cp's OEIS Frontend

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

%I A118118 #48 Feb 24 2024 01:15:33
%S A118118 200,204,206,208,320,322,324,325,326,328,510,512,514,515,516,518,530,
%T A118118 532,534,535,536,538,620,622,624,625,626,628,840,842,844,845,846,848,
%U A118118 890,892,894,895,896,898,1070,1072,1074,1075,1076,1078,1130
%N A118118 Composite numbers that always remain composite when a single decimal digit of the number is changed.
%C A118118 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
%C A118118 Also indices n such that A209252(n) is zero. - _Ray G. Opao_, Aug 01 2020
%H A118118 Paolo P. Lava, <a href="/A118118/b118118.txt">Table of n, a(n) for n = 1..10000</a>
%H A118118 G. L. Honaker, Jr. and Chris Caldwell, <a href="https://t5k.org/curios/cpage/639.html">Prime Curios! 200</a>
%H A118118 Project Euler, <a href="http://projecteuler.net/problem=200">Problem 200. Find the 200th prime-proof sqube containing the contiguous sub-string "200"</a> (2008)
%e A118118 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.
%t A118118 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 A118118 (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 A118118 (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
%o A118118 (Python)
%o A118118 from sympy import isprime
%o A118118 def selfplusneighs(n):
%o A118118     s = str(n); d = "0123456789"; L = len(s)
%o A118118     yield from (int(s[:i]+c+s[i+1:]) for c in d for i in range(L))
%o A118118 def ok(n): return all(not isprime(k) for k in selfplusneighs(n))
%o A118118 print([k for k in range(1131) if ok(k)]) # _Michael S. Branicky_, Jun 19 2022
%Y A118118 Cf. A143641, A050249, A209252.
%K A118118 easy,nonn,base
%O A118118 1,1
%A A118118 Adam Panagos (adam.panagos(AT)gmail.com), May 12 2006
%E A118118 Edited by _Charles R Greathouse IV_, Aug 05 2010