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.

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A299869 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 6.

Original entry on oeis.org

6, 60, 594, 5945, 59454, 594535, 5945351, 59453514, 594535135, 5945351351, 59453513510, 594535135104, 5945351351035, 59453513510351, 594535135103509, 5945351351035091, 59453513510350914, 594535135103509135, 5945351351035091351, 59453513510350913508, 594535135103509135082, 5945351351035091350820
Offset: 1

Views

Author

Eric Angelini and Jean-Marc Falcoz, Feb 21 2018

Keywords

Comments

The sequence starts with a(1) = 6 and is always extended with the smallest integer not yet present in the sequence and not leading to a contradiction.

Examples

			6 + 60 = 66 which is the concatenation of 6 and 6.
6 + 60 + 594 = 660 which is the concatenation of 6, 6 and 0.
6 + 60 + 594 + 5945 = 6605 which is the concatenation of 6, 6, 0 and 5.
From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 660 - 66 = 594, a(4) = 6605 - 660 = 5945, etc. - _M. F. Hasler_, Feb 22 2018

Links

Crossrefs

A300000 is the lexicographically first sequence of this type, with a(1) = 1.
Cf. A299865, ..., A299872 for variants with a(1) = 2, ..., 9.

Programs

  • PARI
    a(n,show=1,a=6,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018

Formula

a(n) = c(n) - c(n-1), where c(n) = concatenation of the first n digits, c(n) ~ 0.66*10^n, a(n) ~ 0.59*10^n. See A300000 for the proof. - M. F. Hasler, Feb 22 2018

A299870 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 7.

Original entry on oeis.org

7, 70, 693, 6936, 69363, 693624, 6936243, 69362433, 693624324, 6936243243, 69362432430, 693624324303, 6936243243024, 69362432430243, 693624324302427, 6936243243024273, 69362432430242733, 693624324302427324, 6936243243024273243, 69362432430242732426, 693624324302427324262, 6936243243024273242622
Offset: 1

Views

Author

Eric Angelini and Jean-Marc Falcoz, Feb 21 2018

Keywords

Comments

The sequence starts with a(1) = 7 and is always extended with the smallest integer not yet present in the sequence and not leading to a contradiction.

Examples

			7 + 70 = 77 which is the concatenation of 7 and 7.
7 + 70 + 693 = 770 which is the concatenation of 7, 7 and 0.
7 + 70 + 693 + 6936 = 7706 which is the concatenation of 7, 7, 0 and 6.
From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 770 - 77 = 693, a(4) = 7706 - 770 = 6936, etc. - _M. F. Hasler_, Feb 22 2018

Links

Crossrefs

A300000 is the lexicographically first sequence of this type, with a(1) = 1.
Cf. A299865, ..., A299872 for variants with a(1) = 2, ..., 9.

Programs

  • PARI
    a(n,show=1,a=7,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018

Formula

a(n) = c(n) - c(n-1), where c(n) = concatenation of the first n digits, c(n) ~ 0.77*10^n, a(n) ~ 0.69*10^n. See A300000 for the proof. - M. F. Hasler, Feb 22 2018

A359482 Lexicographically earliest sequence of distinct terms > 0 such that the sum a(n) + a(n+1) is a substring of the concatenation (a(n), a(n+1)).

Original entry on oeis.org

1, 10, 99, 889, 8009, 1101, 9089, 80718, 100284, 183899, 206021, 396118, 215703, 354632, 108578, 469891, 229021, 61195, 34146, 7321, 13817, 3536, 1825, 749, 167, 508, 324, 2096, 4337, 2958, 2870, 4171, 12941, 16470, 30560, 25465, 21056, 35296, 17665, 35927, 23345, 10106, 548, 279, 516, 1094, 3228, 5302
Offset: 1

Views

Author

Eric Angelini and Hans Havermann, Jul 03 2023

Keywords

Comments

Is this sequence a permutation of the integers > 0?
I conjecture that it isn't, and more specifically, that a(1) = 1 is the only single-digit term, and a(2) = 10 the only multiple of 10 below 100. See Examples for other terms of the form x*10^k, 1 <= x <= 9. - M. F. Hasler, Jul 03 2023

Examples

			1 + 10 = 11 and 11 is a substring of concat(1, 10) = 110.
10 + 99 = 109 and 109 is a substring of concat(10, 99) = 1099.
99 + 889 = 988 and 988 is a substring of concat(99, 889) = 99889.
889 + 8009 = 8898 and 8898 is a substring of 8898009.
8009 + 1101 = 9110 and 9110 is a substring of 80091101, etc.
Some examples of terms of the form x*10^k, x < 10: a(2136) = 800, a(4204) = 1000, a(6246) = 900, a(6618) = 100, a(11268) = 400, a(17446) = 10000, a(39292) = 600, a(44989) = 700, a(91359) = 300, ... - _M. F. Hasler_, Jul 03 2023

Links

Crossrefs

Cf. A300000.

Programs

  • PARI
    A359482_first(n)={my(ok(a,k)=my(c=a*10^logint(k*10,10)+k); k=10^logint(10*a+=k,10); until(a>c\=10, c%k==a&& return(1)), U=[], a=0); vector(n,n, my(k=1); while(setsearch(U,k)|| !ok(a,k), k++); U=setunion(U,[k]); a=k)} \\ Becomes slow for n > 10. - M. F. Hasler, Jul 03 2023

A347260 Lexicographically earliest sequence S of distinct nonnegative terms such that the digits of (a(n) + a(n+1)) are the first n digits of S.

Original entry on oeis.org

1, 0, 10, 91, 919, 9190, 91901, 919018, 9190173, 91901746, 919017453, 9190174538, 91901745381, 919017453809, 9190174538100, 91901745380991, 919017453809928, 9190174538099262, 91901745380992639, 919017453809926380, 9190174538099263811, 91901745380992638108, 919017453809926381082, 9190174538099263810819, 91901745380992638108199, 919017453809926381081990
Offset: 1

Views

Author

Eric Angelini and Carole Dubois, Sep 30 2021

Keywords

Comments

A self-describing sequence.

Examples

			a(1) + a(2) = 1 + 0 = 1 and this 1 is the first digit of S;
a(2) + a(3) = 0 + 10 = 10 and 1, 0 are the first 2 digits of S;
a(3) + a(4) = 10 + 91 = 101 and 1, 0, 1 are the first 3 digits of S;
a(4) + a(5) = 91 + 919 = 1010 and 1, 0, 1, 0 are the first 4 digits of S;
a(5) + a(6) = 919 + 9190 = 10109 and 1, 0, 1, 0, 9 are the first 5 digits of S;
etc.

Links

Crossrefs

Cf. A300000.

Programs

  • Python
    a=[1]
while len(a)<20:a.append(int("".join(map(str,a))[:len(a)])-a[-1])
print(a) # Dominic McCarty, Mar 21 2025

A364201 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) in binary is a substring of the concatenation of all terms a(1)..a(n) in binary.

Original entry on oeis.org

1, 2, 3, 5, 11, 7, 16, 9, 6, 18, 4, 13, 10, 15, 12, 23, 20, 8, 27, 19, 36, 26, 22, 17, 21, 31, 25, 14, 29, 28, 30, 57, 24, 32, 39, 43, 40, 34, 38, 46, 33, 35, 42, 37, 55, 44, 58, 48, 56, 52, 41, 45, 64, 63, 54, 61, 60, 49, 50, 51, 65, 47, 67, 88, 132, 73, 76, 68, 109, 59, 82, 87, 62, 98, 69, 70
Offset: 1

Views

Author

Scott R. Shannon, Jul 13 2023

Keywords

Comments

In the first 10000 terms the smallest number that has not yet appeared is 7026; it is conjectured all numbers eventually appear.
The fixed points begin 1, 2, 3, 29, 48, 68, 96, 182, 471, 839, ... . It is likely there are infinitely more.

Examples

			a(2) = 2 as a(1) + 2 = 1 + 2 = 3 = 11_2, which is a substring of "a(1)"_2 + "2"_2 = "1" + "10" = "110".
a(4) = 5 as a(1) + a(2) + a(3) + 5 = 1 + 2 + 3 + 5 = 11 = 1011_2, which is a substring "a(1)"_2 + "a(2)"_2 + "a(3)"_2 + "5"_2 = "1" + "10" + "11" + "101" = "11011101".
a(5) = 11 as a(1) + a(2) + a(3) + a(4) + 11 = 1 + 2 + 3 + 5 + 11 = 22 = 10110_2, which is a substring "a(1)"_2 + "a(2)"_2 + "a(3)"_2 + "a(4)"_2 + "11"_2 = "1" + "10" + "11" + "101" + "1011" = "110111011011".

Links

Crossrefs

Cf. A363186 (base 10), A359482, A007088, A300000, A339144, A357082.
Previous Showing 11-15 of 15 results.

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