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A382121 Minimal polynomials of nimbers *(2^(2^n)-1), evaluated at 2.

7, 25, 425, 101021, 7158330089, 27971386341277386797, 557019405516812760530014815489825522433, 200070165806576462487855236097886014378133571492030310620129377307348366314169

Offset: 1

Simon Tatham, Mar 16 2025

nonn

Each of these polynomials from n=1 up to n=12 is primitive: if you make a finite field of order 2^(2^n) as GF(2)[x]/ then x generates the field's multiplicative group. A natural conjecture is that this is true for all n.

			For n=3, giving 2^n=8 and 2^(2^n)=256: let x be the nimber *255. Then the powers of x (under nim-multiplication) are *1, *255, *156, *61, *205, *200, *38, *71, *179. Under nim-addition, the subset of these powers *1 + *61 + *200 + *71 + *179 sum to *0. That is, 1+x^3+x^5+x^7+x^8 = 0. No sum of the powers up to and including x^7 is zero. So the polynomial 1+x^3+x^5+x^7+x^8 over GF(2) is the minimal polynomial of *255. Therefore the sequence entry for n=3 is the integer obtained by reinterpreting this polynomial as one over the integers and evaluating it at 2, i.e. 1+2^3+2^5+2^7+2^8 = 425.

Simon Tatham, Table of n, a(n) for n = 1..11

Cf. A051775 for definition of nim-multiplication.

A159009 Numerator of the integral of x^n times the Cantor function, from 0 to 1.

Original entry on oeis.org

1, 5, 11, 233, 97, 36377, 10637, 8885119, 18040327, 107868664309, 19821442673, 2657527033463249, 412093696402361, 28353905269136197727, 57058882710461852501, 30872757660805358101602571

Offset: 0

Simon Tatham (anakin(AT)pobox.com), Apr 02 2009

			I(0) is obviously 1/2 by symmetry.

A095844/A095845 give the integrals of powers of the Cantor function itself.

A159010 gives the corresponding denominators. [From Simon Tatham (anakin(AT)pobox.com), Apr 02 2009]

for n from 0 to 20 do CI[n] := 1/(2*(n+1)) + 1/(2*(3^(n+1)-1)) * add(binomial(n,i)*2^(n-i)*CI[i],i=0..n-1); end do;

I(n) = 1/(2*(n+1)) + 1/(2*3^(n+1)-1) * sum_{i=0}{n-1} (n choose i) 2^(n-i) I(i)

A159010 Denominator of the integral of x^n times the Cantor function, from 0 to 1.

Original entry on oeis.org

2, 16, 48, 1280, 640, 279552, 93184, 87326720, 196485120, 1289117040640, 257823408128, 37368456004239360, 6228076000706560, 458287510968422367232, 982044666360905072640, 563650690404417328113516544

Offset: 0

Simon Tatham (anakin(AT)pobox.com), Apr 02 2009

			I(0) is obviously 1/2 by symmetry.

A159009 gives the corresponding numerators. A095844/A095845 give the integrals of powers of the Cantor function itself.

I(n) = 1/(2*(n+1)) + 1/(2*3^(n+1)-1) * sum_{i=0}{n-1} (n choose i) 2^(n-i) I(i)

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User: Simon Tatham

A382121 Minimal polynomials of nimbers *(2^(2^n)-1), evaluated at 2.

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A159009 Numerator of the integral of x^n times the Cantor function, from 0 to 1.

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A159010 Denominator of the integral of x^n times the Cantor function, from 0 to 1.

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