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Numbers k such that k and k+1 both have an equal number of even and odd digits.
2

%I #14 Jun 07 2024 05:22:40

%S 29,49,69,89,1009,1029,1049,1069,1089,1209,1229,1249,1269,1289,1409,

%T 1429,1449,1469,1489,1609,1629,1649,1669,1689,1809,1829,1849,1869,

%U 1889,2109,2129,2149,2169,2189,2309,2329,2349,2369,2389,2509,2529,2549,2569,2589,2709

%N Numbers k such that k and k+1 both have an equal number of even and odd digits.

%C The terms are of the form 100*m + j, where m is either 0 or a term of A227870 and j is in {29, 49, 69, 89} if m = 0 or in {9, 29, 49, 69, 89} if m > 0.

%H Amiram Eldar, <a href="/A373460/b373460.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 100 * A227870(floor(n/5)) + 20 * (n mod 5) + 9, for n > 4.

%e 29 is a term since it has one even digit (2) and one odd digit (9), and 29+1 = 30 also has one even digit (0) and one odd digit (3).

%t q[n_] := Module[{d = Differences[Tally[Mod[IntegerDigits[n], 2]]]}, d != {} && d[[1, 2]] == 0]; Select[Range[3000], q[#] && q[# + 1] &]

%o (PARI) iseq(n) = {my(o = 0, e = 0); while(n > 0, if((n%10) % 2 == 0, e++, o++); n \= 10); e == o;}

%o lista(kmax) = {my(q1 = 0, q2); for(k = 1, kmax, q2 = iseq(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

%Y Subsequence of A017377 and A227870.

%Y Cf. A337238 (binary analog), A373505.

%K nonn,base,easy

%O 1,1

%A _Amiram Eldar_, Jun 07 2024