A006287 -id:A006287 - cp's OEIS frontend

A239320 Ternary happy numbers.

1, 3, 9, 13, 17, 23, 25, 27, 31, 35, 37, 39, 47, 51, 53, 59, 61, 65, 69, 71, 73, 75, 77, 79, 81, 85, 89, 91, 93, 101, 105, 107, 109, 111, 117, 137, 141, 143, 153, 155, 159, 161, 167, 169, 173, 177, 179, 181, 183, 185, 187, 191, 195, 197, 207, 209, 213

Offset: 1

Jiri Klepl, Apr 13 2014

Numbers where the trajectory of iterated application of A006287 ends at the fixed point 1.

			13 is a ternary happy number because 13=111_3 -> 1 + 1 + 1 = 3 = 10_3 -> 1 + 0 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
H. G. Grundmann, Semihappy Numbers, J. Int. Seq. 13 (2010), 10.4.8.

Cf. A007770, A240849.

isA239320 := proc(n)
    t := A006287(n) ;
    tloo := {} ;
    for i from 1 do
        if t = 1 then
            return true;
        end if;
        if t in tloo then
            return false;
        end if;
        tloo := tloo union {t} ;
        t := A006287(t) ;
    end do:
end proc:
for n from 1 to 300 do
    if isA239320(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jun 13 2014

happyQ[n_, b_] := NestWhile[Plus @@ (IntegerDigits[#, b]^2) &, n, UnsameQ, All] == 1; Select[Range[213], happyQ[#, 3] &] (* Amiram Eldar, May 28 2020 *)

A276191 Sum of the squares of the digits of the base-5 representation of n.

Original entry on oeis.org

0, 1, 4, 9, 16, 1, 2, 5, 10, 17, 4, 5, 8, 13, 20, 9, 10, 13, 18, 25, 16, 17, 20, 25, 32, 1, 2, 5, 10, 17, 2, 3, 6, 11, 18, 5, 6, 9, 14, 21, 10, 11, 14, 19, 26, 17, 18, 21, 26, 33, 4, 5, 8, 13, 20, 5, 6, 9, 14, 21, 8, 9, 12, 17, 24, 13, 14, 17, 22, 29, 20, 21, 24, 29, 36, 9, 10, 13, 18, 25, 10

Offset: 0

R. J. Mathar, Aug 24 2016

Cf. A000120 (base 2), A006287 (base 3), A276190 (base 4).

A276191 := proc(n)
    local d ;
    add(d^2, d= convert(n, base, 5) );
end proc:

Table[Total[IntegerDigits[n,5]^2],{n,0,80}] (* Harvey P. Dale, Aug 15 2017 *)

A276190 Sum of the squares of the digits of the base-4 representation of n.

Original entry on oeis.org

0, 1, 4, 9, 1, 2, 5, 10, 4, 5, 8, 13, 9, 10, 13, 18, 1, 2, 5, 10, 2, 3, 6, 11, 5, 6, 9, 14, 10, 11, 14, 19, 4, 5, 8, 13, 5, 6, 9, 14, 8, 9, 12, 17, 13, 14, 17, 22, 9, 10, 13, 18, 10, 11, 14, 19, 13, 14, 17, 22, 18, 19, 22, 27, 1, 2, 5, 10, 2, 3, 6, 11, 5, 6, 9, 14, 10, 11, 14, 19, 2

Offset: 0

R. J. Mathar, Aug 24 2016

Cf. A000120 (base 2), A006287 (base 3), A276191 (base 5).

A276190 := proc(n)
    local d ;
    add(d^2, d= convert(n, base, 4) );
end proc:

Table[Total[IntegerDigits[n,4]^2 ],{n,0,80}] (* Harvey P. Dale, Jun 26 2022 *)

A375212 a(n) is the product of the leading base-3 digit of n and the sum of the squares of its base-3 digits.

Original entry on oeis.org

1, 8, 1, 2, 5, 8, 10, 16, 1, 2, 5, 2, 3, 6, 5, 6, 9, 8, 10, 16, 10, 12, 18, 16, 18, 24, 1, 2, 5, 2, 3, 6, 5, 6, 9, 2, 3, 6, 3, 4, 7, 6, 7, 10, 5, 6, 9, 6, 7, 10, 9, 10, 13, 8, 10, 16, 10, 12, 18, 16, 18, 24, 10, 12, 18, 12, 14, 20, 18, 20, 26, 16, 18, 24, 18, 20

Offset: 1

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 16 2024

Nathan Fox, Table of n, a(n) for n = 1..100000
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.

Cf. A006287, A376270

A079495 Numbers k such that the sum of the squares of the digits of k in base 3 is 0 (mod 3).

Original entry on oeis.org

0, 13, 14, 16, 17, 22, 23, 25, 26, 31, 32, 34, 35, 37, 38, 39, 42, 46, 47, 48, 51, 58, 59, 61, 62, 64, 65, 66, 69, 73, 74, 75, 78, 85, 86, 88, 89, 91, 92, 93, 96, 100, 101, 102, 105, 109, 110, 111, 114, 117, 126, 136, 137, 138, 141, 144, 153, 166, 167, 169, 170, 172, 173

Offset: 1

Carlos Alves, Jan 20 2003

In base 2 this gives the "Evil Numbers" (cf. A001969) and slope 2. One may conjecture that in base b the asymptotic slope will be b and might suspect asymptotic density 1/b for each result (mod b). For nonprime b larger variations occur and "very big" numbers must be considered to believe in the conjecture (1 million or more...). (Related to A006287, here mod b is considered)

			59 is a member because 59 = 2013_3 and 2^2+0^2+1^2+1^2 = 6 = 0 (mod 3).

Cf. A001969, A006287, A075311.

Ev = Function[{b, x}, vx = IntegerDigits[x, b]; Mod[vx.vx, b]]; Seq = Function[{b, n}, Flatten[Position[Table[Ev[b, k], {k, 1, n}], 0]]]; Seq[3, 1000]

Revised by Sean A. Irvine, Aug 17 2025

A377082 a(0) = 1, a(1) = 3, a(2) = 13; for n >= 2, a(n+1) = 2*3^((a(n)-1)/4) - 1.

Original entry on oeis.org

1, 3, 13, 53, 3188645

Offset: 0

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024

nonn

The next term is too large to include.
a(n) is the least positive integer that requires n steps to reach 1 under iteration of the sum of squares of digits function in base 3 (A006287).
a(n) is also the least positive integer that requires n steps to reach 1 under iteration of the map A375212.

N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.
G. Lapointe, On finding the smallest happy numbers of any heights, arXiv:1904.12032 [math.NT], 2019.

Cf. A006287, A007013, A375212.

cp's OEIS Frontend

A239320 Ternary happy numbers.

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A276191 Sum of the squares of the digits of the base-5 representation of n.

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A276190 Sum of the squares of the digits of the base-4 representation of n.

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A375212 a(n) is the product of the leading base-3 digit of n and the sum of the squares of its base-3 digits.

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A079495 Numbers k such that the sum of the squares of the digits of k in base 3 is 0 (mod 3).

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A377082 a(0) = 1, a(1) = 3, a(2) = 13; for n >= 2, a(n+1) = 2*3^((a(n)-1)/4) - 1.

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