A119509 -id:A119509 - cp's OEIS frontend

A162999 Positions of primes in A119509.

2, 3, 5, 7, 10, 13, 15, 16, 21, 22, 26, 28, 34, 38, 39, 43, 45, 48, 53, 59, 69, 83, 84, 87, 88, 92, 103, 109, 112, 124, 130, 132, 135, 145, 146, 150, 153, 156, 162, 175, 177, 178, 183, 187, 190, 196, 200, 201, 217, 230, 238, 240, 250, 267, 272, 279, 280, 286, 290, 294

Offset: 1

Zak Seidov, Jul 20 2009

There are exactly 87 primes in A119509. Cf. A162950 primes in A119509 Numbers whose square contains no repeated digit.

			Rest 27 terms after a(60)=294 are: 309, 316, 318, 320, 321, 330, 337, 355, 362, 367, 368, 378, 385, 394, 395, 397, 401, 408, 411, 413, 418, 429, 430, 435, 451, 477, 481.

A162950 Primes whose square contains no repeated digit.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 43, 53, 59, 61, 71, 73, 79, 89, 113, 137, 179, 181, 191, 193, 199, 233, 269, 281, 311, 353, 367, 397, 463, 487, 509, 557, 571, 593, 647, 661, 677, 709, 733, 757, 797, 829, 839, 929, 1117, 1277, 1307, 1433, 1873, 1949

Offset: 1

Claudio Meller, Jul 18 2009

Or: primes in A119509.

There are only 87 terms, the last being 21397.

Nathaniel Johnston, Table of n, a(n) for n = 1..87 (full sequence)

Cf. A119509.

[p: p in PrimesUpTo(21397) | #Set(d) eq #d where d is Intseq(p^2)]; // Bruno Berselli, Aug 02 2011

t={}; Do[p=Prime[n]; If[Max[DigitCount[p^2]]<= 1,AppendTo[t,p]],{n,300}]; t   (* Jayanta Basu, May 10 2013 *)

is(n)=if(n<21398 && isprime(n), n=digits(n^2); #v==#vecsort(v,,8), 0) \\ Charles R Greathouse IV, May 10 2013

Keyword:base added by R. J. Mathar, Jul 19 2009

A226796 Number of nonnegative numbers x < 10^n such that the digits of x^2 occur with an equal frequency of 1.

Original entry on oeis.org

10, 59, 221, 441, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611, 611

Offset: 1

T. D. Noe, Jun 21 2013

			All numbers 0 to 9 have squares containing only digits of frequency 1: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81. See A119509 for the positive terms.

Cf. A119509 (positive terms x), A225428 (for 2 digits).

cnt = 0; x = 0; Table[While[x < 10^n, If[Union[Last[Transpose[Tally[IntegerDigits[x^2]]]]] == {1}, cnt++]; x++]; cnt, {n, 5}]

A376897 Positive numbers k such that all the digits in the octal expansion of k^2 are distinct.

Original entry on oeis.org

1, 2, 4, 5, 7, 13, 14, 15, 18, 20, 21, 28, 30, 37, 39, 43, 44, 45, 53, 55, 63, 78, 84, 103, 110, 113, 117, 127, 149, 155, 156, 161, 162, 172, 173, 174, 175, 179, 220, 236, 242, 270, 286, 293, 299, 301, 340, 343, 350, 356, 361, 395, 407, 412, 425, 439, 461, 475, 499, 674, 819, 1001, 1211, 1230, 1244, 1323, 1764, 2450, 2751, 3213

Offset: 1

Kalle Siukola, Oct 08 2024

There are no terms >= 2^12 because 2^24-1 is the largest eight-digit octal number.

			110 is in the sequence because 110^2 = 12100 = 27504_8 and no octal digit occurs more than once.

Cf. A007094, A119509, A376898.

Select[Range[2^12], DuplicateFreeQ[IntegerDigits[#^2, 8]] &] (* Michael De Vlieger, Oct 12 2024 *)

for k in range(1, 2**12):
    octal = format(k**2, "o")
    if len(octal) == len(set(octal)): print(k, end=",")

A225565 Numbers with repeated digits, whose squares do not contain any repeated digits.

Original entry on oeis.org

33, 44, 55, 66, 99, 113, 116, 117, 118, 133, 144, 181, 191, 199, 224, 226, 228, 232, 233, 252, 272, 282, 292, 299, 311, 322, 323, 353, 442, 445, 557, 616, 626, 661, 677, 686, 717, 733, 737, 757, 777, 778, 797, 799, 855, 884, 889, 929, 1017, 1113, 1117

Offset: 1

Jayanta Basu, May 10 2013

Subsequence of A119509. There are a total of 274 such terms. a(274)=99066.

			117 is a member since it has repeated digits but 117^2=13689 contains no repeated digits.

T. D. Noe, Table of n, a(n) for n = 1..274

Cf. A119509.

mx[n_]:=Max[DigitCount[n]]; Select[Range[1150],mx[#]>1 && mx[#^2]<=1 &]

A259187 Primes p such that both p and p^2 are distinct-digit numbers.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 43, 53, 59, 61, 71, 73, 79, 89, 137, 179, 193, 269, 281, 367, 397, 463, 487, 509, 571, 593, 647, 709, 829, 839, 1307, 1873, 2069, 2731, 2801, 3041, 4157, 4967, 4987, 6043, 7549, 7621, 8623, 21397

Offset: 1

Zak Seidov, Jun 20 2015

Corresponding squares are 4, 9, 25, 49, 169, 289, 361, 529, 841, 961, 1369, 1849, 2809, 3481, 3721, 5041, 5329, 6241, 7921, 18769, 32041, 37249, 72361, 78961, 134689, 157609, 214369, 237169, 259081, 326041, 351649, 418609, 502681, 687241, 703921, 1708249, 3508129, 4280761, 7458361, 7845601, 9247681, 17280649, 24671089, 24870169, 36517849, 56987401, 58079641, 74356129, 457831609 (subsequence of A078255).

Subsequence of A029743 and of A119509. Cf. A078255.

Select[Prime[Range[2500]],Max[DigitCount[#]]<2&&Max[DigitCount[#^2]]<2&] (* Harvey P. Dale, May 25 2020 *)

A376814 a(n) is the number of squares that have all digits distinct in base n.

Original entry on oeis.org

2, 2, 7, 7, 21, 42, 71, 268, 611, 1352, 3099, 8471, 23877, 63564, 182771, 527001, 1671752, 5055853

Offset: 2

Robert Israel, Oct 09 2024

			a(4) = 7 because the only squares with distinct digits in base 4 are 0^2 = 0_4, 1^2 = 1_4, 2^2 = 10_4, 3^2 = 21_4, 6^2 = 210_4, 7^2 = 301_4 and 15^2 = 3201_4.

Cf. A119509, A376897.

f:= proc(b) local k,t,F;
 t:= 0;
 for k from 0 to floor(sqrt(b^b-1)) do
   F:= convert(k^2, base, b);
   if nops(F) = nops(convert(F,set)) then t:= t+1 fi;
 od;
 t
end proc:
map(f, [$2..12]);

from math import isqrt
from sympy.ntheory import digits
def A376814(n): return sum(1 for k in range(isqrt(n**n-1)+1) if len(s:=digits(k**2,n)[1:])==len(set(s))) # Chai Wah Wu, Oct 09 2024

a(15)-a(16) from Michael S. Branicky, Oct 09 2024
a(17) from Michael S. Branicky, Oct 10 2024
a(18) from Michael S. Branicky, Oct 14 2024
a(19) from Michael S. Branicky, Oct 31 2024

cp's OEIS Frontend

A162999 Positions of primes in A119509.

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A162950 Primes whose square contains no repeated digit.

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A226796 Number of nonnegative numbers x < 10^n such that the digits of x^2 occur with an equal frequency of 1.

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Mathematica

A376897 Positive numbers k such that all the digits in the octal expansion of k^2 are distinct.

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Mathematica

Python

A225565 Numbers with repeated digits, whose squares do not contain any repeated digits.

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Mathematica

A259187 Primes p such that both p and p^2 are distinct-digit numbers.

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Mathematica

A376814 a(n) is the number of squares that have all digits distinct in base n.

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Maple

Python

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