A032737 - OEIS

%I #21 Jan 01 2025 23:32:39

%S 5620,7358,13308,13332,13650,14612,26302,27971,28472,28838,29542,

%T 29650,31328,33027,33170,35914,35970,36186,39608,40078,41165,41528,

%U 42422,47172,47382,48046,48052,48454,50774,52735,55553,60222,60806

%N Composite numbers k such that all the decimal concatenations ik and ikj (i, j = 1...9) are also composite.

%C The old definition was that a(n) must be composite and "cannot be prefixed or followed by any digit to form a prime ('empty' suffixes are allowed)".

%H Harvey P. Dale, <a href="/A032737/b032737.txt">Table of n, a(n) for n = 1..1000</a>

%e 55553 prefixed with a digit from (1,2,3,4,5,6,7,8,9) and followed by a digit from ('',1,3,7,9) never yields a prime: '3'55553'_' = 11 x 32323; '2'5620'9' = 3 x 41 x 2083.

%p # Naive program to test for membership -  _N. J. A. Sloane_, Jan 01 2025:

%p isA032737 := proc(x) local S,y,L1,L2,i,j;

%p L1:=[seq(i,i=1..9)]; L2:=[1,3,7,9];

%p S:=[x];

%p for i in L1 do y:=parse(cat(i,x)); S:=[op(S),y]; od:

%p for i in L1 do for j in L2 do y:=parse(cat(i,x,j)); S:=[op(S),y]; od: od:

%p for i in S do if isprime(i) then return('false', i,"is prime"); break; fi; od:

%p 'true';

%p end;

%t pfdQ[n_]:=CompositeQ[n]&&NoneTrue[Flatten[Table[10(d1*10^IntegerLength[n]+n)+d2,{d1,Range[9]},{d2,{1,3,7,9}}]],PrimeQ] && NoneTrue[ Flatten[Table[d1*10^IntegerLength[n]+n,{d1,Range[9]}]],PrimeQ]; Select[Range[61000],pfdQ] (* _Harvey P. Dale_, Jan 01 2025 *)

%Y Cf. A032682-A032685 and A032702-A032733.

%K nonn,base

%O 1,1

%A _Patrick De Geest_, May 15 1998

%E Offset changed by _Andrew Howroyd_, Aug 13 2024

%E Definition revised by _N. J. A. Sloane_, Jan 01 2025

%E Terms corrected and extended by _Harvey P. Dale_, Jan 01 2025