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A373505
Numbers k such that k and k+1 both have an equal number of odd and even digits in their factorial-base representations.
2
25, 29, 37, 41, 55, 67, 73, 77, 85, 89, 103, 115, 727, 739, 745, 749, 757, 761, 775, 787, 793, 797, 805, 809, 823, 835, 841, 845, 853, 857, 889, 893, 901, 905, 937, 941, 949, 953, 967, 979, 985, 989, 997, 1001, 1015, 1027, 1033, 1037, 1045, 1049, 1063, 1075, 1081
OFFSET
1,1
COMMENTS
If m is the sum of the first k odd-indexed factorial numbers (A000142), for k >= 2, then m-1 is a term, since the factorial-base representation of m is 1010...10, with the block "10" repeated k times, and the factorial-base representation of m-1 is the 1010...1001, with the block "10" repeated k-1 times and followed by "01" (these numbers are 25, 745, 41065, 3669865, 482671465, ...).
EXAMPLE
25 is a term since the factorial-base representations of 25 and 26 are 1001 and 1010, respectively, and both have 2 odd digits and 2 even digits.
MATHEMATICA
With[{max = 7}, fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; s = Select[Range[1, max!], EvenQ[Length[(d = fctBaseDigits[#])]] && Count[d, _?EvenQ] == Length[d]/2 &]; ind = Position[Differences[s], 1] // Flatten; s[[ind]]]
PROG
(PARI) iseq(n) = {my(p = 2, o = 0, e = 0); while(n > 0, if((n%p) %2 == 0, e++, o++); n \= p; p++); e == o; }
lista(kmax) = {my(q1 = 0, q2); for(k = 1, kmax, q2 = iseq(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2); }
CROSSREFS
Subsequence of A351895.
Similar sequences: A337238, A373460.
Sequence in context: A259028 A358425 A317392 * A234640 A219258 A044861
KEYWORD
nonn,base,easy
AUTHOR
Amiram Eldar, Jun 07 2024
STATUS
approved