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Self-locating numbers within the decimal expansion of log(2): strings k beginning at position k (first digit after decimal point is position 2).
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%I #19 Feb 22 2020 07:03:11

%S 7,23,52,54,86,303,389,5112,60392,87491,94788,97115,616916,672938066

%N Self-locating numbers within the decimal expansion of log(2): strings k beginning at position k (first digit after decimal point is position 2).

%C a(15) > 10^9.

%C 52 and 54 are self-locating numbers which appear consecutively in log(2) = 0.693...5254....

%e For 0.693147..., if 6 appears at position 2, then 7 appears at position 7.

%t dlog2 = RealDigits[Log[2], 10, 1000010][[1]]; Select[Range[2, 1000000], FromDigits[Take[dlog2, {# - 1, # - 2 + IntegerLength[#]}]] == # &] (* _Vaclav Kotesovec_, Feb 18 2020 *)

%Y Decimal expansion of log(2): A002162.

%Y Self-locating digits of Pi: A057679, A153220.

%K nonn,base,more

%O 1,1

%A _S. Alwin Mao_, Feb 14 2020