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Numbers k such that k is a substring of 2^k.
+10
11
6, 10, 35, 36, 37, 44, 49, 51, 60, 67, 72, 73, 82, 85, 89, 93, 179, 188, 190, 191, 226, 234, 252, 297, 312, 321, 356, 373, 391, 425, 429, 430, 438, 445, 451, 475, 478, 479, 486, 516, 519, 521, 526, 549, 551, 581, 582, 583, 598, 601, 603, 609, 613, 619, 627, 632, 642, 652, 653, 655, 660
EXAMPLE
2^93 = 99035203142830421991929_93_792.
MATHEMATICA
d[n_] := IntegerDigits[n]; parQ[n_] := MemberQ[Partition[d[2^n], Length[x = d[n]], 1], x]; Select[Range[660], parQ] (* Jayanta Basu, Jun 17 2013 *) Select[Range[700], SequenceCount[IntegerDigits[2^#], IntegerDigits[#]]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 10 2019 *)
PROG
(Haskell)
import Data.List (isInfixOf)
a032740 n = a032740_list !! (n-1)
a032740_list = [x | x <- [0..], show x `isInfixOf` (show $ 2 ^ x)]
Numbers k such that k is a substring of 3^k.
+10
8
7, 9, 24, 28, 57, 61, 62, 69, 71, 72, 77, 78, 80, 83, 87, 89, 95, 111, 162, 170, 174, 185, 191, 218, 222, 225, 229, 232, 249, 255, 259, 266, 267, 286, 288, 298, 314, 315, 322, 328, 329, 330, 332, 338, 351, 352, 362, 373, 376, 381, 386, 387, 414, 421, 435, 438
MATHEMATICA
ssQ[n_]:=MemberQ[Partition[IntegerDigits[3^n], IntegerLength[n], 1], IntegerDigits[ n]]; Select[Range[500], ssQ] (* Harvey P. Dale, Jul 16 2013 *)
Numbers k such that k is a substring of 4^k.
+10
3
6, 10, 17, 25, 36, 42, 50, 59, 60, 61, 72, 73, 78, 79, 81, 84, 86, 87, 89, 92, 93, 95, 96, 160, 200, 212, 222, 225, 227, 239, 260, 261, 269, 290, 291, 296, 300, 301, 304, 311, 313, 315, 324, 326, 327, 330, 336, 344, 345, 348, 350, 355, 362, 372, 378, 379, 381
Numbers k such that k is a substring of 6^k.
+10
3
6, 7, 9, 13, 21, 22, 23, 29, 39, 40, 42, 44, 45, 48, 53, 55, 56, 60, 63, 64, 65, 67, 68, 69, 70, 73, 74, 75, 76, 77, 79, 82, 83, 87, 89, 92, 93, 94, 98, 105, 107, 127, 129, 131, 134, 137, 143, 147, 152, 163, 165, 167, 174, 179, 184, 189, 197, 224, 226, 227, 234, 240
EXAMPLE
9 is in the sequence because 6^9 = 10077696 contains 9 as a substring. - David A. Corneth, Aug 13 2021
MATHEMATICA
Select[Range[250], SequenceCount[IntegerDigits[6^#], IntegerDigits[#]]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 03 2018 *)
PROG
(Python)
def ok(n): return str(n) in str(6**n)
(PARI) is(n) = { my(digs6n, digsn, streak, i, j); digs6n = digits(6^n); digsn = digits(n); for(i = 1, #digs6n + 1 - #digsn, streak = 0; for(j = 1, #digsn, if(digs6n[i + j - 1] == digsn[j], streak++ , next(2) ) ); if(streak == #digsn, return(1) ) ); 0 } \\ David A. Corneth, Aug 13 2021
Numbers k such that k is a substring of 8^k.
+10
3
4, 6, 7, 10, 13, 17, 18, 28, 31, 33, 36, 38, 42, 44, 47, 48, 49, 52, 54, 56, 58, 60, 63, 64, 67, 68, 69, 76, 77, 79, 81, 82, 83, 85, 86, 89, 90, 91, 94, 97, 112, 115, 124, 130, 135, 165, 173, 176, 178, 189, 193, 195, 206, 208, 215, 221, 225, 249, 251, 252, 253, 256
MATHEMATICA
Select[Range[300], SequenceCount[IntegerDigits[8^#], IntegerDigits[#]]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 16 2018 *)
PROG
(Python)
def ok(n): return str(n) in str(8**n)
Numbers k such that k is a substring of 5^k.
+10
2
2, 5, 6, 7, 9, 19, 25, 32, 34, 36, 41, 54, 55, 56, 59, 62, 64, 67, 69, 70, 71, 75, 80, 81, 82, 84, 86, 87, 89, 92, 93, 95, 96, 111, 115, 125, 128, 140, 163, 166, 177, 178, 189, 192, 205, 212, 219, 221, 226, 233, 236, 242, 258, 259, 267, 294, 303, 309, 323, 327, 329
MATHEMATICA
ss5nQ[n_]:=Module[{len=IntegerLength[n]}, MemberQ[Partition[ IntegerDigits[ 5^n], len, 1], IntegerDigits[n]]]; Select[Range[400], ss5nQ] (* Harvey P. Dale, Jan 06 2013 *)
Numbers k such that k is a substring of 7^k.
+10
2
3, 4, 6, 8, 12, 15, 20, 40, 42, 43, 50, 53, 55, 59, 60, 61, 62, 69, 72, 73, 74, 75, 78, 79, 80, 81, 83, 86, 87, 88, 89, 93, 94, 95, 96, 97, 99, 100, 103, 111, 113, 114, 118, 164, 165, 185, 193, 200, 207, 210, 215, 220, 230, 232, 238, 241, 243, 250, 253, 254, 255
PROG
(Python)
def ok(n): return str(n) in str(7**n)
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