cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A167521 Positive integers not occurring in A167520.

Original entry on oeis.org

11, 29, 47, 67, 85, 105, 123, 143, 161, 181, 182, 184, 187, 190, 193, 196, 199, 202, 205, 209, 239, 266, 296, 323, 353, 380, 410, 445, 446, 448, 451, 454, 457, 460, 463, 467, 497, 527, 554, 584, 614, 641, 671, 701, 727, 728, 730, 733, 736, 739, 742, 745, 748
Offset: 1

Views

Author

M. F. Hasler, Nov 05 2009

Keywords

Comments

Equivalently, positions of zero digits in A167520 (when all terms are concatenated).
The first differences are (18, 18, 20, 18, 20, 18, 20, 18, 20, 1, 2, 3, 3, 3, 3, 3, 3, 3, 4, 30, 27, 30, 27, 30, 27, 30, 35, 1, 2, 3, 3, 3, 3, 3, 4, ...)

Examples

			The first occurrence of the digit '0' in A167520 is as the least significant digit of A167520(10)=10, which occurs at position 11, thus a(1)=11. Equivalently, this is the least positive integer missing in A167520.
The next occurrence of the digit '0' in A167520 is at position 29 (and 29 is the second positive integer not occurring in A167520), thus a(2)=29.

Crossrefs

Cf. A167500-A167503, A167520-A167523.

Programs

  • PARI
    base(n,b=10) = { my( a=[ n%b ]); while( 0=n & for(i=a[n-1]+1,#b,b[i] & (a=concat(a,i)) & break); #a

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

Views

Author

M. F. Hasler, Nov 19 2009

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A098645, A167519, A167520.

Programs

  • PARI
    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

Views

Author

M. F. Hasler, Nov 19 2009

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A098645, A167519, A167520, A167452.

Programs

  • PARI
    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

Views

Author

M. F. Hasler, Nov 19 2009

Keywords

Comments

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.

Examples

			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.

Crossrefs

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

Programs

  • PARI
    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

Views

Author

M. F. Hasler, Nov 19 2009

Keywords

Comments

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.

Examples

			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.

Crossrefs

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

Programs

  • PARI
    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

Views

Author

M. F. Hasler, Nov 19 2009

Keywords

Comments

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.

Examples

			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.

Crossrefs

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

Programs

  • PARI
    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

Views

Author

M. F. Hasler, Nov 19 2009

Keywords

Comments

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.

Examples

			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.

Crossrefs

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

Programs

  • PARI
    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

Views

Author

M. F. Hasler, Nov 19 2009

Keywords

Comments

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.

Examples

			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.

Crossrefs

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

Programs

  • PARI
    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

Views

Author

M. F. Hasler, Nov 19 2009

Keywords

Comments

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.

Examples

			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.

Crossrefs

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

Programs

  • PARI
    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)) ])

A167522 Positive integers not occurring in A167500.

Original entry on oeis.org

3, 5, 6, 11, 12, 13, 15, 16, 19, 21, 25, 27, 28, 29, 32, 33, 35, 37, 39, 40, 42, 45, 47, 53, 54, 55, 58, 60, 65, 72, 73, 74, 76, 78, 79, 81, 82, 84, 85, 88, 90, 92, 93, 96, 98, 102, 105, 106, 108, 112, 115, 116, 117, 118, 121, 122, 123, 127, 128, 130, 133, 134, 139, 141
Offset: 1

Views

Author

M. F. Hasler, Nov 05 2009

Keywords

Comments

Equivalently, positions of zero digits in A167502 (when all terms are concatenated).

Crossrefs

Cf. A167500-A167503, A167520-A167523.

Programs

  • PARI
    {a=b=[]; for(n=1,99, #b>=n & for(i=a[n-1]+1,#b,b[i] & (a=concat(a,i)) & break); #a
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