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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A285470 Numbers k where "2" appears as the second digit of the decimal representation.

Original entry on oeis.org

12, 22, 32, 42, 52, 62, 72, 82, 92, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 620, 621, 622, 623, 624, 625, 626, 627
Offset: 1

Views

Author

Jamie Robert Creasey, Apr 19 2017

Keywords

Comments

To find a(n), concatenate the first digit of n with 2 and then the other digits (if any) from n. See example. - David A. Corneth, Jun 12 2017

Examples

			a(21) = 221, a(36) = 326.
As the first digit of 983 is 9, and the others are 83, a(983) = 9283. - _David A. Corneth_, Jun 12 2017

Links

Crossrefs

Cf. A011532 (containing 2), A052404 (without 2), A217394 (starting with 2).

Programs

  • Maple
    seq(seq(seq(a*10^d + 2*10^(d-1)+c, c=0..10^(d-1)-1),a=1..9),d=1..2); # Robert Israel, Jun 12 2017
  • Mathematica
    Table[FromDigits@ Apply[Join, {{First@ #}, {2}, Rest@ #}] &@ IntegerDigits@ n, {n, 67}] (* Michael De Vlieger, Jun 12 2017 *)
Select[Range[700],NumberDigit[#,IntegerLength[#]-2]==2&] (* Harvey P. Dale, Aug 15 2025 *)
  • PARI
    isok(n) = (n>9) && digits(n)[2] == 2; \\ Michel Marcus, Jun 12 2017
  • PARI
    a(n) = my(d = digits(n)); fromdigits(concat([d[1], [2], vector(#d-1, i, d[i+1])])) \\ David A. Corneth, Jun 12 2017
  • PARI
    nxt(n) = if(isok(n+1), n+1, d = digits(n); t = 9*10^(#d-2); if(d[1]==9,t*=3); n+=t++) \\ David A. Corneth, Jun 12 2017
  • Python
    def a(n): s = str(n); return int(s[0] + "2" + s[1:])
print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Dec 22 2021

Formula

From Robert Israel, Jun 12 2017: (Start)
a(10*n+j) = 10*a(n)+j for 0<=j<=9 and n >= 1.
G.f. g(x) satisfies g(x) = 10*(1-x^10)*g(x^10)/(1-x) + (x + 2*x + ... + 9*x^9)*x^10/(1-x^10) + 12*x + 22*x^2 + ... + 92*x^9. (End)

A331046 Numbers k such that floor(k/10^m) is a prime number for some m >= 0.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 83, 89, 97, 101, 103, 107, 109, 110, 111, 112, 113
Offset: 1

Views

Author

Rémy Sigrist, Jan 08 2020

Keywords

Comments

In other words, these are the numbers with a prime prefix.
For any m > 0:
- let f(m) be the proportion of positive numbers <= 10^m belonging to this sequence,
- we have f(m) = Sum_{p < 10^m in A069090} 1/10^A055642(p),
- also f(m) <= f(m+1) <= 1,
- so {f(m)} has a limit, say F, as m tends to infinity,
- what is the value of F?

Examples

			The number 2 is prime, so every number in A217394 belongs to this sequence.

Links

Crossrefs

Cf. A055642, A069090, A202259 (complement), A217394, A331044, A331045.

Programs

  • PARI
    is(n,base=10) = while (n, if (isprime(n), return (1), n\=base)); return (0)

A341909 a(0) = 0; for n > 0, a(n) is the smallest positive integer not yet in the sequence such that the first digit of a(n) differs by 1 from the last digit of a(n-1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 80, 10, 11, 20, 12, 13, 21, 22, 14, 30, 15, 40, 16, 50, 17, 60, 18, 70, 19, 81, 23, 24, 31, 25, 41, 26, 51, 27, 61, 28, 71, 29, 82, 32, 33, 42, 34, 35, 43, 44, 36, 52, 37, 62, 38, 72, 39, 83, 45, 46, 53, 47, 63, 48, 73, 49, 84, 54, 55, 64, 56, 57, 65, 66, 58, 74, 59
Offset: 0

Views

Author

Scott R. Shannon, Feb 23 2021

Keywords

Examples

			a(10) = 80 as the last digit of a(9) = 9 is 9, thus the first digit of a(10) must be 8. As 8 has already been used the next smallest number starting with 8 is 80.
a(16) = 21 as the last digit of a(15) = 13 is 3, thus the first digit of a(16) must be 2 or 4. As 2, 4 and 20 have already been used the next smallest number starting with 2 is 21.

Links

Crossrefs

Cf. A001477, A000027, A341692, A341002, A331163, A331375.
Cf. A131835, A217394, A217395, A217397, A217398, A217399, A217400, A217401, A217402.

Programs

  • Mathematica
    Block[{a = {0}, k}, Do[k = 1; While[Nand[FreeQ[a, k], Abs[First@ IntegerDigits[k] - Mod[a[[-1]], 10]] == 1], k++]; AppendTo[a, k], {i, 76}]; a] (* Michael De Vlieger, Feb 23 2021 *)
  • Python
    def nextd(strn, d):
  n = int(strn) if strn != "" else 0
  return n+1 if str(n+1)[0] == str(d) else int(str(d)+'0'*len(strn))
def aupton(term):
  alst, aset = [0], {0}
  lastdstr = ["" for d in range(10)]
  for n in range(1, term+1):
    lastdig = alst[-1]%10
    firstdigs = set([max(lastdig-1, 0), min(lastdig+1, 9)]) - {0}
    cands = [nextd(lastdstr[d], d) for d in firstdigs]
    m = min(cands)
    argmin = cands.index(m)
    alst.append(m)
    strm = str(m)
    lastdstr[int(strm[0])] = strm
  return alst
print(aupton(76)) # Michael S. Branicky, Feb 23 2021
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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