A345904 Numbers ending in 1, 3, 7 or 9 that yield no primes if their first digit is changed to another nonzero digit.

1071, 1149, 1183, 1401, 1509, 1611, 1771, 1773, 1809, 1891, 1921, 2071, 2149, 2183, 2401, 2509, 2611, 2771, 2773, 2809, 2891, 2921, 3071, 3149, 3183, 3401, 3509, 3611, 3771, 3773, 3809, 3891, 3921, 4071, 4149, 4183, 4401, 4509, 4611, 4771, 4773, 4809, 4891, 4921, 5071

Offset: 1

Author

Tanya Khovanova, Jun 29 2021

Keywords

• nonn
• base

Comments

If a number is in this sequence, then all the numbers with the first digit changed to another nonzero digit are also in this sequence.

Numbers ending in 0, 2, 4, 5, 6, and 8 are not included by definition, because they are composite independently of the other digits.

Examples

			1071, 2071, 3071, 4071, 5071, 6071, 7071, 8071 and 9071 are all composite numbers. Thus, all of them are in this sequence.
		

Links

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

Crossrefs

Subsequence of A045572.

Programs

Maple

q:= n-> (l-> l[1] in [1, 3, 7, 9] and andmap(not isprime, [seq(parse
   (cat(j, seq(l[-i], i=2..nops(l)))), j=1..9)]))(convert(n, base, 10)):
select(q, [$1..5080])[];  # Alois P. Heinz, Jun 29 2021

Mathematica

Select[Range[1, 6000, 2], Take[IntegerDigits[#], -1] != {5} && CompositeQ[Table[FromDigits[Prepend[Rest[IntegerDigits[#]], n]], {n, 9}]] == {True, True, True, True, True, True, True, True, True} &]
Select[Range[1,5100,2],NumberDigit[#,0]!=5&&NoneTrue[FromDigits/@Table[ PadRight[ {d},IntegerLength[#],IntegerDigits[#]],{d,9}],PrimeQ]&] (* Harvey P. Dale, Sep 24 2021 *)

Python

from sympy import isprime
def ok(n):
    if n < 10 or n%10 not in {1, 3, 7, 9}: return False
    s = str(n)[1:]
    return all(not isprime(int(d+s)) for d in "123456789")
print(list(filter(ok, range(1, 6000, 2)))) # Michael S. Branicky, Jun 29 2021