A373460 -id:A373460 - cp's OEIS frontend

A373505 Numbers k such that k and k+1 both have an equal number of odd and even digits in their factorial-base representations.

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

Amiram Eldar, Jun 07 2024

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, ...).

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation..

Subsequence of A351895.
Cf. A000142, A007623.
Similar sequences: A337238, A373460.

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]]]

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);}

cp's OEIS Frontend

A373505 Numbers k such that k and k+1 both have an equal number of odd and even digits in their factorial-base representations.

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