A342575 a(n) is the exponent of the least power of 2 such that the concatenated digits of the decimal expansion of 2^n are a proper substring of the concatenated digits of the decimal expansion of 2^a(n).

4, 5, 6, 7, 14, 15, 26, 102, 103, 104, 224, 103, 104, 105, 506, 507, 452, 1169, 1170, 1171, 8228, 10419, 15186, 5227, 16619, 16620, 16621, 25102, 130090, 62640, 330791, 330792, 351403, 273100, 681504, 649069, 352375, 3045104, 3045105, 3635007, 9532211, 7819691, 3091425, 3091426

Offset: 0

Author

Hugo Pfoertner, Mar 16 2021

Keywords

• nonn
• base

Examples

			   n   2^n  a(n)  2^a(n)
   0     1    4  _1_6
   1     2    5   3_2_
   2     4    6   6_4_
   3     8    7   12_8_
   4    16   14  _16_384
   5    32   15  _32_768
   6    64   26   671088_64_
   7   128  102   50706024009129176059868_128_21504
   8   256  103   101412048018258352119736_256_43008
   9   512  104   202824096036516704239472_512_86016
  10  1024  224   2695994666715 ... 06736371444225405724811036_1024_9216
  11  2048  103   10141_2048_01825835211973625643008
  12  4096  104   20282_4096_03651670423947251286016
  13  8192  105   40564_8192_07303340847894502572032
		

Crossrefs

Cf. A342601.

Programs

Maple

a:= proc(n) local k, p; p:= ""||(2^n); for k
      from n+1 while searchtext(p, ""||(2^k))=0 do od; k
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Mar 17 2021

PARI

vecmatch(vshort,vlong)={my(l=#vshort,L=#vlong);for(i=0,L-l,if(vshort==vlong[i+1..i+l],return(i+1)));0}
for(n=0,26,my(vx=digits(2^n));for(y=n+1,oo,my(vy=digits(2^y));if(vecmatch(vx,vy)>0,print1(y,", ");break)))

PARI

a(n) = my(k=n+1, s=Str(2^n)); while (#strsplit(Str(2^k), s) <=1, k++); k; \\ Michel Marcus, Mar 17 2021

Python

def a(n):
  k, twok, target = n+1, 2**(n+1), str(2**n)
  while target not in str(twok): k, twok = k+1, twok*2
  return k
print([a(n) for n in range(27)]) # Michael S. Branicky, Mar 16 2021

Extensions

a(28)-a(36) from Jon E. Schoenfield, confirmed by Michael S. Branicky, Mar 17 2021

a(37)-a(43) from Bert Dobbelaere, Mar 19 2021