A066686 -id:A066686 - cp's OEIS frontend

A322131 In the decimal expansion of n, replace each digit d with 2*d.

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 210, 212, 214, 216, 218, 40, 42, 44, 46, 48, 410, 412, 414, 416, 418, 60, 62, 64, 66, 68, 610, 612, 614, 616, 618, 80, 82, 84, 86, 88, 810, 812, 814, 816, 818, 100, 102, 104, 106, 108, 1010, 1012, 1014

Offset: 0

Rémy Sigrist, Nov 27 2018

This is an operation on digit strings: 1066 becomes 201212, for example. 86420 becomes 1612840. The result is always even - see A330336. - N. J. A. Sloane, Dec 17 2019

This sequence is a variant of A124108 in decimal base.

			For n = 109:
- we replace "1" with "2", "0" with "0" and "9" with "18",
- hence a(109) = 2018.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A048385, A061581, A066686, A124108, A330336.

f:= proc(n) option remember;
local m,d;
d:= n mod 10; m:= floor(n/10);
if d >= 5 then 100*procname(m) + 2*d
else 10*procname(m)+2*d
fi
end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Nov 28 2018

a[n_] := FromDigits@Flatten@IntegerDigits[2*IntegerDigits[n]]; Array[a, 60, 0] (* Amiram Eldar, Nov 28 2018 *)

a(n, base=10) = my (d=digits(n, base), v=0); for (i=1, #d, v = v*base^max(1,#digits(2*d[i],base)) + 2*d[i]); v

def A322131(n):
   return int(''.join(str(int(d)*2) for d in str(n))) # Chai Wah Wu, Nov 29 2018

A061581(n+1) = a(A061581(n)).
A066686(a(n), a(k)) = a(A066686(n, k)) for any n > 0 and k > 0.
a(10*n + d) = 10*a(n) + 2*d for any n >= 0 and d = 0..4.
a(10*n + d) = 100*a(n) + 2*d for any n >= 0 and d = 5..9.
G.f. g(x) satisfies g(x) = (2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 16*x^8 + 18*x^9)/(1-x^10) + (10 + 10*x + 10*x^2 + 10*x^3 + 10*x^4 + 100*x^5 + 100*x^6 + 100*x^7 + 100*x^8 + 100*x^9)*g(x^10). - Robert Israel, Nov 28 2018

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

Reinhard Zumkeller, Jun 09 2003

Indranil Ghosh, Rows 1..100 of triangle, flattened

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

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

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

T(n, 1) = A017281(n); T(n, n) = A020338(n).

A067574 Array T(i,j) read by ascending antidiagonals, where T(i,j) is the concatenation of i and j (1<=i, 1<=j).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jan 30 2002

The antidiagonals are read in the opposite direction to those in A066686.

			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 ...

Cf. A055642, A066686, A084854.

a = {}; Do[ a = Append[a, ToExpression[ StringJoin[ ToString[i - j], ToString[j]]]], {i, 2, 13}, {j, 1, i - 1} ]; a

def T(i, j): return int(str(i) + str(j))
def auptodiag(maxd):
    return [T(d+1-j, j) for d in range(1, maxd+1) for j in range(1, d+1)]
print(auptodiag(12)) # Michael S. Branicky, Nov 21 2021

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

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

Juri-Stepan Gerasimov, Oct 30 2019

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

			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), ...

Alois P. Heinz, Table of n, a(n) for n = 0..20000

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

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

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

More terms from Alois P. Heinz, Oct 30 2019

A083678 Numbers m = d_1 d_2 ... d_k (in base 10) with properties that k is even and d_i + d_{k+1-i} = 10 for all i.

Original entry on oeis.org

19, 28, 37, 46, 55, 64, 73, 82, 91, 1199, 1289, 1379, 1469, 1559, 1649, 1739, 1829, 1919, 2198, 2288, 2378, 2468, 2558, 2648, 2738, 2828, 2918, 3197, 3287, 3377, 3467, 3557, 3647, 3737, 3827, 3917, 4196, 4286, 4376, 4466, 4556, 4646, 4736, 4826, 4916

Offset: 1

Zak Seidov Jun 15 2003

The two-digit terms here occur in many sequences, e.g., A066686, A081926, A017173, A030108, A043457, A052224, A061388, A084364.

			1469 and 6284 are members because 1+9=4+6=10 and 6+4=2+8=10.

Harvey P. Dale, Table of n, a(n) for n = 1..819 (All terms through 999999.)

Cf. A066686, A081926, A017173, A030108, A043457, A052224, A061388, A084364.

ok10Q[n_]:=Module[{idn=IntegerDigits[n]},idn[[1]]+idn[[4]]==idn[[2]]+idn[[3]]==10]; Join[ Select[ Range[10,99],Total[IntegerDigits[#]]==10&],Select[Range[1000,9999],ok10Q]] (* Harvey P. Dale, Oct 14 2023 *)

isok(n) = {digs = digits(n); if (#digs % 2 == 0, for (i = 1, #digs/2, if ((digs[i] + digs[#digs+1-i]) ! = 10, return (0));); return (1);); return (0);} \\ Michel Marcus, Oct 05 2013

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

Reinhard Zumkeller, Jun 09 2003

Indranil Ghosh, Rows 1..100 of triangle, flattened

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

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

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

T(n, n) = A020338(n).

A318225 Lexicographically earliest sequence of positive terms such that the concatenation of two consecutive terms is always unique.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 3, 11, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 4, 10, 4, 11, 5, 5, 6, 5, 7, 5, 8, 5, 9, 5, 10, 5

Offset: 1

Rémy Sigrist, Aug 21 2018

In other words, for any distinct i and j, A066686(a(i), a(i+1)) <> A066686(a(j), a(j+1)).
The sequence b such that for any n > 0, b(n) = A066686(a(n), a(n+1)), satisfies b(n) = A303446(n+1) for n = 1..116 and b(117) <> A303446(118).
The sequence contains long runs of consecutive terms where one term out of two equals 1.

			The first terms, alongside the concatenation of a(n) and of a(n+1), are:
  n  a(n)   A066686(a(n), a(n+1))
  -- ----   ---------------------
   1    1    11
   2    1    12
   3    2    21
   4    1    13
   5    3    31
   6    1    14
   7    4    41
   8    1    15
   9    5    51
  10    1    16
  11    6    61
  12    1    17
  13    7    71
  14    1    18
  15    8    81
  16    1    19
  17    9    91
  18    1   110
  19   10   101
  20    1   111
  21   11   112
  22    2    22
  23    2    23
  24    3    32
  25    2    24

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Logarithmic density plot of the first 300000000 terms
Rémy Sigrist, C++ program for A318225

Cf. A066686, A303446.

A328932 Number of primes that are a concatenation of two positive integers whose sum is prime(n).

Original entry on oeis.org

1, 0, 2, 2, 4, 3, 2, 6, 5, 9, 6, 7, 10, 11, 12, 14, 12, 16, 14, 17, 12, 15, 16, 20, 19, 19, 20, 17, 23, 23, 18, 27, 28, 24, 30, 25, 26, 26, 28, 30, 27, 30, 32, 27, 25, 27, 37, 42, 38, 32, 32, 33, 30, 39, 38, 36, 43, 38, 43, 42, 36, 36, 47, 47, 49, 38, 45, 48, 51, 50

Offset: 1

Juri-Stepan Gerasimov, Oct 31 2019

			a(3) = 2 because primes 23, 41 are concatenations of prime(3) = 5 = 2 + 3 = 4 + 1.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Subsequence of A328903.

Cf. A000040, A066686, A067754.

a:= n-> (p-> add(`if`(isprime(parse(cat(i,
        p-i))), 1, 0), i=1..p-1))(ithprime(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 31 2019

a(n) = A328903(A000040(n)).

A358596 a(n) is the least prime p such that the concatenation p|n has exactly n prime factors with multiplicity.

Original entry on oeis.org

3, 2, 83, 2, 67, 41, 947, 4519, 15659081, 2843, 337957, 389, 1616171, 6132829, 422116888343, 24850181, 377519743, 194486417892947, 533348873, 324403, 980825013273164555563, 25691144027, 273933405157, 1238831928746353181, 311195507789, 129917586781, 2159120477658983490299

Offset: 1

Zak Seidov and Robert Israel, Feb 24 2023

From Jon E. Schoenfield, Feb 26 2023: (Start)
For 2-digit indices n, the following rules can be applied to expedite the search for a(n):
Let P(n) be the concatenation of a(n) and n. Then P(n) is the product of n primes (counted with multiplicity), P(n) mod 100 = n, a(n) = floor(P(n)/100) is prime, and it can be shown that the following constraints apply to the prime factors of P(n):
If n is odd, then 2 cannot appear among the prime factors of P(n).
If n == 2 (mod 4), then 2 must appear with multiplicity exactly 1.
If n == 4 (mod 8), then 2 must appear with multiplicity >= 3.
If n == 0 (mod 8), then 2 must appear with multiplicity exactly 2.
If n != 0 (mod 5), then 5 cannot appear among the prime factors of P(n).
If n == 0 (mod 5) but n != 0 (mod 25), then 5 must appear with multiplicity 1.
If n == 0 (mod 25), then 5 must appear with multiplicity >= 2.
Any prime q that divides n but does not divide 10 cannot appear among the prime factors of P(n).
For example, for n = 24, the following constraints apply to the primes that appear among the 24 prime factors of P(24):
  since 8 | n, exactly two are 2's;
  since 5 !| n, none are 5's;
  since 3 | n, none are 3's;
so P(24) >= 2^2 * 7^22. As it turns out, P(24) = 123883192874635318124 = 2^2 * 7^19 * 11 * 13 * 19. (End)
a(924) = floor(2^2 * 13^907 * 17^9 * 19^5 * 31 / 1000) = 8.0881...*10^1026. - Jon E. Schoenfield, Mar 03 2023

			a(5) = 67 because 67 is prime and 675 = 3^3 * 5^2 with A001222(675) = 3+2 = 5, and no smaller prime works.

Jon E. Schoenfield, Table of n, a(n) for n = 1..923

Cf. A001222, A066686.

icat:= proc(a,b) 10^(1+ilog10(b))*a+b end proc:
f:= proc(n) local p;
p:= 1;
do
  p:= nextprime(p);
  if numtheory:-bigomega(icat(p,n)) = n then return p fi;
od
end proc:
map(f, [$1..17]); # Robert Israel, Feb 24 2023

A001222(A066686(a(n),n)) = n.

a(18) and a(21)-a(27) from Jon E. Schoenfield, Feb 24 2023

cp's OEIS Frontend

A322131 In the decimal expansion of n, replace each digit d with 2*d.

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

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A067574 Array T(i,j) read by ascending antidiagonals, where T(i,j) is the concatenation of i and j (1<=i, 1<=j).

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

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A083678 Numbers m = d_1 d_2 ... d_k (in base 10) with properties that k is even and d_i + d_{k+1-i} = 10 for all i.

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

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A318225 Lexicographically earliest sequence of positive terms such that the concatenation of two consecutive terms is always unique.

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A328932 Number of primes that are a concatenation of two positive integers whose sum is prime(n).

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