A032737 Composite numbers k such that all the decimal concatenations ik and ikj (i, j = 1...9) are also composite.

5620, 7358, 13308, 13332, 13650, 14612, 26302, 27971, 28472, 28838, 29542, 29650, 31328, 33027, 33170, 35914, 35970, 36186, 39608, 40078, 41165, 41528, 42422, 47172, 47382, 48046, 48052, 48454, 50774, 52735, 55553, 60222, 60806

Offset: 1

Author

Patrick De Geest, May 15 1998

Keywords

• nonn
• base

Comments

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)".

Examples

			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.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A032682-A032685 and A032702-A032733.

Programs

Maple

# Naive program to test for membership -  N. J. A. Sloane, Jan 01 2025:
isA032737 := proc(x) local S,y,L1,L2,i,j;
L1:=[seq(i,i=1..9)]; L2:=[1,3,7,9];
S:=[x];
for i in L1 do y:=parse(cat(i,x)); S:=[op(S),y]; od:
for i in L1 do for j in L2 do y:=parse(cat(i,x,j)); S:=[op(S),y]; od: od:
for i in S do if isprime(i) then return('false', i,"is prime"); break; fi; od:
'true';
end;

Mathematica

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 *)

Extensions

Offset changed by Andrew Howroyd, Aug 13 2024

Definition revised by N. J. A. Sloane, Jan 01 2025

Terms corrected and extended by Harvey P. Dale, Jan 01 2025