Happy number
In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because , and . On the other hand, 4 is not a happy number because the sequence starting with and eventually reaches , the number that started the sequence, and so the process continues in an infinite cycle without ever reaching 1. A number which is not happy is called sad or unhappy.
More generally, a -happy number is a
The origin of happy numbers is not clear. Happy numbers were brought to the attention of
Happy numbers and perfect digital invariants
Formally, let be a natural number. Given the perfect digital invariant function
- .
for base , a number is -happy if there exists a such that , where represents the -th iteration of , and -unhappy otherwise. If a number is a nontrivial perfect digital invariant of , then it is -unhappy.
For example, 19 is 10-happy, as
For example, 347 is 6-happy, as
There are infinitely many -happy numbers, as 1 is a -happy number, and for every , ( in base ) is -happy, since its sum is 1. The happiness of a number is preserved by removing or inserting zeroes at will, since they do not contribute to the cross sum.
Natural density of b-happy numbers
By inspection of the first million or so 10-happy numbers, it appears that they have a natural density of around 0.15. Perhaps surprisingly, then, the 10-happy numbers do not have an asymptotic density. The upper density of the happy numbers is greater than 0.18577, and the lower density is less than 0.1138.[2]
Happy bases
Are
A happy base is a number base where every number is -happy. The only happy integer bases less than 5×108 are
Specific b-happy numbers
4-happy numbers
For , the only positive perfect digital invariant for is the trivial perfect digital invariant 1, and there are no other cycles. Because all numbers are preperiodic points for , all numbers lead to 1 and are happy. As a result,
6-happy numbers
For , the only positive perfect digital invariant for is the trivial perfect digital invariant 1, and the only cycle is the eight-number cycle
- 5 → 41 → 25 → 45 → 105 → 42 → 32 → 21 → 5 → ...
and because all numbers are preperiodic points for , all numbers either lead to 1 and are happy, or lead to the cycle and are unhappy. Because base 6 has no other perfect digital invariants except for 1, no positive integer other than 1 is the sum of the squares of its own digits.
In base 10, the 74 6-happy numbers up to 1296 = 64 are (written in base 10):
- 1, 6, 36, 44, 49, 79, 100, 160, 170, 216, 224, 229, 254, 264, 275, 285, 289, 294, 335, 347, 355, 357, 388, 405, 415, 417, 439, 460, 469, 474, 533, 538, 580, 593, 600, 608, 628, 638, 647, 695, 707, 715, 717, 767, 777, 787, 835, 837, 847, 880, 890, 928, 940, 953, 960, 968, 1010, 1018, 1020, 1033, 1058, 1125, 1135, 1137, 1168, 1178, 1187, 1195, 1197, 1207, 1238, 1277, 1292, 1295
10-happy numbers
For , the only positive perfect digital invariant for is the trivial perfect digital invariant 1, and the only cycle is the eight-number cycle
- 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4 → ...
and because all numbers are preperiodic points for , all numbers either lead to 1 and are happy, or lead to the cycle and are unhappy. Because base 10 has no other perfect digital invariants except for 1, no positive integer other than 1 is the sum of the squares of its own digits.
In base 10, the 143 10-happy numbers up to 1000 are:
- 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, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496, 536, 556, 563, 565, 566, 608, 617, 622, 623, 632, 635, 637, 638, 644, 649, 653, 655, 656, 665, 671, 673, 680, 683, 694, 700, 709, 716, 736, 739, 748, 761, 763, 784, 790, 793, 802, 806, 818, 820, 833, 836, 847, 860, 863, 874, 881, 888, 899, 901, 904, 907, 910, 912, 913, 921, 923, 931, 932, 937, 940, 946, 964, 970, 973, 989, 998, 1000 (sequence A007770 in the OEIS).
The distinct combinations of digits that form 10-happy numbers below 1000 are (the rest are just rearrangements and/or insertions of zero digits):
- 1, 7, 13, 19, 23, 28, 44, 49, 68, 79, 129, 133, 139, 167, 188, 226, 236, 239, 338, 356, 367, 368, 379, 446, 469, 478, 556, 566, 888, 899. (sequence A124095 in the OEIS).
The first pair of consecutive 10-happy numbers is 31 and 32.[4] The first set of three consecutive is 1880, 1881, and 1882.[5] It has been proven that there exist sequences of consecutive happy numbers of any natural number length.[6] The beginning of the first run of at least n consecutive 10-happy numbers for n = 1, 2, 3, ... is[7]
- 1, 31, 1880, 7839, 44488, 7899999999999959999999996, 7899999999999959999999996, ...
As Robert Styer puts it in his paper calculating this series: "Amazingly, the same value of N that begins the least sequence of six consecutive happy numbers also begins the least sequence of seven consecutive happy numbers."[8]
The number of 10-happy numbers up to 10n for 1 ≤ n ≤ 20 is[9]
- 3, 20, 143, 1442, 14377, 143071, 1418854, 14255667, 145674808, 1492609148, 15091199357, 149121303586, 1443278000870, 13770853279685, 130660965862333, 1245219117260664, 12024696404768025, 118226055080025491, 1183229962059381238, 12005034444292997294.
Happy primes
A -happy prime is a number that is both -happy and prime. Unlike happy numbers, rearranging the digits of a -happy prime will not necessarily create another happy prime. For instance, while 19 is a 10-happy prime, 91 = 13 × 7 is not prime (but is still 10-happy).
All prime numbers are 2-happy and 4-happy primes, as
6-happy primes
In
- 211, 1021, 1335, 2011, 2425, 2555, 3351, 4225, 4441, 5255, 5525
10-happy primes
In
- 7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487 (sequence A035497 in the OEIS).
The palindromic prime 10150006 + 7426247×1075000 + 1 is a 10-happy prime with 150007 digits because the many 0s do not contribute to the sum of squared digits, and 12 + 72 + 42 + 22 + 62 + 22 + 42 + 72 + 12 = 176, which is a 10-happy number. Paul Jobling discovered the prime in 2005.[10]
As of 2010[update], the largest known 10-happy prime is 242643801 − 1 (a Mersenne prime).[dubious ] Its decimal expansion has 12837064 digits.[11]
12-happy primes
In
- 11031, 1233E, 13011, 1332E, 16377, 17367, 17637, 22E8E, 2331E, 233E1, 23955, 25935, 25X8E, 28X5E, 28XE5, 2X8E5, 2E82E, 2E8X5, 31011, 31101, 3123E, 3132E, 31677, 33E21, 35295, 35567, 35765, 35925, 36557, 37167, 37671, 39525, 4878E, 4X7X7, 53567, 55367, 55637, 56357, 57635, 58XX5, 5X82E, 5XX85, 606EE, 63575, 63771, 66E0E, 67317, 67371, 67535, 6E60E, 71367, 71637, 73167, 76137, 7XX47, 82XE5, 82EX5, 8487E, 848E7, 84E87, 8874E, 8X1X7, 8X25E, 8X2E5, 8X5X5, 8XX17, 8XX71, 8E2X5, 8E847, 92355, 93255, 93525, 95235, X1X87, X258E, X285E, X2E85, X85X5, X8X17, XX477, XX585, E228E, E606E, E822E, EX825, ...
Programming example
The examples below implement the perfect digital invariant function for and a default base described in the definition of happy given at the top of this article, repeatedly; after each time, they check for both halt conditions: reaching 1, and repeating a number.
A simple test in Python to check if a number is happy:
def pdi_function(number, base: int = 10):
"""Perfect digital invariant function."""
total = 0
while number > 0:
total += pow(number % base, 2)
number = number // base
return total
def is_happy(number: int) -> bool:
"""Determine if the specified number is a happy number."""
seen_numbers = set()
while number > 1 and number not in seen_numbers:
seen_numbers.add(number)
number = pdi_function(number)
return number == 1
See also
References
- ^ "Sad Number". Wolfram Research, Inc. Retrieved 16 September 2009.
- Bibcode:2011arXiv1110.3836G.
- ^ Sloane, N. J. A. (ed.). "Sequence A161872 (Smallest unhappy number in base n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A035502 (Lower of pair of consecutive happy numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 8 April 2011.
- ^ Sloane, N. J. A. (ed.). "Sequence A072494 (First of triples of consecutive happy numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 8 April 2011.
- arXiv:math/0607213.
- ^ Sloane, N. J. A. (ed.). "Sequence A055629 (Beginning of first run of at least n consecutive happy numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Styer, Robert (2010). "Smallest Examples of Strings of Consecutive Happy Numbers". Journal of Integer Sequences. 13: 5. 10.6.3 – via University of Waterloo. Cited in Sloane "A055629".
- ^ Sloane, N. J. A. (ed.). "Sequence A068571 (Number of happy numbers <= 10^n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Chris K. Caldwell. "The Prime Database: 10150006 + 7426247 · 1075000 + 1". utm.edu.
- ^ Chris K. Caldwell. "The Prime Database: 242643801 − 1". utm.edu.
Literature
- ISBN 0-387-20860-7.
External links
- Schneider, Walter: Mathews: Happy Numbers.
- Weisstein, Eric W. "Happy Number". MathWorld.
- calculate if a number is happy Archived 25 January 2019 at the Wayback Machine
- Happy Numbers at The Math Forum.
- 145 and the Melancoil at Numberphile.
- Symonds, Ria. "7 and Happy Numbers". Numberphile. Brady Haran. Archived from the original on 15 January 2018. Retrieved 2 April 2013.