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A118118 Composite numbers that always remain composite when a single decimal digit of the number is changed. 4
200, 204, 206, 208, 320, 322, 324, 325, 326, 328, 510, 512, 514, 515, 516, 518, 530, 532, 534, 535, 536, 538, 620, 622, 624, 625, 626, 628, 840, 842, 844, 845, 846, 848, 890, 892, 894, 895, 896, 898, 1070, 1072, 1074, 1075, 1076, 1078, 1130 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..10000

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 200

Project Euler, Problem 200 (2008)

EXAMPLE

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.

MAPLE

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

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

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

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

MATHEMATICA

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 *)

PROG

(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

(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

CROSSREFS

Cf. A143641, A050249.

Sequence in context: A180104 A114984 A192545 * A124472 A247399 A078492

Adjacent sequences:  A118115 A118116 A118117 * A118119 A118120 A118121

KEYWORD

easy,nonn,base

AUTHOR

Adam Panagos (adam.panagos(AT)gmail.com), May 12 2006

EXTENSIONS

Edited by Charles R Greathouse IV, Aug 05 2010

STATUS

approved

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Last modified September 21 17:46 EDT 2019. Contains 327273 sequences. (Running on oeis4.)