A192545 -id:A192545 - cp's OEIS frontend

A048853 Number of primes (different from n) that can be produced by altering one digit of decimal expansion of n (without changing the number of digits).

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

Offset: 1

G. L. Honaker, Jr. and Patrick De Geest

a(A192545(n)) = 0. - Reinhard Zumkeller, Jul 05 2011

			Altering the number 13 gives eight primes: 11, 17, 19, 23, 43, 53, 73, 83, so a(13)=8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A050652-A050673.

Cf. A010051, A158124.

import Data.List (inits, tails, nub)
a048853 n = (sum $ map (a010051 . read) $ tail $ nub $ concat $ zipWith
  (\its tls -> map ((\xs ys d -> xs ++ (d:ys)) its tls) "0123456789")
    (map init $ tail $ inits $ show n) (tail $ tails $ show n)) - a010051 n
-- Reinhard Zumkeller, Jul 05 2011

A048853 := proc(n::integer) local resul,ddigs,d,c,tmp ; resul := 0 ; ddigs := convert(n,base,10) ; for d from 1 to nops(ddigs) do for c from 0 to 9 do if c = 0 and d = nops(ddigs) then continue ; else if c <> op(d,ddigs) then tmp := [op(1..d-1,ddigs),c,op(d+1..nops(ddigs),ddigs)] ; tst := sum(op(i,tmp)*10^(i-1),i=1..nops(tmp)) ; if isprime(tst) then resul := resul+1 ; fi ; fi ; fi ; od : od ; RETURN(resul) ; end: for n from 1 to 90 do printf("%d,",A048853(n)) ; od ; # R. J. Mathar, Apr 25 2006

a[n_] := Module[{idn = IntegerDigits[n], id, np = 0}, Do[id = idn; If[ id[[j]] != k, id[[j]] = k; If[ id[[1]] != 0 && PrimeQ[ FromDigits[id]], np = np + 1]], {j, 1, Length[idn]}, {k, 0, 9}]; np]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Dec 01 2011 *)

from sympy import isprime
def h1(n): # hamming distance 1 neighbors of n, not starting with 0
    s = str(n); d = "0123456789"; L = len(s)
    yield from (int(s[:i]+c+s[i+1:]) for c in d for i in range(L) if c!=s[i] and not (i==0 and c=="0"))
def a(n): return sum(1 for k in h1(n) if isprime(k))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Jul 31 2022

A158124 Weakly prime numbers (or isolated primes): changing any one decimal digit always produces a composite number, with restriction that first digit may not be changed to a 0.

Original entry on oeis.org

294001, 505447, 584141, 604171, 929573, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3070663, 3085553, 3326489, 4393139, 5152507, 5285767, 5564453, 5575259, 5974249, 6173731, 6191371, 6236179, 6463267, 6712591, 7204777, 7469789, 7469797, 7810223

Offset: 1

Eric W. Weisstein, Mar 13 2009

The definition could be restated as "primes p with d digits such that there is no prime q with at most d digits at Hamming distance 1 from p (in base 10)". - N. J. A. Sloane, May 06 2019

For the following values of k, 5, 6, 7, 8, 9, 10, the number of terms < 10^k in this sequence is 0, 6, 43, 406, 3756, 37300. - Jean-Marc Rebert, Nov 10 2015

Jean-Marc Rebert, Table of n, a(n) for n = 1..3756
C. Rivera, Weakly Primes
Eric Weisstein's World of Mathematics, Weakly Prime

Cf. A050249, A158125 (weakly primes), A186995, A192545.

filter:= proc(n)
  local L,i,d,ds;
  if not isprime(n) then return false fi;
  L:= convert(n,base,10);
  for i from 1 to nops(L) do
    if i = nops(L) then ds:= {$1..9} minus {L[i]}
    elif i = 1 then ds:= {1,3,7,9} minus {L[i]}
    else ds:= {$0..9} minus {L[i]}
    fi;
    for d in ds do
      if isprime(n + (d - L[i])*10^(i-1)) then return false fi;
    od
  od;
  true
end proc:
select(filter, [seq(i,i=11..10^6,2)]); # Robert Israel, Dec 15 2015

Select[Prime@ Range[10^5], Function[n, Function[w, Total@ Map[Boole@ PrimeQ@ # &, DeleteCases[#, n]] &@ Union@ Flatten@ Map[Function[d, FromDigits@ ReplacePart[w, d -> #] & /@ If[d == 1, #, Prepend[#, 0]] &@ Range@ 9], Range@ Length@ w] == 0]@ IntegerDigits@ n]] (* Michael De Vlieger, Dec 13 2016 *)

isokp(n) = {v = digits(n); for (k=1, #v, w = v; if (k==1, idep = 1, idep=0); for (j=idep, 9, if (j != v[k], w[k] = j; ntest = subst(Pol(w), x, 10); if (isprime(ntest), return(0));););); return (1);}
lista(nn) = {forprime(p=2, nn, if (isokp(p), print1(p, ", ")););} \\ Michel Marcus, Dec 15 2015

from sympy import isprime
def h1(n): # hamming distance 1 neighbors of n, not starting with 0
    s = str(n); d = "0123456789"; L = len(s)
    yield from (int(s[:i]+c+s[i+1:]) for c in d for i in range(L) if c!=s[i] and not (i==0 and c=="0"))
def ok(n): return isprime(n) and all(not isprime(k) for k in h1(n))
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 31 2022

Edited by Charles R Greathouse IV, Aug 02 2010

Missing a(3385) inserted into b-file by Andrew Howroyd, Feb 23 2018

A382137 Smallest integer that cannot be converted to a multiple of n by changing at most one of its decimal digit.

Original entry on oeis.org

545, 51, 44, 31, 21, 21, 22, 21, 21, 11, 15, 11, 11, 11, 11, 11, 12, 11, 11, 11, 14, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12

Offset: 11

Mickaël Launay, Mar 27 2025

a(n) is not defined for n<11 since any integer can be converted to a multiple of n by changing its last digit.
For all n, a(n)>=11, since any number <=10 ca be converted to 0 which is a multiple of n.
For n>=42, a(n)=11, except for a(51)=a(61)=a(71)=a(81)=a(91)=12. Especially a(n)=11 for all n>=100 because 11 is the first integer that cannot be converted to 0 or a 3-digit number.

			For n=12, any number<50 can be converted to a multiple of 12 by changing its last digit. Since a) there is no multiple of 12 between 50 and 59 so changing the last digit is useless, b) 50 can be converted to 60=5*12 and c) changing the "5" of 51 gives only odd (non multiples of 12) numbers, then a(12)=51.

Mickaël Launay, Le démon des multiples, L’énigme maths du "Monde" n°48. In French.

Cf. A353023, A192545.

import math
def test(N,M):
   t = True
   for i in range(0,math.floor(math.log(N,10))+1):
      for j in range(0,10): t = t and ((N+(j-N // 10**i % 10)*10**i)%M != 0)
   return t
for M in range(11,100):
   N = 1
   while (not test(N,M)): N += 1
   print(N, end=', ')

cp's OEIS Frontend

A048853 Number of primes (different from n) that can be produced by altering one digit of decimal expansion of n (without changing the number of digits).

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A158124 Weakly prime numbers (or isolated primes): changing any one decimal digit always produces a composite number, with restriction that first digit may not be changed to a 0.

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Maple

Mathematica

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A382137 Smallest integer that cannot be converted to a multiple of n by changing at most one of its decimal digit.

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Author

Keywords

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Examples

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Programs

Python