A128889 -id:A128889 - cp's OEIS frontend

A252491 a(n) = (10^(n^2) - 1)/(10^n - 1).

1, 101, 1001001, 1000100010001, 100001000010000100001, 1000001000001000001000001000001, 1000000100000010000001000000100000010000001, 100000001000000010000000100000001000000010000000100000001, 1000000001000000001000000001000000001000000001000000001000000001

Offset: 1

M. F. Hasler, Jan 08 2015

When written in base 10, the terms consist of n digits '1' separated by strings of n-1 digits '0'.
This sequence is relevant for counterexamples to a conjecture in A086766: If p is prime and a(p) is not prime, then A086766(10^(p-1)) = 0.
a(n) is the product of A019328(d) for all d that divide n^2 but not n. - Robert Israel, Jan 08 2015
If a(n) is a prime then n is a prime. What is the smallest prime term greater than 101 in this sequence? - Farideh Firoozbakht, Jan 08 2015
According to what precedes, a(n) is prime iff A019328(d) is prime, where d is the only divisor of n^2 which is not a divisor of n, i.e., iff n is a prime and n^2 is in A138940. No such term is known, except for n=2. - M. F. Hasler, Jan 09 2015

Cf. A128889 (for 2 instead of 10).

seq((10^(n^2)-1)/(10^n-1), n=1..20); # Robert Israel, Jan 08 2015

PARI
```
A252491(n)=(10^(n^2)-1)\(10^n-1)
```

A119408 Decimal equivalent of the binary string generated by the n X n identity matrix.

Original entry on oeis.org

1, 9, 273, 33825, 17043521, 34630287489, 282578800148737, 9241421688590303745, 1210107565283851686118401, 634134936313486520338360567809, 1329552593586084350528447794605199361, 11151733894906779683522195341810241573494785

Offset: 1

Lynn R. Purser, Jul 25 2006

a(n) is divisible by 2^n - 1. a(n) == n mod 2^(n+1) - 1. - Robert Israel, Jun 09 2015

			n=2: [1 0; 0 1] == 1001_2 = 9;
n=3: [1 0 0; 0 1 0; 0 0 1] == 100010001_2 = 273;
n=4: [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1] == 1000010000100001_2 = 33825.

Harvey P. Dale, Table of n, a(n) for n = 1..57

Cf. A128889.

for n = 1:10 bi2de((reshape(eye(n),length(eye(n))^2,1))') end
% Kyle Stern, Dec 14 2011

For[n=2,n<=10,Print[n," ",Sum[2^((n+1)(k-1)), {k,1,n}]];n++ ]
Table[FromDigits[Flatten[IdentityMatrix[n]],2],{n,15}] (* Harvey P. Dale, Dec 31 2021 *)

a(n)=(2^n*2^(n^2)-1)/(2*2^n-1) \\ Charles R Greathouse IV, Jun 09 2015

a(n) = 2^((n+1)(n-1)) + 2^((n+1)(n-2)) + ... + 1 where n=2,3,...

a(n) = (2^n*2^(n^2)-1)/(2*2^n-1). - Stuart Bruff, Jun 08 2015

A275779 a(n) = (2^(n^2) - 1)/(1 - 1/2^n).

Original entry on oeis.org

2, 20, 584, 69904, 34636832, 69810262080, 567382630219904, 18519084246547628288, 2422583247133816584929792, 1268889750375080065623288448000, 2659754699919401766201267083003561984, 22306191045953951743035482794815064402563072

Offset: 1

Olivier Gérard, Aug 08 2016

Sum of the geometric progression of ratio 2^n.

Number of all partial binary matrices with rows of length n: A partial binary matrix has 1<=k<=n rows of length n. The number of different partial matrices with k rows is 2^(k*n). a(n) is the sum for k between 1 and n.

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A128889 (accepting the null matrix and excluding the full n*n matrices)

Cf. A096131, A057524.

Table[(2^(n^2) - 1)/(1 - 1/2^n), {n, 1, 10}]

a(n) = {(2^(n^2) - 1)/(1 - 1/2^n)} \\ Andrew Howroyd, Apr 26 2020

a(n) = Sum_{k=1..n} 2^(k*n).

Terms a(11) and beyond from Andrew Howroyd, Apr 26 2020

A356274 a(n) is the number whose base-(n+1) expansion equals the binary expansion of n.

Original entry on oeis.org

1, 3, 5, 25, 37, 56, 73, 729, 1001, 1342, 1741, 2366, 2941, 3615, 4369, 83521, 104977, 130340, 160021, 194922, 234741, 280393, 332377, 406250, 474553, 551151, 636637, 732511, 837901, 954304, 1082401, 39135393, 45435425, 52521910, 60466213, 69345326, 79236613

Offset: 1

Thomas Scheuerle, Aug 02 2022

If the binary expansion of n is n = bit0*2^0 + bit1*2^1 + bit2*2^2 + ... then a(n) = bit0*(n+1)^0 + bit1*(n+1)^1 + bit2*(n+1)^2 + ... . In other words: Interpret the binary expansion of n as digits in base n+1.

			n=4 in binary is 100 and interpreting those digits as base n+1 = 5 is a(4) = 25.

Cf. A000523, A007814, A104257, A104258, A128889, A136516.

a[n_] := FromDigits[IntegerDigits[n, 2], n + 1]; Array[a, 40] (* Amiram Eldar, Aug 19 2022 *)

PARI
```
a(n) = fromdigits(digits(n, 2), n+1)
```

def a(n): return sum((n+1)**i*int(bi) for i, bi in enumerate(bin(n)[2:][::-1]))
print([a(n) for n in range(1, 39)]) # Michael S. Branicky, Aug 02 2022

a(2^n) = (2^n + 1)^n = A136516(n).
a(2^n - 1) = (2^(n^2) - 1)/(2^n - 1) = A128889(n).
a(2^n + 1) = 1 + (2^n + 2)^n.
a(n) = A104257(n+1, n).
a(n) = (1/n)*Sum_{j>=1} floor((n + 2^(j-1))/2^j) * ((n-1)*(n+1)^(j-1) + 1).
a(n) = (1/n)*Sum_{j=1..n} ((n-1)*(n+1)^A007814(j) + 1).
a(2*n) = A104258(2*n+1) - 1.
(2*m+1)^n divides a((2*m+1)^n-1) for positive m and n > 0.

cp's OEIS Frontend

A252491 a(n) = (10^(n^2) - 1)/(10^n - 1).

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A119408 Decimal equivalent of the binary string generated by the n X n identity matrix.

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A275779 a(n) = (2^(n^2) - 1)/(1 - 1/2^n).

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A356274 a(n) is the number whose base-(n+1) expansion equals the binary expansion of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula