If p is a decimal sturdy prime, then p satisfies this property:

* For all x>log_10(p), 1+A007953(p-(10^x mod p)) >= A007953(p).

Almost all primes do not satisfy this property. Here is a list of all primes < 2*10^11 that satisfy the property, and whether each prime is decimal sturdy.

Prime p       Decimal sturdy?  Comment
2             no               Divisor of 10
3             yes              a(1)
5             no               Divisor of 10
11            yes              a(2)
41            yes              a(3)
101           yes              a(4)
271           no               Divisor of 11111
4003          no               Divisor of 1 + 10^29 + 10^58
21401         no               A007953(21401) = 8 and A077196(21401) = 5
62003         no               Divisor of 310015
1000003       no               Divisor of 1 + 10^713 + 10^1211
1003001       no               Divisor of 1 + 10^287 + 10^411
4023001       no               Divisor of 1 + 10^1341 + 10^2682
12004721      no               Divisor of 122112022012
110301001     no               Divisor of 1 + 10^7188 + 10^7913
200001001     no               Divisor of 1 + 10^12766 + 10^13690
230100001     no               Divisor of 1 + 10^476 + 10^20130
1001112001    no               Divisor of 1 + 10^15500 + 10^25742
1100010001    no               Divisor of 1 + 10^4049 + 10^24112
1373000003    no               Divisor of 15103000033
2000011001    no               Divisor of 1 + 10^57868 + 10^70650
5363222357    no               Divisor of 48269001213
30105031001   no               Divisor of 1 + 10^2951 + 10^1898 + 10^2896
40001100001   no               Divisor of 1 + 10^241365 + 10^242490
182521213001  no               Divisor of 1015001013062200003

Therefore, a(5) > 2*10^11.