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Numbers k such that 2^k contains 2^10 as a substring.
1

%I #20 Mar 18 2021 05:45:30

%S 10,224,278,286,452,473,502,510,645,656,698,744,871,889,909,921,955,

%T 960,966,972,1010,1062,1086,1113,1121,1163,1182,1200,1201,1208,1271,

%U 1273,1282,1315,1327,1328,1377,1431,1444,1510,1541,1550,1564,1570,1583,1610,1626,1630,1674,1677,1693,1706,1719,1720,1726,1738

%N Numbers k such that 2^k contains 2^10 as a substring.

%C This sequence includes no 1-digit numbers, only 1.111% of the 2-digit numbers, 2.111% of the 3-digit numbers, 15.744% of the 4-digit numbers, and 74.734% of the 5-digit numbers. 6-digit numbers not in the sequence become increasingly scarce. The only numbers in the interval [300000, 500000] that are not in the sequence are 304702, 328762, 329873, 344218, and 384135. Is 384135 the largest number that is not in the sequence? - _Jon E. Schoenfield_, Mar 16 2021

%e The last few digits of 2^224 are 610249216. They contain 1024 as a substring.

%p filter:= n -> StringTools:-Search("1024",sprintf("%d",2^n)) > 0:

%p select(filter, [$1..2000]); # _Robert Israel_, Mar 16 2021

%t Select[Range[2000], StringContainsQ[ToString[2^#], ToString[2^10]] &]

%o (PARI) isok(k) = #strsplit(Str(2^k), Str(2^10)) > 1; \\ _Michel Marcus_, Mar 16 2021

%o (Python)

%o A342601_list, k, m, s = [], 1, 2, str(2**10)

%o while k < 10**6:

%o if s in str(m):

%o A342601_list.append(k)

%o k += 1

%o m *= 2 # _Chai Wah Wu_, Mar 17 2021

%Y Cf. A000079, A046296.

%K nonn,base

%O 1,1

%A _Tanya Khovanova_, Mar 16 2021