A185679 -id:A185679 - cp's OEIS frontend

A366721 Number of digits left of the radix point of n when written in base Pi using a greedy algorithm representation.

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5

Offset: 0

Paolo Xausa, Oct 17 2023

More than the usual number of terms are shown to distinguish this sequence from A185679.

			a(10) = 3 because 10 in base Pi (100.01022...) has 3 digits before the radix point.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Non-integer base of numeration.

Cf. A000796, A055642, A185679, A344939, A362692, A363832.

A366721[n_]:=Floor[Log[Pi,Max[n,1]]]+1;Array[A366721,200,0]

a(0) = 1; for n >= 1, a(n) = floor(log_Pi(n)) + 1.

A360803 Numbers whose squares have a digit average of 8 or more.

Original entry on oeis.org

3, 313, 94863, 298327, 987917, 3162083, 9893887, 29983327, 99477133, 99483667, 197483417, 282753937, 314623583, 315432874, 706399164, 773303937, 894303633, 947047833, 948675387, 989938887, 994927133, 994987437, 998398167, 2428989417, 2754991833, 2983284917, 2999833327

Offset: 1

Dmitry Kamenetsky, Feb 21 2023

This sequence is infinite. For example, numbers floor(30*100^k - (5/3)*10^k) beginning with 2 followed by k 9s, followed by 8 and k 3s, have a square whose digit average converges to (but never equals) 8.25. [Corrected and formula added by M. F. Hasler, Apr 11 2023]
Only a few examples are known whose square has a digit average of 8.25 and above: 3^2 = 9, 707106074079263583^2 = 499998999999788997978888999589997889 (digit average 8.25), 94180040294109027313^2 = 8869879989799999999898984986998979999969 (digit average 8.275).
This is the union of A164772 (digit average = 8) and A164841 (digit average > 8). - M. F. Hasler, Apr 11 2023

			94863 is in the sequence, because 94863^2 = 8998988769, which has a digit average of 8.1 >= 8.

Michael S. Branicky, Python program for OEIS A360822 and searching for squares with < given number of digits < 8
Matthieu Dufour and Silvia Heubach, Squares with Large Digit Average, 2020.
Dmitry Kamenetsky, The first 68 terms and their digit average

Cf. A004159, A185679.

Cf. A164772 (digit average = 8), A164841 (digit average > 8).

isok(k) = my(d=digits(k^2)); vecsum(d)/#d >= 8; \\ Michel Marcus, Feb 22 2023

def ok(n): d = list(map(int, str(n**2))); return sum(d) >= 8*len(d)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Feb 22 2023

A185076 a(n) is the least number k such that (sum of digits of k^2) + (number of digits of k^2) = n, or 0 if no such k exists.

Original entry on oeis.org

0, 1, 0, 10, 2, 100, 11, 1000, 4, 3, 6, 8, 19, 35, 7, 16, 34, 106, 13, 41, 24, 17, 37, 107, 323, 43, 124, 317, 67, 113, 63, 114, 134, 343, 83, 133, 367, 1024, 167, 374, 264, 314, 386, 1043, 313, 583, 1303, 3283, 707, 1183, 3316, 836, 1333, 3286, 10133

Offset: 1

Carmine Suriano, Feb 23 2011

a(n) < sqrt(10^(n-1)). 0 < a(2m) <= 10^(m-1) with the upper bound reached for 1<=m<=4. - Chai Wah Wu, Mar 15 2023

			a(7)=11 since 7 = sumdigits(121) + numberdigits(121) = 4 + 3.

Chai Wah Wu, Table of n, a(n) for n = 1..185

Cf. A004159, A185679.

Table[k=1; While[d=IntegerDigits[k^2]; n>Length[d] && n != Total[d] + Length[d], k++]; If[Length[d] >= n, k=0]; k, {n, 50}]

from itertools import count
def A185076(n):
    for k in count(1):
        if n == (t:=len(s:=str(k**2)))+sum(map(int,s)):
            return k
        if t >= n:
            return 0 # Chai Wah Wu, Mar 15 2023

n = A004159(a(n)) + A185679(a(n)).

A309432 Number of distinct digits in decimal representation of n^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 4, 4, 3, 3, 4, 4, 2, 3, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 3, 4, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 4, 4, 4, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 3, 4

Offset: 0

Hauke Löffler, Aug 01 2019

			a(0) = 1 because 0^2 = 0 has 1 distinct digit (0).
a(5) = 2 because 5^2 = 25 has 2 distinct digits (2, 5).
a(10) = 2 because 10^2 = 100 has 2 distinct digits (0, 1).

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000290 (squares), A043537, A054039, A137214, A185679, A309431.

Array[Count[DigitCount[#^2], ?(# > 0 &)] &, 105, 0] (* _Michael De Vlieger, Jun 22 2022 *)

a(n) = if(n==0, 1, #Set(digits(n^2))); \\ Jinyuan Wang, Aug 02 2019, Jun 22 2022 [a(0)=1 corrected by Georg Fischer, Jun 22 2022]

def a(n): return len(set(str(n*n)))
print([a(n) for n in range(87)]) # Michael S. Branicky, Jun 22 2022

[len(set(Integer((n^2)).digits(padto=1))) for n in range(25) ]

cp's OEIS Frontend

A366721 Number of digits left of the radix point of n when written in base Pi using a greedy algorithm representation.

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A360803 Numbers whose squares have a digit average of 8 or more.

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A185076 a(n) is the least number k such that (sum of digits of k^2) + (number of digits of k^2) = n, or 0 if no such k exists.

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A309432 Number of distinct digits in decimal representation of n^2.

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