Adam J.T. Partridge - cp's OEIS frontend

A262433 Quater-imaginary representation of the Gaussian primes with an even imaginary part.

3, 11, 13, 21, 31, 101, 111, 113, 123, 133, 201, 211, 213, 223, 233, 301, 321, 323, 331, 1003, 1011, 1021, 1031, 1033, 1101, 1113, 1123, 1131, 1133, 1201, 1203, 1213, 1223, 1231, 1233, 1311, 1321, 1323, 2001, 2011, 2031, 2033, 2103, 2113, 2131, 2133, 2203

Offset: 1

Adam J.T. Partridge, Sep 22 2015

Not all Gaussian primes will be in this list as complex numbers with an odd imaginary part require a value after the radix point (".") in the quater-imaginary number system.

			1231_(2i) = 1(2i)^3 + 2(2i)^2 + 3(2i)^1 + 1(2i)^0 = -7-2i which is a Gaussian prime.

Donald Knuth, An imaginary number system, Communications of the ACM 3 (4), April 1960, pp. 245-247.
OEIS Wiki, Quater-imaginary base
OEIS Wiki, Gaussian primes
Wikipedia, Quater-imaginary base

A002145 when translated using A212494 is a subsequence.

Real and imaginary parts of the Gaussian primes A103431, A103432.

A262222 Decimal representations of hexadecimal numbers that can be misinterpreted as decimal numbers in scientific E notation.

Original entry on oeis.org

480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 992, 993, 994, 995, 996, 997, 998, 999, 1000, 1001, 1248, 1249, 1250, 1251, 1252, 1253, 1254, 1255, 1256, 1257, 1504, 1505, 1506, 1507, 1508, 1509, 1510, 1511

Offset: 1

Adam J.T. Partridge, Sep 16 2015

Hexadecimal numbers containing no alpha digits other than a single "e", where the "e" is not the first or last digit (e.g., 3e12), may be misinterpreted as numbers in scientific E notation.
These numbers are especially troublesome when importing hexadecimal numbers from CSV files into Microsoft Excel.
These numbers can be written as 16^(i+1)a + 14*16^i + b where a and b are members of A102489 with a>0, b<16^i and i>0.
Numbers whose hexadecimal representation matches the regular expression [1-9][0-9]*e[0-9]+. - Eric M. Schmidt, Sep 28 2023

			480_10 = 1e0_16 and 1e0 = 1 in E notation.
739_10 = 2e3_16 and 2e3 = 2000_10 in E notation.

Christian Perfect, Table of n, a(n) for n = 1..99999
Microsoft Community, Unable to disable scientific notation in Microsoft Excel.
Wikipedia, Scientific E notation.

A subset of A102490.

#include 
#define DIGIT_E 14
bool isA262222(int k)
{
    if (k <= 0 || k % 16 == DIGIT_E) return false;
    bool foundE = false;
    int digit;
    while (k > 0) {
        digit = k % 16;
        if (digit == DIGIT_E) {
            if (foundE) return false;
            foundE = true;
        }
        else if (digit > 9) return false;
        k /= 16;
    }
    return foundE && digit != DIGIT_E;
} // Eric M. Schmidt, Sep 28 2023

from itertools import count,product
# every string of d characters with exactly one 'e' in it, and all the other characters digits 0-9, in ascending lexicographic order
def mids(d):
    if d<1:
        raise Exception("d<1")
    if d==1:
        yield 'e'
        return
    for i in range(0,10):
        for m in mids(d-1):
            yield str(i)+m
    for i in range(10**(d-1)):
        yield 'e'+str(i).zfill(d-1)
def a_generator():
    for d in count(1):
        for start in range(1,10): # for each leading digit 1-9
            for mid in mids(d): # for all possible middles made of d characters, containing exactly one 'e'
                for end in range(10): #for each possible final digit, 0-9
                    s = '{}{}{}'.format(start,''.join(mid),end)
                    i = int(s,16)
                    yield i
a262222 = a_generator()
[next(a262222) for  in range(48)] # _Christian Perfect, Oct 20 2015

A259368 Number of digits in n^n when written in binary.

Original entry on oeis.org

1, 3, 5, 9, 12, 16, 20, 25, 29, 34, 39, 44, 49, 54, 59, 65, 70, 76, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 148, 154, 161, 167, 173, 180, 187, 193, 200, 207, 213, 220, 227, 234, 241, 248, 255, 262, 269, 276, 283, 290, 297, 304, 311, 318, 326, 333

Offset: 1

Adam J.T. Partridge, Jun 25 2015

			For n=3, 3^3=11011_2 so a(3)=5.

Cf. A000312, A066022, A070939.

[Floor(n*Log(n)/Log(2)) + 1: n in [1..70]]; // Vincenzo Librandi, Jul 15 2015

Array[IntegerLength[#^#, 2] &, 60] (* or *)
Array[Floor[# Log[2, #]] + 1 &, 60] (* Michael De Vlieger, Jul 03 2015 *)

a(n) = #binary(n^n); \\ Michel Marcus, Jul 03 2015

a(n) = floor(n*log(n)/log(2)) + 1.

a(n) = A070939(A000312(n)). - Michel Marcus, Jul 03 2015

A258371 Triangle read by rows: T(n,k) is number of ways of arranging n indistinguishable points on an n X n square grid such that k rows contain at least one point.

Original entry on oeis.org

1, 2, 4, 3, 54, 27, 4, 408, 1152, 256, 5, 2500, 22500, 25000, 3125, 6, 13830, 315900, 988200, 583200, 46656, 7, 72030, 3709545, 25882780, 40588905, 14823774, 823543, 8, 360304, 39024384, 535754240, 1766195200, 1657012224, 411041792, 16777216

Offset: 1

Adam J.T. Partridge, May 28 2015

Row sums give A014062, n >= 1.
Leading diagonal is A000312, n >= 1.
The triangle t(n,k) = T(n,k)/binomial(n,k) gives the number of ways to place n stones into the k X n grid of squares such that each of the k rows contains at least one stone. See A259051. One can use a partition array for this (and the T(n,k)) problem. See A258152. - Wolfdieter Lang, Jun 17 2015

			The number of ways of arranging eight pawns on a standard chessboard such that two rows contain at least one pawn is T(8,2)=360304.
Triangle T(n,k) begins:
n\k 1      2        3       4        5       6 ...
1:  1
2:  2      4
3:  3     54      27
4:  4    408    1152      256
5:  5   2500   22500    25000     3125
6:  6  13830  315900   988200   583200   46656
...
n = 7:  7  72030 3709545 25882780  40588905 14823774 823543,
n = 8:  8 360304 39024384 535754240 1766195200 1657012224 411041792 16777216.

Giovanni Resta, Table of n, a(n) for n = 1..1830 (first 60 rows)

Cf. A000312, A014062, A258152, A259051.

T[n_,k_]:= Binomial[n,k] * Sum[Multinomial@@ (Last/@ Tally[e]) * Times@@ Binomial[n,e], {e, IntegerPartitions[n, {k}]}]; Flatten@ Table[ T[n,k],{n,9}, {k,n}] (* Giovanni Resta, May 28 2015 *)

T(n,2) = binomial(n,2)*(binomial(2*n,n)-2). - Giovanni Resta, May 28 2015

A258103 Number of pandigital squares (containing each digit exactly once) in base n.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 4, 26, 87, 47, 87, 0, 547, 1303, 3402, 0, 24192, 187562

Offset: 2

Adam J.T. Partridge, May 20 2015

For n = 18, the smallest and largest pandigital squares are 2200667320658951859841 and 39207739576969100808801. For n = 19, they are 104753558229986901966129 and 1972312183619434816475625. For n = 20, they are 5272187100814113874556176 and 104566626183621314286288961. - Chai Wah Wu, May 20 2015
When n is even, (n-1) is a factor of the pandigital squares.  When n is odd, (n-1)/2 is a factor with the remaining factors being odd.  Therefore, when n is odd and (n-1)/2 has an odd number of 2s as prime factors there are no pandigital squares in base n (e.g. 5, 13, 17 and 21). - Adam J.T. Partridge, May 21 2015
If n is odd and (n-1)/2 has an odd 2-adic valuation, then there are no squares in base n using all the digits from 1 to n-1 once, or all the digits from 0 to n-2 once or all the digits from 1 to n-2 once. This can be proved using the same argument as in the linked blogposts. - Chai Wah Wu, Feb 25 2024

			For n=4 there is one pandigital square, 3201_4 = 225 = 15^2.
For n=6 there is one pandigital square, 452013_6 = 38025 = 195^2.
For n=10 there are 87 pandigital squares (A036745).
There are no pandigital squares in bases 2, 3, 5 or 13.
Hexadecimal has 3402 pandigital squares, the largest is FED5B39A42706C81.

A. J. T. Partridge, Why there are no pandigital squares in base 13
Chai Wah Wu, Square pandigital numbers
Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 2.

Cf. A036745, A054038, A071519.

a(n) = if(n%2==1 && valuation(n-1,2)%2==0, 0, my(lim=sqrtint(n^n - (n^n-n)/(n-1)^2), count=0); for(m=sqrtint((n^n-n)/(n-1)^2 + n^(n-2)*(n-1) - 1), lim, if(#Set(digits(m^2,n))==n, count++)); count) \\ Jianing Song, Feb 23 2024. Note that the searching range for m is [sqrt(A049363(n)), sqrt(A062813(n))]

from gmpy2 import isqrt, mpz, digits
def A258103(n): # requires 2 <= n <= 62
    c, sm, sq = 0, mpz(''.join([digits(i, n) for i in range(n-1, -1, -1)]), n), mpz(''.join(['1', '0']+[digits(i, n) for i in range(2, n)]), n)
    m = isqrt(sq)
    sq = m*m
    m = 2*m+1
    while sq <= sm:
        if len(set(digits(sq, n))) == n:
            c += 1
        sq += m
        m += 2
    return c # Chai Wah Wu, May 20 2015

a(17)-a(19) from Giovanni Resta, May 20 2015

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User: Adam J.T. Partridge

A262433 Quater-imaginary representation of the Gaussian primes with an even imaginary part.

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A262222 Decimal representations of hexadecimal numbers that can be misinterpreted as decimal numbers in scientific E notation.

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A259368 Number of digits in n^n when written in binary.

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A258371 Triangle read by rows: T(n,k) is number of ways of arranging n indistinguishable points on an n X n square grid such that k rows contain at least one point.

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A258103 Number of pandigital squares (containing each digit exactly once) in base n.

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PARI

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