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

1,1

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.

Links

Table of n, a(n) for n=1..50.

Example

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).

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 *)

Prog

(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

Crossrefs

Cf. A329147, A329149, A329150.

Sequence in context: A254870 A110028 A117608 * A038514 A244766 A125386

Adjacent sequences: A361747 A361748 A361749 * A361751 A361752 A361753

Keyword

nonn,base

Author

Bernard Schott, Mar 23 2023

Status

approved