A361750 Terms of A329150 that have several preimages.

23, 223, 230, 232, 233, 235, 237, 323, 523, 723, 1123, 1323, 1723, 1923, 2023, 2223, 2230, 2232, 2233, 2235, 2237, 2300, 2302, 2303, 2305, 2307, 2311, 2313, 2317, 2319, 2320, 2322, 2323, 2325, 2327, 2330, 2332, 2333, 2335, 2337, 2350, 2352, 2353, 2355, 2357, 2370, 2372, 2373, 2375, 2377

Offset: 1

Author

Bernard Schott, Mar 23 2023

Keywords

• nonn
• base

Comments

From k = 1 to 9, A329147(k) has more than one preimage only for k = 9 since A329147(9) = A329147(12) = 23; so, all terms of this sequence have in their decimal expansion the two-digit integer '23'; it follows that when || denotes concatenation, the terms of this sequence are of the form:

X||23 as 223, 323, 523, 723, 1123, 1323, 1723, 1923, 2023, 2223, ... or,

23||Y as 230, 232, 233, 235, 237, 2300, 2302, 2303, 2305, 2307, 2311, ... or,

X||23||Y as 2230, 2232, 2233, 2235, 2237, 3230, 3232, 3233, ...

Every term of this sequence has 2^q preimages, and the smallest terms with exactly 2^q preimages for q >= 1 are 23, 2323, 232323, ... with q times the 2-digit number 23 in its decimal expansion; hence, the 8 preimages of 232323 are 999, 9912, 9129, 91212, 1299, 12912, 12129 and 121212.

Examples

			One example for each type:
23||2 = 232 = A329147(91) = A329147(121).
7||23 = 723 = A329147(49) = A329147(412).
2||23||5 = 2235 = A329147(193) = A329147(1123).
23||23 = 2323 = A329147(99) = A329147(129) = A329147(1212) = A329147(912).
		

Crossrefs

Cf. A329147, A329149, A329150.

Programs

Mathematica

p[n_] := If[n > 0, Prime[n], 0]; seq[ndigmax_] := Module[{t = Table[FromDigits[ Flatten @ IntegerDigits @ (p /@ IntegerDigits[n]) ], {n, 1, 10^ndigmax - 1}]}, Sort @ Select[Tally[t], First[#] < 10^ndigmax && Last[#] > 1 &][[;; , 1]]]; seq[4] (* Amiram Eldar, Mar 23 2023 *)

PARI

f(n) = if (n, fromdigits(concat(apply(d -> if (d, digits(prime(d)), [0]), digits(n)))), 0); \\ A329147
lista(nn) = {my(map = Map()); for (n=1, nn, my(m = f(n)); if (mapisdefined(map, m), mapput(map, m, 1+mapget(map, m)), mapput(map, m, 0));); my(list = List()); for (n=1, nn, if (mapisdefined(map, n), my(v = mapget(map, n)); if (#v > 0, listput(list, n)););); Vec(list);} \\ Michel Marcus, Mar 24 2023