A316347 -id:A316347 - cp's OEIS frontend

A339023 Replace each digit d in the decimal representation of n with the digital root of n*d.

0, 1, 4, 9, 7, 7, 9, 4, 1, 9, 10, 22, 36, 43, 52, 63, 76, 82, 99, 19, 40, 63, 88, 16, 36, 58, 73, 99, 28, 49, 90, 34, 61, 99, 31, 64, 99, 37, 67, 99, 70, 25, 63, 13, 55, 99, 46, 85, 36, 79, 70, 36, 85, 46, 99, 55, 13, 63, 25, 79, 90, 67, 37, 99, 64, 31, 99, 61

Offset: 0

Sebastian Karlsson, Jan 18 2021

			a(23) = 16 because 2*23 = 46 and 3*23 = 69 and the digital roots of 46 and 69 are 1 and 6.

Sebastian Karlsson, Table of n, a(n) for n = 0..10000
Sebastian Karlsson, The sequence generalized and plotted in arbitrary bases

Cf. A010888, A056992, A059729, A106649, A106746, A139337, A316347, A322131.

dr(n) = if(n, (n-1)%9+1); \\ A010888
a(n) = if (n==0, return(0)); my(d=digits(n), s=""); for (k=1, #d, s=concat(s, dr(n*d[k]))); eval(s); \\ Michel Marcus, Jan 18 2021

def digitalroot(n):
    return 0 if n == 0 else (n-1)%9 + 1
def a(n):
    return int(''.join([str(digitalroot(n*int(d))) for d in str(n)]))
for n in range(0, 68):
    print(a(n), end=', ')

a(9*n + 1) = 9*n + 1.

a(10*n) = 10*a(n). - Sebastian Karlsson, Feb 14 2021

A344851 a(n) = (n^2) mod (2^A070939(n)).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 16, 25, 4, 17, 0, 17, 4, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 0, 17, 36, 57, 16, 41, 4, 33, 0, 33, 4, 41, 16, 57, 36, 17, 0, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Offset: 0

Rémy Sigrist, May 30 2021

Informally, if n has w binary digits, a(n) is obtained by keeping the w final binary digits of n^2.
For n > 0, a(n) is the final digit of n^2 in base A062383(n).
This sequence has interesting graphical features (see illustration in Links section).

			For n = 42:
- A070939(42) = 6,
- a(42) = (42^2) mod (2^6) = 1764 mod 64 = 36.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
John S. McCaskill and Peter R.Wills, On permutations derived from integer powers x^n, arXiv:1907.01890 [math.NT], 2019.
Rémy Sigrist, Scatterplot of the sequence for n = 2^18..2^19

Cf. A000290, A048152, A062383, A070939, A086341, A116882, A316347 (decimal analog).

{0}~Join~Table[Mod[n^2,2^(1+Floor@Log2@n)],{n,100}] (* Giorgos Kalogeropoulos, Jun 02 2021 *)

PARI
```
a(n) = (n^2) % 2^#binary(n)
```

def a(n): return (n**2) % (2**n.bit_length())
print([a(n) for n in range(75)]) # Michael S. Branicky, May 30 2021

a(n) = 0 iff n = 0 or n > 1 and n belongs to A116882.
a(n) = 1 iff n belongs to A086341.
a(2^k + m) = a(2^(k+1)-m) for any k > 0 and m = 0..2^k.

cp's OEIS Frontend

A339023 Replace each digit d in the decimal representation of n with the digital root of n*d.

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