A165425 -id:A165425 - cp's OEIS frontend

A329050 Square array A(n,k) = prime(n+1)^(2^k), read by descending antidiagonals (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...; Fermi-Dirac primes (A050376) in matrix form, sorted into rows by their prime divisor.

2, 4, 3, 16, 9, 5, 256, 81, 25, 7, 65536, 6561, 625, 49, 11, 4294967296, 43046721, 390625, 2401, 121, 13, 18446744073709551616, 1853020188851841, 152587890625, 5764801, 14641, 169, 17, 340282366920938463463374607431768211456, 3433683820292512484657849089281, 23283064365386962890625, 33232930569601, 214358881, 28561, 289, 19

Offset: 0

Antti Karttunen and Peter Munn, Nov 02 2019

This sequence is a permutation of A050376, so every positive integer is the product of a unique subset, S_factors, of its terms. If we restrict S_factors to be chosen from a subset, S_0, consisting of numbers from specified rows and/or columns of this array, there are notable sequences among those that may be generated. See the examples. Other notable sequences can be generated if we restrict the intersection of S_factors with specific rows/columns to have even cardinality. In any of the foregoing cases, the numbers in the resulting sequence form a group under the binary operation A059897(.,.).
Shares with array A246278 the property that columns grow downward by iterating A003961, and indeed, this array can be obtained from A246278 by selecting its columns 1, 2, 8, 128, ..., 2^((2^k)-1), for k >= 0.
A(n,k) is the image of the lattice point with coordinates X=n and Y=k under the inverse of the bijection f defined in the first comment of A306697. This geometric relationship can be used to construct an isomorphism from the polynomial ring GF(2)[x,y] to a ring over the positive integers, using methods similar to those for constructing A297845 and A306697. See A329329, the ring's multiplicative operator, for details.

			The top left 5 X 5 corner of the array:
  n\k |   0     1       2           3                   4
  ----+-------------------------------------------------------
   0  |   2,    4,     16,        256,              65536, ...
   1  |   3,    9,     81,       6561,           43046721, ...
   2  |   5,   25,    625,     390625,       152587890625, ...
   3  |   7,   49,   2401,    5764801,     33232930569601, ...
   4  |  11,  121,  14641,  214358881,  45949729863572161, ...
Column 0 continues as a list of primes, column 1 as a list of their squares, column 2 as a list of their 4th powers, and so on.
Every nonnegative power of 2 (A000079) is a product of a unique subset of numbers from row 0; every squarefree number (A005117) is a product of a unique subset of numbers from column 0. Likewise other rows and columns generate the sets of numbers from sequences:
Row 1:                 A000244 Powers of 3.
Column 1:              A062503 Squares of squarefree numbers.
Row 2:                 A000351 Powers of 5.
Column 2:              A113849 4th powers of squarefree numbers.
Union of rows 0 and 1:     A003586 3-smooth numbers.
Union of columns 0 and 1:  A046100 Biquadratefree numbers.
Union of row 0 / column 0: A122132 Oddly squarefree numbers.
Row 0 excluding column 0:  A000302 Powers of 4.
Column 0 excluding row 0:  A056911 Squarefree odd numbers.
All rows except 0:         A005408 Odd numbers.
All columns except 0:      A000290\{0} Positive squares.
All rows except 1:         A001651 Numbers not divisible by 3.
All columns except 1:      A252895 (have odd number of square divisors).
If, instead of restrictions on choosing individual factors of the product, we restrict the product to be of an even number of terms from each row of the array, we get A262675. The equivalent restriction applied to columns gives us A268390; applied only to column 0, we get A028260 (product of an even number of primes).

Antti Karttunen, Table of n, a(n) for n = 0..77; the first 12 antidiagonals of array
Wikipedia, Polynomial ring

Transpose: A329049.
Permutation of A050376.
Rows 1-4: A001146, A011764, A176594, A165425 (after the two initial terms).
Columns 1-5: A000040, A001248, A030514, A179645, A030635.
Antidiagonal products: A191555.
Subtable of A182944, A242378, A246278, A329332.
A000290, A003961, A225546 are used to express relationship between terms of this sequence.
Related binary operations: A059897, A306697, A329329.
See also the table in the example section.

Table[Prime[#]^(2^k) &[m - k + 1], {m, 0, 7}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Dec 28 2019 *)

up_to = 105;
A329050sq(n,k) = (prime(1+n)^(2^k));
A329050list(up_to) = { my(v = vector(up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A329050sq(col, a-col))); (v); };
v329050 = A329050list(up_to);
A329050(n) = v329050[1+n];
for(n=0,up_to-1,print1(A329050(n),", ")); \\ Antti Karttunen, Nov 06 2019

A(0,k) = 2^(2^k), and for n > 0, A(n,k) = A003961(A(n-1,k)).
A(n,k) = A182944(n+1,2^k).
A(n,k) = A329332(2^n,2^k).
A(k,n) = A225546(A(n,k)).
A(n,k+1) = A000290(A(n,k)) = A(n,k)^2.

Example annotated for clarity by Peter Munn, Feb 12 2020

A078888 Decimal expansion of Sum {n>=0} 1/7^(2^n).

Original entry on oeis.org

1, 6, 3, 6, 8, 1, 9, 7, 2, 7, 1, 6, 8, 6, 8, 0, 1, 7, 9, 1, 1, 7, 2, 9, 7, 2, 5, 8, 9, 3, 9, 0, 9, 2, 0, 0, 6, 0, 5, 2, 4, 4, 8, 5, 4, 1, 5, 9, 3, 3, 6, 8, 2, 5, 3, 2, 7, 8, 6, 2, 2, 1, 0, 3, 5, 9, 7, 2, 5, 1, 1, 8, 5, 9, 2, 9, 2, 3, 5, 7, 5, 0, 2, 5, 1, 1, 7, 3, 9, 7, 8, 4, 0, 1, 2, 7, 2, 9, 4, 3, 8, 1, 8, 4, 7

Offset: 0

Robert G. Wilson v, Dec 11 2002

			0.163681972716868017911...

Aubrey J. Kempner, On Transcendental Numbers, Transactions of the American Mathematical Society, volume 17, number 4, October 1916, pages 476-482.
Index entries for transcendental numbers

Cf. A165425.

Similar sums: A007404, A078885, A078585, A078886, A078887, A078889, A078890, A036987.

RealDigits[ N[ Sum[1/7^(2^n), {n, 0, Infinity}], 110]][[1]]

suminf(n=0, 1/7^(2^n)) \\ Michel Marcus, Nov 11 2020

Equals -Sum_{k>=1} mu(2*k)/(7^k - 1), where mu is the Möbius function (A008683). - Amiram Eldar, Jul 12 2020

A225159 Denominators of the sequence of fractions f(n) defined recursively by f(1) = 7/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

Original entry on oeis.org

1, 6, 43, 2143, 5211907, 30351298460743, 1016966398053911225889737707, 1130815308619683511655208290917557601522304473342184143

Offset: 1

Martin Renner, Apr 30 2013

nonn

Numerators of the sequence of fractions f(n) is A165425(n+1), hence sum(A165425(i+1)/a(i),i=1..n) = product(A165425(i+1)/a(i),i=1..n) = A165425(n+2)/A225166(n).

			f(n) = 7, 7/6, 49/43, 2401/2143, ...
7 + 7/6 = 7 * 7/6 = 49/6; 7 + 7/6 + 49/43 = 7 * 7/6 * 49/43 = 2401/258; ...

Cf. A100441, A165425, A225166.

b:=n->7^(2^(n-2)); # n > 1
b(1):=7;
p:=proc(n) option remember; p(n-1)*a(n-1); end;
p(1):=1;
a:=proc(n) option remember; b(n)-p(n); end;
a(1):=1;
seq(a(i),i=1..9);

a(n) = 7^(2^(n-2)) - product(a(i),i=1..n-1), n > 1 and a(1) = 1.

a(n) = 7^(2^(n-2)) - p(n) with a(1) = 1 and p(n) = p(n-1)*a(n-1) with p(1) = 1.

A152580 a(n) = 7^(2^n) + 2.

Original entry on oeis.org

9, 51, 2403, 5764803, 33232930569603, 1104427674243920646305299203

Offset: 0

Cino Hilliard, Dec 08 2008

These numbers are divisible by 3. This follows by expanding the binomial (6+1)^(2^n) + 2 to get 6h + 1 + 2 for some h. Therefore 3 divides 7^(2^n) + 2.

Cf. A165425.

A152580:=n->7^(2^n) + 2: seq(A152580(n), n=0..8); # Wesley Ivan Hurt, Jan 22 2017

g(a,n) = if(a%2,b=2,b=1);for(x=0,n,y=a^(2^x)+b;print1(y","))

a(n) = 7^(2^n) + 2; \\ Michel Marcus, Jan 24 2017

a(n) = A165425(n+3) + 2. - R. J. Mathar, Sep 10 2016

A225166 Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 7/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

Original entry on oeis.org

1, 6, 258, 552894, 2881632108858, 87461276190009420415561494, 88945179016152188483365571645414219233310820789054258

Offset: 1

Martin Renner, Apr 30 2013

nonn

Numerators of the sequence s(n) of the sum resp. product of fractions f(n) is A165425(n+2), hence sum(A165425(i+1)/A225159(i),i=1..n) = product(A165425(i+1)/A225159(i),i=1..n) = A165425(n+2)/a(n).

			f(n) = 7, 7/6, 49/43, 2401/2143, ...
7 + 7/6 = 7 * 7/6 = 49/6; 7 + 7/6 + 49/43 = 7 * 7/6 * 49/43 = 2401/258; ...
s(n) = 1/b(n) = 7, 49/6, 2401/258, ...

Cf. A076628, A165425, A225159.

b:=proc(n) option remember; b(n-1)-b(n-1)^2; end:
b(1):=1/7;
a:=n->7^(2^(n-1))*b(n);
seq(a(i),i=1..8);

a(n) = 7^(2^(n-1))*b(n) where b(n)=b(n-1)-b(n-1)^2 with b(1)=1/7.

cp's OEIS Frontend

A329050 Square array A(n,k) = prime(n+1)^(2^k), read by descending antidiagonals (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...; Fermi-Dirac primes (A050376) in matrix form, sorted into rows by their prime divisor.

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A078888 Decimal expansion of Sum {n>=0} 1/7^(2^n).

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A225159 Denominators of the sequence of fractions f(n) defined recursively by f(1) = 7/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

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A152580 a(n) = 7^(2^n) + 2.

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A225166 Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 7/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

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