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 1-5 of 5 results.

A066686 Array T(i,j) read by antidiagonals, where T(i,j) is the concatenation of i and j (1<=i, 1<=j).

Original entry on oeis.org

11, 12, 21, 13, 22, 31, 14, 23, 32, 41, 15, 24, 33, 42, 51, 16, 25, 34, 43, 52, 61, 17, 26, 35, 44, 53, 62, 71, 18, 27, 36, 45, 54, 63, 72, 81, 19, 28, 37, 46, 55, 64, 73, 82, 91, 110, 29, 38, 47, 56, 65, 74, 83, 92, 101, 111, 210, 39, 48, 57, 66, 75, 84, 93, 102, 111
Offset: 1

Views

Author

Robert G. Wilson v, Jan 11 2002

Keywords

Comments

The element at T(i,j) is the {(i+j-1)(i+j-2)/2 + i}-th element read in the sequence.

Examples

			The array begins
11 12 13 14 15 16 17 18 19 110 ...
21 22 23 24 25 26 27 28 29 210 ...
31 32 33 34 35 36 37 38 39 310 ...
41 42 43 44 45 46 47 48 49 410 ...

Links

Crossrefs

Cf. A055642, A067574, A084854.

Programs

  • Mathematica
    a = {}; Do[ a = Append[a, ToExpression[ StringJoin[ ToString[k], ToString[n - k]]]], {n, 2, 13}, {k, 1, n - 1} ]; a
  • Python
    def T(i, j): return int(str(i) + str(j))
def auptodiag(maxd):
    return [T(i, d+1-i) for d in range(1, maxd+1) for i in range(1, d+1)]
print(auptodiag(12)) # Michael S. Branicky, Nov 21 2021

Formula

T(i, j) = i*10^A055642(i) + j. - Michael S. Branicky, Nov 21 2021

A084854 Triangular array, read by rows: T(n,k) = concatenated decimal representations of n and k, 1<=k<=n.

Original entry on oeis.org

11, 21, 22, 31, 32, 33, 41, 42, 43, 44, 51, 52, 53, 54, 55, 61, 62, 63, 64, 65, 66, 71, 72, 73, 74, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 1010, 111, 112, 113, 114, 115, 116, 117, 118, 119, 1110, 1111
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 09 2003

Keywords

Links

Crossrefs

Cf. A017281, A020338, A055642, A066686, A067574, A084855.

Programs

  • Python
    def T(n, k): return int(str(n) + str(k))
def auptorow(maxrow):
    return [T(n, k) for n in range(1, maxrow+1) for k in range(1, n+1)]
print(auptorow(11)) # Michael S. Branicky, Nov 21 2021

Formula

T(n, k) = n*10^A055642(k) + k.
T(n, 1) = A017281(n); T(n, n) = A020338(n).

A328903 Number of primes that are a concatenation of two positive integers whose sum is n.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 2, 3, 0, 3, 4, 0, 3, 5, 0, 4, 2, 0, 6, 5, 0, 4, 5, 0, 6, 7, 0, 6, 9, 0, 6, 8, 0, 9, 8, 0, 7, 7, 0, 9, 10, 0, 11, 2, 0, 12, 12, 0, 10, 11, 0, 11, 14, 0, 3, 10, 0, 10, 12, 0, 16, 12, 0, 16, 14, 0, 14, 19, 0, 13, 17, 0, 12, 16, 0, 15, 2, 0
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Oct 30 2019

Keywords

Comments

First n > 1 with n != 0 (mod 3) and a(n) = 0 is n = 4477. - Alois P. Heinz, Oct 30 2019

Examples

			1(-), 2(11), 3(-), 4(13, 31), 5(23, 41), 6(-), 7(43, 61), 8(17, 53, 71), 9(-), 10(19, 37, 73), 11(29, 47, 83, 101), 12(-), 13(67, 103, 211), ...

Links

Crossrefs

Cf. A000040, A008486, A008585, A066686, A067574, A154530, A328932.

Programs

  • Maple
    a:= proc(n) option remember; `if`(irem(n, 3)=0, 0, add(
     `if`(isprime(parse(cat(i, n-i))), 1, 0), i=1..n-1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 30 2019

Formula

a(n) = 0 if n = 1 or n == 0 (mod 3). - Alois P. Heinz, Oct 30 2019

Extensions

More terms from Alois P. Heinz, Oct 30 2019

A084855 Triangular array, read by rows: T(n,k) = concatenated decimal representations of k and n, 1<=k<=n.

Original entry on oeis.org

11, 12, 22, 13, 23, 33, 14, 24, 34, 44, 15, 25, 35, 45, 55, 16, 26, 36, 46, 56, 66, 17, 27, 37, 47, 57, 67, 77, 18, 28, 38, 48, 58, 68, 78, 88, 19, 29, 39, 49, 59, 69, 79, 89, 99, 110, 210, 310, 410, 510, 610, 710, 810, 910, 1010, 111, 211, 311, 411, 511, 611, 711, 811, 911, 1011, 1111
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 09 2003

Keywords

Links

Crossrefs

Cf. A020338, A055642, A066686, A067574, A084854, A349520.

Programs

  • Python
    def T(n, k): return int(str(k) + str(n))
def auptorow(maxrow):
    return [T(n, k) for n in range(1, maxrow+1) for k in range(1, n+1)]
print(auptorow(11)) # Michael S. Branicky, Nov 21 2021

Formula

T(n, k) = k*10^A055642(n) + n.
T(n, n) = A020338(n).

A362023 a(n) is the least positive integer whose decimal expansion is the concatenation of the decimal expansions of two numbers whose product is n.

Original entry on oeis.org

11, 12, 13, 14, 15, 16, 17, 18, 19, 25, 111, 26, 113, 27, 35, 28, 117, 29, 119, 45, 37, 112, 123, 38, 55, 126, 39, 47, 129, 56, 131, 48, 113, 134, 57, 49, 137, 138, 133, 58, 141, 67, 143, 114, 59, 146, 147, 68, 77, 105, 151, 134, 153, 69, 115, 78, 157, 158
Offset: 1

Views

Author

Rémy Sigrist, Apr 05 2023

Keywords

Comments

For any prime number p, a(p) is the least of the concatenation of p with 1 or the concatenation of 1 with p.

Examples

			The first terms, alongside an appropriate way to split them into two factors, are:
  n   a(n)  a(n)
  --  ----  ----
   1    11   1*1
   2    12   1*2
   3    13   1*3
   4    14   1*4
   5    15   1*5
   6    16   1*6
   7    17   1*7
   8    18   1*8
   9    19   1*9
  10    25   2*5
  11   111  11*1
  12    26   2*6
  13   113  1*13
  14    27   2*7
  15    35   3*5

Links

Crossrefs

Cf. A055642, A067574, A347471, A362022 (binary variant).

Programs

  • Mathematica
    Table[Min@ Map[FromDigits[Join @@ #] &, Join @@ {#, Reverse /@ #}] &@ Map[IntegerDigits[#] &, Transpose@{#, n/#}, {2}] &@ TakeWhile[Divisors[n], # <= Sqrt[n] &], {n, 60}] (* Michael De Vlieger, Apr 07 2023 *)
  • PARI
    a(n, base = 10) = { my (v = oo); fordiv (n, d, v = min(v, n/d * base^#digits(d, base) + d);); return (v); }
  • Python
    from sympy import divisors
def a(n): return min(int(str(d)+str(n//d)) for d in divisors(n))
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Apr 05 2023

Formula

a(n) <= 10*n + 1.
a(n) <= 10^A055642(n) + n.
a(n) = Min_{d | n} A067574(n/d, d).
Showing 1-5 of 5 results.

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