 760, 770, 771, 772, 773, 774, 775, 777, 778, 779, 780, 781, 782, 783, 784, 785, 787, 788, 789, 790, 791, 792, 793, 794, 795, 797, 798, 799, 800

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 "6" in the string obtained by concatenating all these terms, written in base 10.

			We cannot have a(1)=1 (since then there's no "6" in the first place), but a(1)=2 is possible.
Then a(2) must start with a digit "6", so a(2)=6 is the smallest possible choice.
This allows us to go on with a(3)=7, a(4)=8, a(5)=9, but then must be follow 4 digits "6" (the 6th through 9th digit of the sequence), so a(6)=66 and a(7)=660 are the smallest possible choices.
Then the reasoning continues in analogy with A167452-A167455.

Cf. A098645, A167519, A167520, A167452 - A167455.

concat([ [2,6,7,8,9,66,660], vector((66-9)\3-1,i,700-(i<=6)+i+(i>=16)), [760], select(x->x%10-6 & x\10%10-6,vector((660-66)\3+10,i,770+i-1)) ])

cp's OEIS Frontend

A210414 List the positions of all digits 0 in the concatenation of all terms, not necessarily in order. This is the lexicographically earliest such sequence.

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A210415 List the positions of all digits 1 in the concatenation of all terms, not necessarily in order. This is the lexicographically earliest such sequence.

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A167452 Smallest sequence which lists the position of digits "2" in the sequence.

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A167453 Smallest sequence which lists the position of digits "3" in the sequence.

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A167457 Smallest sequence which lists the position of digits "7" in the sequence.

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PARI

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

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PARI

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

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PARI

A167454 Smallest sequence which lists the position of digits "4" in the sequence.

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PARI

A167455 Smallest sequence which lists the position of digits "5" in the sequence.

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