A055629 Beginning of first run of at least n consecutive happy numbers.

1, 31, 1880, 7839, 44488, 7899999999999959999999996, 7899999999999959999999996

Offset: 1

Author

David W. Wilson, Jun 05 2000

Keywords

• base
• nonn
• hard

Comments

The next term a(8) is too large to include.

This sequence is infinite - see Theorem 3.1 of El-Sedy and Siksek.

With notation {9:repeat_count_of_digit_nine}, a(8) = 58{9:11}6{9:144}5, a(9) = 26{9:137}7{9:74}5, a(10) = 38{9:560}0{9:87}5. - Lambert Klasen (lambert.klasen(AT)postmaster.co.uk), Oct 17 2004 [a(9)-a(10) were corrected, using Styer's paper, by Amiram Eldar, Aug 03 2025]

From Amiram Eldar, Aug 03 2025: (Start)

a(11) = 27{9:280}0{9:1287}4, a(12) = 388{9:158021}8{9:136 nines}4, and a(13) = 288{9:218491}3{9:385203}3 (Styer, 2010).

a(14) = 7888{9:160493827157}1{9:34569}3 and a(15) = 77{9:2222222222222220}3{9:97388}3 (Lyons, 2013). (End)

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10
• Esam El-Sedy and Samir Siksek, On happy numbers, Rocky Mountain J. Math. 30 (2000), 565-570.
• H. G. Grundman and E. A. Teeple, Sequences of consecutive happy numbers, Rocky Mountain J. Math. 37 (6) (2007), 1905-1916.
• Daniel Lyons, Smallest numbers beginning sequences of 14 and 15 consecutive happy numbers, Involve, A Journal of Mathematics, Vol. 6, No. 4 (2013), pp. 461-466.
• Hao Pan, On consecutive happy numbers, J. Numb. Theory 128 (6) (2008), 1646-1654.
• Robert Styer, Smallest Examples of Strings of Consecutive Happy Numbers, J. Int. Seq. 13 (2010), Article 10.6.3.
• Robert Styer, Strings of Consecutive Happy Numbers, 2012.

Crossrefs

Cf. A007770.

Extensions

Entry corrected by Sergio Pimentel, Dec 10 2005