A356412 First differences of A007770 (happy numbers).

6, 3, 3, 6, 4, 5, 3, 1, 12, 5, 19, 2, 9, 3, 4, 5, 3, 3, 3, 3, 6, 20, 1, 3, 6, 28, 9, 12, 2, 2, 1, 10, 5, 11, 7, 4, 6, 3, 23, 1, 17, 11, 2, 8, 1, 8, 3, 6, 1, 6, 3, 2, 7, 18, 6, 3, 2, 1, 8, 3, 4, 3, 5, 1, 5, 7, 5, 31, 6, 18, 5, 9, 9, 3, 6, 40, 20, 7, 2, 1, 42, 9, 5, 1, 9, 3, 2, 1

Offset

1,1

Comments

El-Sedy and Siksek show that there exist arbitrarily long runs of consecutive integers that are happy numbers. So this sequence contains arbitrarily long runs of 1's.

Links

Table of n, a(n) for n=1..88.

E. El-Sedy and S. Siksek, On happy numbers, Rocky Mountain J. Math. 30 (2000), 565-570.

Formula

a(n) = A007770(n+1) - A007770(n), where A007770(n) is the n-th happy number.

Mathematica

happyQ[n_] := NestWhile[Plus @@ (IntegerDigits[#]^2) &, n, UnsameQ, All] == 1; Differences @ Select[Range[700], happyQ] (* Amiram Eldar, Aug 06 2022 *)

Prog

(Python)
from itertools import count, islice
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/cycle
    return n==1
def agen(): # generator of terms
    prevk = 1
    for k in count(2):
        if ok(k): yield k - prevk; prevk = k
print(list(islice(agen(), 88))) # Michael S. Branicky, Aug 06 2022

Crossrefs

Cf. A007770.

Sequence in context: A327773 A085670 A011410 * A298206 A023407 A153841

Adjacent sequences: A356409 A356410 A356411 * A356413 A356414 A356415

Keyword

nonn,base

Author

Darío D. Devia, Aug 05 2022

Status

approved