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A240849 Quinary happy numbers.

1, 5, 7, 11, 19, 23, 25, 27, 33, 35, 41, 43, 49, 51, 55, 79, 81, 83, 91, 93, 95, 99, 103, 109, 115, 119, 121, 123, 125, 127, 133, 135, 141, 143, 149, 153, 157, 159, 161, 165, 169, 171, 173, 175, 181, 189, 193, 197, 201, 203, 205, 209, 213, 215, 217, 219, 221, 223, 229, 231, 233, 237, 241, 243, 245, 249

Offset: 1

Jiri Klepl, Apr 13 2014

Numbers for which the repeated application of the operation "Sum the squares of the digits of the base-5 representation" is trapped by (ends at) the fixed point 1.

			19 is a quinary happy number because 19=34_5 -> 3^2 + 4^2 = 25 = 100_5 -> 1+0+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, A239320.

isA240849 := proc(n)
    t := SqrdB5(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 := A276191(t) ;
    end do:
end proc:
for n from 1 to 300 do
    if isA240849(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Aug 24 2016

happyQ[n_, b_] := NestWhile[Plus @@ (IntegerDigits[#, b]^2) &, n, UnsameQ, All] == 1; Select[Range[250], happyQ[#, 5] &] (* Amiram Eldar, May 28 2020 *)

A239320 Ternary happy numbers.

Original entry on oeis.org

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 *)

A229470 Positions of 2 in decimal expansion of 0.1231232331232332333..., whose definition is given below.

Original entry on oeis.org

2, 5, 7, 11, 13, 16, 21, 23, 26, 30, 36, 38, 41, 45, 50, 57, 59, 62, 66, 71, 77, 85, 87, 90, 94, 99, 105, 112, 121, 123, 126, 130, 135, 141, 148, 156, 166, 168, 171, 175, 180, 186, 193, 201, 210, 221, 223, 226, 230, 235, 241, 248, 256, 265, 275, 287, 289, 292, 296, 301, 307, 314, 322, 331, 341, 352, 365, 367, 370, 374, 379, 385

Offset: 1

Jiri Klepl, Sep 24 2013

0.1231232331232332333... = Sum_{k>=0} 10^(-(k + 3)! / (3! * k!)) * (1 + 10 * Sum_{l=2..k+2} 10^(-(l^2 + l) / 2) * ((10^l - 1) / 3 - 10^(l - 1))).

a(n)=sum(k=0,n-1,1+k-binomial(round(sqrt(2+2*k)),2)+issquare(8*k+1)*(sqrtint(1+8*k)+1)/2) /* Ralf Stephan, Oct 09 2013 */

a((n^2+n+2m-2)/2) = (n^3+6n^2+3m^2+11n-3m+6)/6; n+2>=m>=2.

a(n) = Sum_{k=0..n-1} ( 1 + A002262(k) + A010054(k)*(sqrt(1+8*k)+1)/2 ).

Formula corrected by Ralf Stephan, Oct 09 2013

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User: Jiri Klepl

A240849 Quinary happy numbers.

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A239320 Ternary happy numbers.

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A229470 Positions of 2 in decimal expansion of 0.1231232331232332333..., whose definition is given below.

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