Displaying 1-10 of 113 results found.
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1, 10, 13, 97, 100, 103, 130, 226, 262, 319, 356, 365, 391, 556, 565, 907, 970, 1000, 1003, 1030, 1122, 1177, 1188, 1212, 1221, 1222, 1277, 1300, 1339, 1393, 1448, 1478, 1484, 1487, 1557, 1575, 1717, 1727, 1748, 1755, 1771, 1772, 1784, 1818, 1844, 1847, 1874
COMMENTS
Every power of 10 is in this sequence, as both the sum of squared digits map ( A003132) and the map A376270 map powers of 10 to 1.
PROG
(Python)
def ssd(n): return sum(int(d)**2 for d in str(n))
def f(n): return (d:=list(map(int, str(n))))[0] * sum(di*di for di in d)
def happy(n):
if n == 1: return True
s = list(map(int, str(n)))
while n not in [1, 4]: n = ssd(n) # iterate until fixed point or cycle
return n == 1
def elated(n):
if n == 1: return True
traj = {n}
while (n:=f(n)) not in traj: traj.add(n)
return 1 in traj
def ok(n): return happy(n) and elated(n)
AUTHOR
N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024
Sum of squares of the decimal digits of the n-th prime.
+0
2
4, 9, 25, 49, 2, 10, 50, 82, 13, 85, 10, 58, 17, 25, 65, 34, 106, 37, 85, 50, 58, 130, 73, 145, 130, 2, 10, 50, 82, 11, 54, 11, 59, 91, 98, 27, 75, 46, 86, 59, 131, 66, 83, 91, 131, 163, 6, 17, 57, 89, 22, 94, 21, 30, 78, 49, 121, 54, 102, 69, 77, 94, 58, 11
EXAMPLE
For n=7, the 7th prime = 17 and those digits 1^2 + 7^2 = 50 = a(7).
MATHEMATICA
a[n_]:=Norm[IntegerDigits[Prime[n]]]^2; Array[a, 64] (* Stefano Spezia, Oct 03 2024 *)
PROG
(PARI) a(n) = norml2(digits(prime(n))); \\ Michel Marcus, Oct 03 2024 (Python)
from sympy import prime
def A376714(n): return sum((0, 1, 4, 9, 16, 25, 36, 49, 64, 81)[int(d)] for d in str(prime(n)) if d>'0') # Chai Wah Wu, Oct 04 2024
Sum of squares of the decimal digits of 2^n.
+0
1
1, 4, 16, 64, 37, 13, 52, 69, 65, 30, 21, 84, 133, 150, 126, 162, 131, 64, 77, 177, 191, 164, 139, 301, 225, 113, 266, 197, 231, 269, 209, 275, 404, 450, 443, 371, 426, 332, 461, 487, 413, 288, 266, 396, 346, 382, 426, 404, 463, 393, 514, 528, 517, 569, 584
EXAMPLE
For n=4, 2^4 = 16 and those digits 1^2 + 6^2 = 37 = a(4).
MATHEMATICA
a[n_]:=Norm[IntegerDigits[2^n]]^2; Array[a, 55, 0] (* Stefano Spezia, Sep 06 2024 *)
PROG
(Python)
def A375976(n): return sum((0, 1, 4, 9, 16, 25, 36, 49, 64, 81)[int(d)] for d in str(1<<n) if d>'0') # Chai Wah Wu, Sep 30 2024
a(n) is the product of the leading digit of n and the sum of the squares of its digits.
+0
9
0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 8, 10, 16, 26, 40, 58, 80, 106, 136, 170, 27, 30, 39, 54, 75, 102, 135, 174, 219, 270, 64, 68, 80, 100, 128, 164, 208, 260, 320, 388, 125, 130, 145, 170, 205, 250, 305, 370, 445, 530, 216, 222, 240, 270, 312, 366
LINKS
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.
MAPLE
a:= n-> (l-> l[-1]*add(i^2, i=l))(convert(n, base, 10)):
MATHEMATICA
a[n_]:=First[d=IntegerDigits[n]]Norm[d]^2; Array[a, 66, 0] (* Stefano Spezia, Sep 18 2024 *)
PROG
(PARI) a(n) = if (n, my(d=digits(n)); d[1]*norml2(d), 0);
(Python)
def a(n): return (d:=list(map(int, str(n))))[0] * sum(di*di for di in d)
Number of iterations of the "x -> sum of squares of digits of x" map ( A003132) for n to converge to either 0, 1 or the 8-cycle (37,58,89,145,42,20,4,16).
+0
1
1, 1, 2, 6, 1, 5, 10, 6, 6, 5, 2, 3, 6, 3, 7, 4, 1, 6, 4, 5, 1, 6, 7, 4, 2, 4, 3, 7, 4, 3, 6, 3, 4, 5, 5, 6, 9, 1, 3, 6, 2, 7, 1, 5, 5, 8, 5, 4, 7, 5, 5, 4, 4, 6, 8, 6, 3, 5, 1, 3, 10, 2, 3, 9, 5, 3, 8, 3, 3, 6, 6, 6, 7, 2, 4, 5, 3, 3, 5, 4, 6, 4, 4, 3, 7, 2
COMMENTS
The initial number counts as iteration 1.
FORMULA
a(n) <= A193995(n) with equality if and only if n is a Happy number ( A007770). a(n) = 1 if and only if n is in A039943.
PROG
(Python)
c = 1
while n not in {0, 1, 37, 58, 89, 145, 42, 20, 4, 16}:
n = sum((0, 1, 4, 9, 16, 25, 36, 49, 64, 81)[ord(d)-48] for d in str(n))
c += 1
return c
The sum of the squares of the digits of n, repeated until reaching a single-digit number.
+0
0
0, 1, 4, 9, 4, 4, 4, 1, 4, 4, 1, 2, 5, 1, 4, 4, 4, 4, 4, 1, 4, 5, 8, 1, 4, 4, 4, 4, 1, 4, 9, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 1, 4, 4, 4, 4, 4, 4, 4, 2, 1, 4, 4, 1, 4, 4, 4, 1, 2, 4, 4, 4, 1, 4, 4, 1, 4, 4, 1, 4, 4, 1
COMMENTS
Square the digits of n, then sum the squares. Repeat the process until the sum is less than 10.
EXAMPLE
For n=28, the sum of the squares of the digits gives 4+64 = 68. Repeating the process gives 36+64 = 100; repeating once more gives 1+0+0 = 1. Therefore a(28) is 1.
a(n) = 4 for 72 of the first 100 n (0 to 99 inclusive.)
MATHEMATICA
f[n_] := Plus @@ (IntegerDigits[n]^2); a[n_] := NestWhile[f, f[n], # > 9 &]; Array[a, 100, 0] (* Amiram Eldar, Feb 17 2023 *)
Sum of squares of digits of primorial base expansion of n.
+0
1
0, 1, 1, 2, 4, 5, 1, 2, 2, 3, 5, 6, 4, 5, 5, 6, 8, 9, 9, 10, 10, 11, 13, 14, 16, 17, 17, 18, 20, 21, 1, 2, 2, 3, 5, 6, 2, 3, 3, 4, 6, 7, 5, 6, 6, 7, 9, 10, 10, 11, 11, 12, 14, 15, 17, 18, 18, 19, 21, 22, 4, 5, 5, 6, 8, 9, 5, 6, 6, 7, 9, 10, 8, 9, 9, 10, 12, 13, 13, 14, 14, 15, 17, 18, 20, 21, 21, 22
FORMULA
For all n >= 0, a(2n+1) = 1 + a(2n).
EXAMPLE
5 in primorial base ( A049345) is written as "21" (because 5 = 2*2 + 1*1), therefore a(5) = 2^2 + 1^2 = 5. 23 in primorial base is written as "321" (because 23 = 3*6 + 2*2 + 1*1), therefore a(23) = 3^2 + 2^2 + 1^2 = 14.
24 in primorial base is written as "400" (because 24 = 4*6 + 0*2 + 0*1), therefore a(24) = 4^2 = 16.
MATHEMATICA
a[n_] := Module[{k = n, p = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, s += r^2; p = NextPrime[p]]; s]; Array[a, 100, 0] (* Amiram Eldar, Mar 06 2024 *)
PROG
(PARI) A360108(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d^2; n = (n-d)/p; p = nextprime(1+p)); (s); };
Irregular triangle read by rows in which row n lists the possible periods for the iterations of the map sum of n-th powers of digits.
+0
0
1, 1, 8, 1, 2, 3, 1, 2, 7, 1, 2, 4, 6, 10, 12, 22, 28, 1, 2, 3, 4, 10, 30, 1, 2, 3, 6, 12, 14, 21, 27, 30, 56, 92, 1, 25, 154, 1, 2, 3, 4, 8, 10, 19, 24, 28, 30, 80, 93, 1, 6, 7, 17, 81, 123
EXAMPLE
Triangle begins:
1;
1, 8;
1, 2, 3;
1, 2, 7;
1, 2, 4, 6, 10, 12, 22, 28;
1, 2, 3, 4, 10, 30;
1, 2, 3, 6, 12, 14, 21, 27, 30, 56, 92;
1, 25, 154;
1, 2, 3, 4, 8, 10, 19, 24, 28, 30, 80, 93;
1, 6, 7, 17, 81, 123;
...
Numbers m such that A257588(m) = 0.
+0
2
0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 220, 330, 354, 440, 453, 550, 660, 770, 880, 990, 1001, 1100, 1111, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 1221, 1331, 1441, 1487, 1551, 1575, 1661, 1771, 1784, 1881, 1991, 2002, 2112, 2200, 2211, 2222, 2233, 2244, 2255, 2266, 2277
COMMENTS
If m is a term, 10*m is also a term; so, terms with no trailing zeros are all primitive terms.
Palindromes with even number of digits ( A056524) are all terms.
EXAMPLE
354 is a term since 3^2 - 5^2 + 4^2 = 0 (with Pythagorean triple (3,4,5)).
1487 is a term since 1^2 - 4^2 + 8^2 - 7^2 = 0.
MATHEMATICA
f[n_] := Abs @ Total[(d = IntegerDigits[n]^2) * (-1)^Range[Length[d]]]; Select[Range[0, 2300], f[#] == 0 &] (* Amiram Eldar, Mar 20 2022 *)
PROG
(Python)
from itertools import count, islice
def A352535_gen(startvalue=0): # generator of terms >= startvalue return filter(lambda m: not sum(int(d)**2*(-1 if i % 2 else 1) for i, d in enumerate(str(m))), count(max(startvalue, 0)))
Happy Niven (or happy harshad) numbers.
+0
1
1, 7, 10, 70, 100, 133, 190, 192, 230, 280, 320, 392, 440, 644, 700, 736, 820, 874, 888, 910, 912, 1000, 1088, 1090, 1092, 1122, 1125, 1128, 1141, 1148, 1152, 1185, 1188, 1212, 1215, 1233, 1251, 1274, 1275, 1300, 1323, 1330, 1332, 1512, 1521, 1547, 1679, 1725
COMMENTS
Numbers that are divisible by the sum of their digits and whose trajectory under iteration of sum of squares of digits map includes 1.
EXAMPLE
133 is a term because 133/7 = 19 and its trajectory under iteration of sum of squares of digits map is 133 -> 19 -> 82 -> 68 -> 100 -> 1.
MAPLE
q:= proc(n) local m, s; m, s:= n, {};
if irem(n, add(i, i=convert(n, base, 10)))>0 then return false fi;
do if m=1 then return true
elif m in s then return false
else s, m:= s union {m}, add(i^2, i=convert(m, base, 10))
fi
od
end:
select(q, [$1..2000])[];
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