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Revision History for A204120

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A204120

Symmetric matrix based on f(i,j) = gcd(prime(i+1),prime(j+1)), by antidiagonals.
(history; published version)

#11 by Susanna Cuyler at Tue Sep 25 20:50:44 EDT 2018

STATUS

proposed

approved

#10 by Antti Karttunen at Tue Sep 25 15:21:50 EDT 2018

STATUS

editing

proposed

#9 by Antti Karttunen at Tue Sep 25 09:13:22 EDT 2018

COMMENTS

Square array with odd primes (A065091) on main diagonal, and 1 at all other entries, ; array A204118 without its top row and the leftmost column. - Antti Karttunen, Sep 25 2018

#8 by Antti Karttunen at Tue Sep 25 09:12:45 EDT 2018

COMMENTS

Square array with odd primes (A065091) on main diagonal, and 1 at all other entries, array A204118 without its top row and the leftmost column. - Antti Karttunen, Sep 25 2018

CROSSREFS

Cf. A065091 (main diagonal), A204118, A204121, A204016, A202453.

#7 by Antti Karttunen at Tue Sep 25 09:09:55 EDT 2018

LINKS

Antti Karttunen, <a href="/A204120/b204120.txt">Table of n, a(n) for n = 1..65703 (the first 362 antidiagonals of array)</a>

#6 by Antti Karttunen at Tue Sep 25 09:08:12 EDT 2018

NAME

Symmetric matrix based on f(i,j) =GCD gcd(prime(i+1),prime(j+1)), by antidiagonals.

COMMENTS

Square array with odd primes (A065091) on main diagonal, and 1 at all other entries. - Antti Karttunen, Sep 25 2018

LINKS

Antti Karttunen, <a href="/A204120/b204120.txt">Table of n, a(n) for n = 1..65703</a>

PROG

(PARI)

up_to = 65703; \\ = binomial(362+1, 2)

A204120sq(row, col) = gcd(prime(1+row), prime(1+col));

A204120list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, if(i++ > up_to, return(v)); v[i] = A204120sq((a-(col-1)), col))); (v); };

v204120 = A204120list(up_to);

A204120(n) = v204120[n]; \\ Antti Karttunen, Sep 25 2018

CROSSREFS

Cf. A065091 (main diagonal), A204121, A204016, A202453.

STATUS

approved

editing

#5 by Russ Cox at Fri Mar 30 18:58:07 EDT 2012

AUTHOR

_Clark Kimberling (ck6(AT)evansville.edu), _, Jan 11 2012

Discussion

Fri Mar 30

18:58

OEIS Server: https://oeis.org/edit/global/285

#4 by T. D. Noe at Wed Jan 11 17:29:42 EST 2012

STATUS

proposed

approved

#3 by Clark Kimberling at Wed Jan 11 17:23:42 EST 2012

STATUS

editing

proposed

#2 by Clark Kimberling at Wed Jan 11 11:22:10 EST 2012

NAME

allocated for Clark KimberlingSymmetric matrix based on f(i,j)=GCD(prime(i+1),prime(j+1)), by antidiagonals.

DATA

3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

OFFSET

1,1

COMMENTS

A204120 represents the matrix M given by f(i,j)=GCD(prime(i+1),prime(j+1)) for i>=1 and j>=1. See A204121 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

EXAMPLE

Northwest corner:

3 1 1 1

1 5 1 1

1 1 7 1

1 1 1 11

MATHEMATICA

f[i_, j_] := GCD[Prime[i + 1], Prime[j + 1]];

m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

TableForm[m[8]] (* 8x8 principal submatrix *)

Flatten[Table[f[i, n + 1 - i],

{n, 1, 15}, {i, 1, n}]] (* A204120 *)

p[n_] := CharacteristicPolynomial[m[n], x];

c[n_] := CoefficientList[p[n], x]

TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

Table[c[n], {n, 1, 12}]

Flatten[%] (* A204121 *)

TableForm[Table[c[n], {n, 1, 10}]]

CROSSREFS

Cf. A204121, A204016, A202453.

KEYWORD

allocated

nonn,tabl

AUTHOR

Clark Kimberling (ck6(AT)evansville.edu), Jan 11 2012

STATUS

approved

editing