A258083 -id:A258083 - cp's OEIS frontend

A258225 a(n) = A258083(n) / 3.

7, 4, 1, 8, 5, 2, 9, 6, 3, 70, 37, 104, 71, 38, 105, 72, 39, 106, 73, 40, 107, 74, 41, 108, 75, 42, 109, 76, 43, 10, 77, 44, 11, 78, 45, 12, 79, 46, 13, 80, 47, 14, 81, 48, 15, 82, 49, 16, 83, 50, 17, 84, 51, 18, 85, 52, 19, 86, 53, 20, 87, 54, 21, 88, 55

Offset: 1

Reinhard Zumkeller, May 27 2015

Permutation of the positive integers, inverse: A258226.

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999
Index entries for sequences that are permutations of the natural numbers

Cf. A258083, A258226, A258329, A258334.

Haskell
```
a258225 = flip div 3 . a258083
```

lista(nn) = {v = []; vs = vecsort(v); for (n=1, nn, k=3; pt = 10^(#digits(n)); while (! (((k % pt) == n) && !vecsearch(vs, k)), k+=3); v = concat(v, k); vs = vecsort(v); print1(k/3, ", "););} \\ Michel Marcus, Jun 26 2015

A258370 Primes as they appear in sequence {A258083(n)/3}.

Original entry on oeis.org

7, 5, 2, 3, 37, 71, 73, 107, 41, 109, 43, 11, 79, 13, 47, 83, 17, 19, 53, 89, 23, 59, 61, 29, 97, 31, 367, 701, 1039, 373, 709, 379, 1049, 383, 1051, 719, 389, 727, 1061, 1063, 397, 733, 67, 401, 739, 409, 743, 751, 419, 1087, 421, 757, 1091, 1093, 761, 1097, 431, 433, 101, 769, 103, 439, 773, 1109, 443, 113, 449, 787, 457, 127, 461

Offset: 1

Vladimir Shevelev, May 28 2015

The sequence is a permutation of the primes.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000040, A258083, A258225.

A258188 Smallest multiple of 7 not appearing earlier that ends with n.

Original entry on oeis.org

21, 42, 63, 14, 35, 56, 7, 28, 49, 210, 511, 112, 413, 714, 315, 616, 217, 518, 119, 420, 721, 322, 623, 224, 525, 126, 427, 728, 329, 630, 231, 532, 133, 434, 735, 336, 637, 238, 539, 140, 441, 742, 343, 644, 245, 546, 147, 448, 749, 350, 651, 252, 553, 154

Offset: 1

Eric Angelini and Reinhard Zumkeller, May 23 2015

a(10*n) = 10*a(n).
The sequence is a permutation of the positive multiples of 7. - Vladimir Shevelev, May 24 2015
A258329(n) = a(n) / 7 is a permutation of the positive integers. - Reinhard Zumkeller, May 27 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999

Cf. A008589, A258083, A258217, A258329.

import Data.List (isPrefixOf, delete)
a258188 n = a258188_list !! (n-1)
a258188_list = f 1 $ tail $ zip
   a008589_list $ map (reverse . show) a008589_list where
   f x ws = g ws where
     g ((u, vs) : uvs) = if isPrefixOf xs vs
                         then u : f (x + 1) (delete (u, vs) ws) else g uvs
     xs = reverse $ show x

a[n_] := a[n] = For[k = 7, True, k = k + 7, If[Divisible[k - n, 10^IntegerLength[n]] && FreeQ[Array[a, n-1], k], Return[k]]]; Array[a, 54] (* Jean-François Alcover, Feb 07 2018 *)

A258217 Smallest multiple of 7 starting with n, that did not appear earlier.

Original entry on oeis.org

14, 21, 35, 42, 56, 63, 7, 84, 91, 105, 112, 126, 133, 140, 154, 161, 175, 182, 196, 203, 210, 224, 231, 245, 252, 266, 273, 28, 294, 301, 315, 322, 336, 343, 350, 364, 371, 385, 392, 406, 413, 420, 434, 441, 455, 462, 476, 483, 49, 504, 511, 525, 532, 546

Offset: 1

Eric Angelini and Reinhard Zumkeller, May 23 2015

The sequence is a permutation of the positive multiples of 7. - Vladimir Shevelev, May 24 2015

A258334(n) = a(n) / 7 is a permutation of the positive integers. - Reinhard Zumkeller, May 27 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Éric Angelini, Multiples of 3 ending with n, SeqFan list, May 23 2015.

Cf. A008589, A258188, A258083, A258334.

import Data.List (isPrefixOf, delete)
a258217 n = a258217_list !! (n-1)
a258217_list = f 1 $ tail $ zip a008589_list $ map show a008589_list where
   f x ws = g ws where
     g ((u, vs) : uvs) = if isPrefixOf (show x) vs
                         then u : f (x + 1) (delete (u, vs) ws) else g uvs

a[1] = 14; a[n_] := a[n] = For[dn = IntegerDigits[n]; k = 7, True, k = k+7, dk = IntegerDigits[k]; lg = Min[Length[dn], Length[dk]]; If[Union[ Take[dk, lg] - Take[dn, lg]] == {0} && FreeQ[Array[a, n-1], k], Return[k]]]; Array[a, 54] (* Jean-François Alcover, Feb 09 2018 *)

A258190 Smallest prime not appearing earlier that ends with A045572(n).

Original entry on oeis.org

11, 3, 7, 19, 211, 13, 17, 419, 421, 23, 127, 29, 31, 233, 37, 139, 41, 43, 47, 149, 151, 53, 157, 59, 61, 163, 67, 269, 71, 73, 277, 79, 181, 83, 487, 89, 191, 193, 97, 199, 101, 103, 107, 109, 2111, 113, 1117, 3119, 3121, 1123, 4127, 1129, 131, 4133, 137, 4139, 2141, 2143, 5147, 11149, 1151, 1153, 4157, 4159, 2161, 1163, 167, 3169

Offset: 1

Vladimir Shevelev, May 23 2015

Using Dirichlet's theorem, we conclude that every term exists. So the sequence is a permutation of the odd primes other than 5. Indeed, an odd prime p other than 5 is either in its natural place in A045572 or appears earlier than that.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000
Index entries for primes involving decimal expansion of n

Cf. A000040, A045572, A258083, A338715.

r:= -1: Used:= 'Used':
for n from 1 to 1000 do
  r:= r+2;
  if r mod 5 = 0 then r:= r+2 fi;
  d:= 10^(1+ilog10(r));
  for x from r by d do
    if isprime(x) and not assigned(Used[x]) then
      a[n]:= x;
      Used[x]:= true;
      break
    fi
  od
od:
seq(a[n],n=1..1000); # Robert Israel, May 27 2015

\\with first line from A045572 by Charles R Greathouse IV
a(n) = {n = 10*(n>>2)+[-1, 1, 3, 7][n%4+1]; my(d = digits(n),m = matrix(#d + 1, 2), w=0); m[1,2] = d[#d] - 10; for(i = 2, matsize(m)[1], m[i,1]=10^(i-2)*d[#d-i+2] + m[i-1,1]; if(m[i-1,1] == m[i,1],m[i,2]=m[i-1,2], j=m[i,1]==m[i-1,2]; while(!isprime(10^(i-1)*j+m[i,1]), j++); m[i,2]=10^(i-1)*j+m[i,1]));m[matsize(m)[1],2]} \\ David A. Corneth, May 25 2015

from sympy import isprime
def aupton(terms):
    alst, aset = [], set()
    for n in range(1, terms+1):
        ending = 2*n - 1 + (n+1)//4 * 2 # A045572
        i, pow10 = ending, 10**len(str(ending))
        while i in aset or not isprime(i): i += pow10
        alst.append(i); aset.add(i)
    return alst
print(aupton(68)) # Michael S. Branicky, Nov 03 2021

a(n) >= A045572(n). The equality holds iff A045572(n) is a prime that did not already appear as a(k), k

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A258225 a(n) = A258083(n) / 3.

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A258370 Primes as they appear in sequence {A258083(n)/3}.

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A258188 Smallest multiple of 7 not appearing earlier that ends with n.

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A258217 Smallest multiple of 7 starting with n, that did not appear earlier.

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A258190 Smallest prime not appearing earlier that ends with A045572(n).

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