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 A118118 Composite numbers that always remain composite when a single decimal digit of the number is changed. 4

%I

%S 200,204,206,208,320,322,324,325,326,328,510,512,514,515,516,518,530,

%T 532,534,535,536,538,620,622,624,625,626,628,840,842,844,845,846,848,

%U 890,892,894,895,896,898,1070,1072,1074,1075,1076,1078,1130

%N Composite numbers that always remain composite when a single decimal digit of the number is changed.

%C The term "prime-proof" for this property is found on projecteuler.net (cf. link). The nontrivial subsequence A143641 is that of odd elements not ending in 5 (i.e. not ending in 0,2,4,5,6 or 8); it starts 212159,595631,872897,... - _M. F. Hasler_, Sep 04 2008

%H Paolo P. Lava, <a href="/A118118/b118118.txt">Table of n, a(n) for n = 1..10000</a>

%H G. L. Honaker, Jr. and Chris Caldwell, <a href="http://primes.utm.edu/curios/cpage/639.html">Prime Curios! 200</a>

%H Project Euler, <a href="http://projecteuler.net/problem=200">Problem 200</a> (2008)

%e a(1) = 200 is in the sequence because changing any digit of 200 (for example 300, 220, or 209) is still composite. The integer 100 is not in the sequence because it can be changed to 107 which is prime.

%p with(numtheory): P:= proc(q) local d,j,k,ok,n; for n from 1 to q do

%p d:=ilog10(n)+1; ok:=1; for k from 1 to d do if ok=1 then for j from 0 to 9 do

%p if isprime((10*trunc(n/10^k)+j)*10^(k-1)+(n mod 10^(k-1)));

%p then ok:=0; break; fi; od; fi; od; if ok=1 then print(n); fi; od; end: P(10^6); # _Paolo P. Lava_, Nov 09 2015

%t unprimeableQ[n_] := Block[{d = IntegerDigits@ n, t = {}}, Do[AppendTo[t, FromDigits@ ReplacePart[d, i -> #] & /@ DeleteCases[Range[0, 9], x_ /; x == d[[i]]]], {i, Length@ d}]; ! AnyTrue[Flatten@ t, PrimeQ]]; Select[Range@ 1200, unprimeableQ] (* _Michael De Vlieger_, Nov 09 2015, Version 10 *)

%o (PARI) /* return 1 if no digit can be changed to make it prime; if d=1, print a prime if n is not prime-proof */ isA118118(n,d=0)={ forstep( k=n\10*10+1, n\10*10+9,2, isprime(k) || next; d && print("prime:",k); return); if( n%2==0 || n%5==0, /* even or ending in 5: no other digit can make it prime, except for the case where the last digit is prime and the first digit is the only other nonzero one */ return( !isprime(n%10) || 9 < n % 10^( log(n+.5)\log(10) ) || (d && print("prime:",n%10)) )); o=10; until( n < o*=10, k=n-o*(n\o%10); for( i=0,9, isprime(k) && return(d && print("prime:",k)); k+=o));1} \\ _M. F. Hasler_, Sep 04 2008

%o (MAGMA) IsA118118:=function(n); D:=Intseq(n); return forall{ <k, j>: k in [1..#D], j in [0..9] | j eq D[k] or not IsPrime(Seqint(S)) where S:=Insert(D, k, k, [j]) }; end function; [ n: n in [1..1200] | IsA118118(n) ]; // _Klaus Brockhaus_, Feb 28 2011

%Y Cf. A143641, A050249.

%K easy,nonn,base

%O 1,1