A001273 -id:A001273 - cp's OEIS frontend

A018785 Terms of A001273 with trailing 9's stripped (at n=13 term becomes periodic with period 49).

1, 10, 13, 23, 1, 7, 356, 78, 3788, 78888, 258, 888888, 157, 2, 688, 26, 15, 128, 57, 188, 15, 2688, 2688, 888888, 277, 37888, 56, 25, 5, 12, 258, 178, 46, 8, 58, 2588, 17, 77, 17, 77, 17, 77, 88888, 7, 5, 12, 1688, 18888, 178, 37888, 46, 38, 18, 36, 6, 2688, 288, 488

Offset: 0

David W. Wilson

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A001273.

Period is 2, 688, 26, 15, 128, 57, 188, 15, 2688, 2688, 888888, 277, 37888, 56, 25, 5, 12, 258, 178, 46, 8, 58, 2588, 17, 77, 17, 77, 17, 77, 88888, 7, 5, 12, 1688, 18888, 178, 37888, 46, 38, 18, 36, 6, 2688, 288, 488, 25, 8888888, 25, 7888.

A007770 Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map (see A003132) includes 1.

Original entry on oeis.org

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338

Offset: 1

N. J. A. Sloane, A.R.McKenzie(AT)bnr.co.uk

Sometimes called friendly numbers, but this usage is deprecated.
Gilmer shows that the lower density of this sequence is < 0.1138 and the upper density is > 0.18577. - Charles R Greathouse IV, Dec 21 2011
Corrected the upper and lower density inequalities in the comment above. - Nathan Fox, Mar 14 2013
Grundman defines the heights of the happy numbers by the number of iterations needed to reach the 1: 0, 5, 1, 2, 4, 3, 3, 2, 3, 4, 4, 2, 5, 3, 3, 2, 4, 4, 3, 1, ... (A090425(n) - 1). E.g., for n=2 the height of 7 is 5 because it needs 5 iterations: 7 -> 49 -> 97 -> 130 -> 10 -> 1. - R. J. Mathar, Jul 09 2017
El-Sedy & Siksek prove that this sequence contains arbitrarily long subsequences of consecutive terms; that is, the upper uniform density of this sequence is 1. - Charles R Greathouse IV, Sep 12 2022

			1 is OK. 2 --> 4 --> 16 --> 37 --> ... --> 4, which repeats with period 8, so never reaches 1, so 2 (and 4) are unhappy.
A correspondent suggested that 98 is happy, but it is not. It enters a cycle 98 -> 145 -> 42 -> 20 -> 4 -> 16 ->37 ->58 -> 89 -> 145 ...

L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999.
R. K. Guy, Unsolved Problems Number Theory, Sect. E34.
J. N. Kapur, Reflections of a Mathematician, Chap. 34 pp. 319-324, Arya Book Depot New Delhi 1996.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 25-26.

Jud McCranie, Table of n, a(n) for n = 1..143071
Breeanne Baker Swart, Susan Crook, Helen G. Grundman, Laura Hall-Seelig, May Mei, and Laurie Zack, Fixed Points of Augmented Generalized Happy Functions II: Oases and Mirages, arXiv:1908.02194 [math.NT], 2019.
T. Cai and X. Zhou, On the height of happy numbers, Rocky Mt. J. Math. 38 (6) (2008) 1921.
E. El-Sedy and S. Siksek, On happy numbers, Rocky Mountain J. Math. 30 (2000), 565-570.
Justin Gilmer, On the density of happy numbers, arXiv:1110.3836 [math.NT], 2011-2015.
H. G. Grundmann, Semihappy Numbers, J. Int. Seq. 13 (2010), 10.4.8.
B. L. Mayer and L. H. A. Monteiro, On the divisors of natural and happy numbers: a study based on entropy and graphs, AIMS Mathematics (2023) Vol. 8, Issue 6, 13411-13424.
Luca Onnis, On a variant of the happy numbers and their generalizations, arXiv:2203.03381 [math.GM], 2022.
Hao Pan, Consecutive happy numbers, arXiv:math/0607213 [math.NT], 2006.
W. Schneider, Happy Numbers (Includes list of terms below 10000)
R. Styer, Smallest Examples of Strings of Consecutive Happy Numbers, J. Int. Seq. 13 (2010), 10.6.3.
Eric Weisstein's World of Mathematics, Happy Number
Eric Weisstein's World of Mathematics, Digitaddition
Wikipedia, Happy number

Cf. A003132 (the underlying map), A001273, A035497 (happy primes), A046519, A031177, A002025, A050972, A050973, A074902, A103369, A035502, A068571, A072494, A124095, A219667, A239320 (base 3), A240849 (base 5).

Cf. A090425 (required iterations including start and end).

a007770 n = a007770_list !! (n-1)
a007770_list = filter ((== 1) . a103369) [1..]
-- Reinhard Zumkeller, Aug 24 2011

f[n_] := Total[IntegerDigits[n]^2]; Select[Range[400], NestWhile[f, #, UnsameQ, All] == 1 &] (* T. D. Noe, Aug 22 2011 *)
Select[Range[1000],FixedPoint[Total[IntegerDigits[#]^2]&,#,10]==1&] (* Harvey P. Dale, Oct 09 2011 *)
(* A example with recurrence formula to test if a number is happy *)
a[1]=7;
a[n_]:=Sum[(Floor[a[n-1]/10^k]-10*Floor[a[n-1]/10^(k+1)]) ^ (2) ,{k, 0,
      Floor[Log[10,a[n-1]]] }]
Table[a[n],{n,1,10}] (* José de Jesús Camacho Medina, Mar 29 2014 *)

ssd(n)=n=digits(n);sum(i=1,#n,n[i]^2)
is(n)=while(n>6,n=ssd(n));n==1 \\ Charles R Greathouse IV, Nov 20 2012

select( {is_A007770(n)=while(6M. F. Hasler, Dec 20 2024

def ssd(n): return sum(int(d)**2 for d in str(n))
def ok(n):
  while n not in [1, 4]: n = ssd(n) # iterate until fixed point or in cycle
  return n==1
def aupto(n): return [k for k in range(1, n+1) if ok(k)]
print(aupto(338)) # Michael S. Branicky, Jan 07 2021

From Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 23 2009: (Start)
1) Every power 10^k is a member of the sequence.
2) If n is member the numbers obtained by placing zeros anywhere in n are members.
3) If n is member each permutation of digits of n gives another member.
4) If the repeated process of summing squared digits give a number which is already a member of sequence the starting number belongs to the sequence.
5) If n is a member the repunit consisting of n 1's is a member.
6) If n is a member delete any digit d, new number consisting of remaining digits of n and d^2 1's placed everywhere to n is a member.
7) It is conjectured that the sequence includes an infinite number of primes (see A035497).
8) For any starting number the repeated process of summing squared digits ends with 1 or gives an "8-loop" which ends with (37,58,89,145,42,20,4,16,37) (End)

A094406 Smallest unhappy number that takes n steps to reach any of the unhappy cycle (4, 16, 37, 58, 89, 145, 42, 20) under iteration of sum of squares of digits map.

Original entry on oeis.org

4, 2, 11, 15, 5, 3, 14, 45, 36, 6, 112, 269, 15999

Offset: 4

Sergio Pimentel, Apr 30 2004

This sequence is an analog of sequence A001273 with the unhappy numbers instead of the happy numbers.

The next term, 577999999999....999999999999 (199 digits) is too large to include.

			15 is the fourth term because: 15: 1^2 + 5^2 = 26 / 26: 2^2 + 6^2 = 40 / 40: 4^2 + 0^2 = 16 and 16 is a member of the unhappy number series (4, 16, 37, 58, 89, 145, 42, 20)

Michel Marcus, Table of n, a(n) for n = 4..17
Walter Schneider, Unhappy Numbers.

Cf. A001273.

Edited by Charles R Greathouse IV, Aug 03 2010

A217705 Smallest number greater than 1 that is happy under bases 2 through n.

Original entry on oeis.org

2, 3, 3, 23, 79, 2207, 58775, 569669, 11814485, 210511543, 73748383237

Offset: 2

Sergio Pimentel, Mar 20 2013

A happy number is a number that after iteration of sum of squares of digits eventually reaches 1 (A007770). The happy property is base-dependent. This sequence lists the smallest number that is happy in bases 2, 3, ..., n.

All numbers are happy in binary and base 4.

			a(8) = 58775 because:
Base 2: 1110010110010111 - 1010 - 10 - 1,
Base 3: 2222121212 - 1011 - 10 - 1,
Base 4: 321121113 - 132 - 32 - 31 - 22 - 20 - 10 - 1,
Base 5: 3340100 - 120 - 10 - 1,
Base 6: 1132035 - 121 - 10 - 1,
Base 7: 333233 - 100 - 1,
Base 8: 162627 - 202 - 10 - 1,
Base 9 fails since the end is the 58 - 108 - 72 cycle and fails to reach 1.

Cf. A007770, A001273, A068571, A161871, A161873, A055629, A161872, A094406.

ssd(n,b)=my(s);while(n,s+=(n%b)^2;n\=b);s
happy(k,b)=my(t=ssd(k,b));k=ssd(t,b);while(t!=k&&k>1,t=ssd(t,b);k=ssd(ssd(k,b),b));k==1
h3(k)=while(k>8, k=ssd(k,3));k==1 || k==3
a(n)=if(n<4,return(n));my(k=2);while(k++, if(!h3(k),next); for(b=5,n, if(!happy(k,b), next(2)));return(k)) \\ Charles R Greathouse IV, Mar 22 2013

a(9)-a(12) from Giovanni Resta, Mar 21 2013

A126973 a(n+1) is the smallest integer greater than a(n) such that the sum of the squares of its decimal digits is equal to a(n).

Original entry on oeis.org

1, 10, 13, 23, 1233, 33999999999999999

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2007; corrected Mar 23 2007

			10 --> 1^2+0^2 = 1+0 =1
13 --> 1^2+3^2 = 1+9 = 10
23 --> 2^2+3^2 = 4+9 =13
1233 --> 1^2+2^2+3^3+3^2 = 1+4+9+9 = 23
33999999999999999 = 3^2*2 + 9^2*15 = 1233

Cf. A001273, A053612.

Next term is greater than 10^419753086419753. [From Charles R Greathouse IV, Nov 13 2010]

A176762 Smallest number that takes n steps to reach a cycle under iteration of sum-of-squares-of-digits map.

Original entry on oeis.org

1, 10, 13, 23, 19, 7, 356, 4, 2, 11, 15, 5, 3, 14, 45, 36, 6, 112, 269, 15999

Offset: 0

Robert G. Wilson v, Apr 25 2010

Cf. A001273, A094406, A045646, A003001.

f[n_] := Plus @@ (IntegerDigits[n]^2); t = Table[0, {25}]; k = 1; While[k < 150000001, a = Length@ NestWhileList[f, k, UnsameQ@## &, All] - 1; If[a < 25 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]

a(n) = minimum of either A001273(n) or A094406(n+3).

A383646 Smallest number that takes n steps to reach 1 under iteration of sum-of-cubes-of-digits map.

Original entry on oeis.org

1, 10, 112, 1189, 778, 13477, 2388889999999999999999

Offset: 0

Shyam Sunder Gupta, May 11 2025

These could also be called the smallest cubic happy numbers of height n.
Subsequent terms are too large to display in full.
a(7) = 1127 * 10^3276941015089163237 - 1 (1126 followed by 3276941015089163237 nines).
a(8) = 35678 * 10^((a(7) - 1054)/729) - 1.

			a(2) = 112, because 112 takes 2 steps to reach 1 (112 --> 1 + 1 + 8 = 10 --> 1 + 0 = 1).

H. G. Grundman and E. A. Teeple, Heights of happy numbers and cubic happy numbers, Fib Quart. 41 (4) (2003) 301-306.
Shyam Sunder Gupta, Happy Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 7, 209-222.

Cf. A007770, A001273, A035504.

cp's OEIS Frontend

A018785 Terms of A001273 with trailing 9's stripped (at n=13 term becomes periodic with period 49).

Views

Author

Keywords

Links

Crossrefs

Formula

A007770 Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map (see A003132) includes 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Python

Formula

A094406 Smallest unhappy number that takes n steps to reach any of the unhappy cycle (4, 16, 37, 58, 89, 145, 42, 20) under iteration of sum of squares of digits map.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A217705 Smallest number greater than 1 that is happy under bases 2 through n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A126973 a(n+1) is the smallest integer greater than a(n) such that the sum of the squares of its decimal digits is equal to a(n).

Views

Author

Keywords

Examples

Crossrefs

Extensions

A176762 Smallest number that takes n steps to reach a cycle under iteration of sum-of-squares-of-digits map.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A383646 Smallest number that takes n steps to reach 1 under iteration of sum-of-cubes-of-digits map.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs