A317777 Numbers whose digital sums in bases 2, 3, 4, and 5 are all equal.

1, 23162843828305, 5722224662500629, 25185954575304707081301, 407805072367801818857674005, 1705412607407578552438012746487125, 1705412607426764386750185803694405, 1705412607426764386750185803695125, 1705412607426764386750186877112645, 1705412607431411795338502226662661

Offset: 1

Author

Giovanni Resta, Aug 06 2018

Keywords

• nonn
• base
• hard

Comments

Comment from Max Alekseyev, Dec 11 2024 (Start)

The beginning of the sequence suggests that the terms may be rather sparse.

However, as terms become larger, they tend to appear in unexpected constellations such as:

89675268900935540640454882129695060

89675268900935540640454882129695061

89675268900935540640454933652788500

89675268900935540640454933652788501

89675273848747799396737823490250005

or

113343080811409868866589092414267671876

113343080811409868866589092414267671877

113343080811409868867743138448975156500

113343080811409868867743138448975156501

113343080811409868867760032719746187600

113343080811409868867760032719746187601

113343080811409868867760239410949735765

113343080811409868867761158326773437508

113343080811409868867761158326773437509

I can explain when two terms differ by 1, where the first one happens to end with a digit < b-1 for each base b in {2,3,4,5}, but the above constellations are a total mystery.

(End)

Examples

			23162843828305 in bases 2, 3, 4, and 5 is equal to 101010001000100000101000101000001000001010001, 10001000100100102010101011001, 11101010011011001001101, and 11014000001010001210, respectively. The sum of the digits is 13 in all four cases.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..28

Crossrefs

Subsequence of A037301 and A135122.

Cf. A317725.

Extensions

Terms a(5) onward from Max Alekseyev, Sep 09 2023