A076757 -id:A076757 - cp's OEIS frontend

A202605 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Fibonacci self-fusion matrix (A202453).

1, -1, 1, -3, 1, 1, -6, 9, -1, 1, -9, 26, -24, 1, 1, -12, 52, -96, 64, -1, 1, -15, 87, -243, 326, -168, 1, 1, -18, 131, -492, 1003, -1050, 441, -1, 1, -21, 184, -870, 2392, -3816, 3265, -1155, 1, 1, -24, 246, -1404, 4871, -10500, 13710

Offset: 1

Clark Kimberling, Dec 21 2011

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive and interlace the zeros of p(n+1). (See the references and examples.)
Following is a guide to sequences (f(n)) for symmetric matrices (self-fusion matrices) and characteristic polynomials. Notation: F(k)=A000045(k) (Fibonacci numbers); floor(n*tau)=A000201(n) (lower Wythoff sequence); "periodic x,y" represents the sequence (x,y,x,y,x,y,...).
f(n)........ symmetric matrix.. char. polynomial
1............... A087062....... A202672
n............... A115262....... A202673
n^2............. A202670....... A202671
2n-1............ A202674....... A202675
3n-2............ A202676....... A202677
n(n+1)/2........ A185957....... A202678
2^n-1........... A202873....... A202767
2^(n-1)......... A115216....... A202868
floor(n*tau).... A202869....... A202870
F(n)............ A202453....... A202605
F(n+1).......... A202874....... A202875
Lucas(n)........ A202871....... A202872
F(n+2)-1........ A202876....... A202877
F(n+3)-2........ A202970....... A202971
(F(n))^2........ A203001....... A203002
(F(n+1))^2...... A203003....... A203004
C(2n,n)......... A115255....... A203005
(-1)^(n+1)...... A003983....... A076757
periodic 1,0.... A203905....... A203906
periodic 1,0,0.. A203945....... A203946
periodic 1,0,1.. A203947....... A203948
periodic 1,1,0.. A203949....... A203950
periodic 1,0,0,0 A203951....... A203952
periodic 1,2.... A203953....... A203954
periodic 1,2,3.. A203955....... A203956
...
In the cases listed above, the zeros of the characteristic polynomials are positive. If more general symmetric matrices are used, the zeros are all real but not necessarily positive - but they do have the interlace property. For a guide to such matrices and polynomials, see A202605.

			The 1st principal submatrix (ps) of A202453 is {{1}} (using Mathematica matrix notation), with p(1) = 1-x and zero-set {1}.
...
The 2nd ps is {{1,1},{1,2}}, with p(2) = 1-3x+x^2 and zero-set {0.382..., 2.618...}.
...
The 3rd ps is {{1,1,2},{1,2,3},{2,3,6}}, with p(3) = 1-6x+9x^2-x^3 and zero-set {0.283..., 0.426..., 8.290...}.
  ...
Top of the array A202605:
  1,   -1;
  1,   -3,    1;
  1,   -6,    9,   -1;
  1,   -9,   26,  -24,    1;
  1,  -12,   52,  -96,   64,   -1;
  1,  -15,   87, -243,  326, -168,    1;

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.
A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Cf. A000045, A202453.

f[k_] := Fibonacci[k];
U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]

A077510 Numbers k such that k + pi(k) is a prime.

Original entry on oeis.org

2, 3, 7, 9, 12, 13, 21, 28, 32, 36, 45, 52, 55, 57, 61, 65, 70, 76, 79, 81, 84, 86, 89, 101, 104, 110, 119, 121, 131, 135, 139, 145, 147, 155, 160, 162, 172, 181, 185, 187, 195, 205, 216, 222, 223, 228, 231, 253, 258, 262, 273, 278, 286, 288, 292, 297, 305, 310

Offset: 1

Amarnath Murthy, Nov 08 2002

nonn

Conjecture: for k > 5, prime(n) <= k < prime(n+1) <= k + pi(k), i.e., the smallest prime greater than k is <= k + pi(k). Equality holds for k = 7.

			21 is a member as 21 + pi(21) = 21 + 8 = 29 is a prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A000720, A061067, A076757.

Select[Range[350],PrimeQ[#+PrimePi[#]]&] (* Harvey P. Dale, Nov 19 2011 *)

for(n=1,200,if(isprime(n+primepi(n)),print1(n,", "))) \\ Derek Orr, Jun 22 2015

pi=0; p=2; forprime(q=3,1e3, pi++; for(n=p,q-1, if(isprime(n+pi), print1(n", "))); p=q) \\ Charles R Greathouse IV, Jun 23 2015

More terms from David Garber, Nov 10 2002

A186098 Primes removed by sieve generating A095117.

Original entry on oeis.org

2, 7, 23, 31, 41, 53, 61, 137, 157, 193, 233, 241, 257, 283, 293, 311, 397, 439, 479, 499, 523, 557, 593, 647, 883, 1061, 1129, 1213, 1303, 1381, 1429, 1439, 1543, 1601, 1847, 1867, 1877, 1931, 2011, 2063, 2129, 2293, 2333, 2347, 2393, 2477, 2551, 2633, 2677, 2687

Offset: 1

Carmine Suriano, Mar 29 2011

nonn

Primes not of the form k + primepi(k). [corrected by Michel Marcus, Oct 27 2021]

Bill McEachen, Table of n, a(n) for n = 1..10000

Complement of A076757 in the primes.

Cf. A095117.

Complement[Prime@ Range@ PrimePi[#], Array[PrimePi[#] + # &, #]] &[2700] (* Michael De Vlieger, Oct 27 2021 *)

genit(maxx=50)={arr=List(); for(n=0,maxx,q=n+prime(n+1); if(ispseudoprime(q),listput(arr,q))); arr} \\ Bill McEachen, Oct 27 2021

A228865 Primes of the form n^2 + pi(n).

Original entry on oeis.org

5, 11, 53, 149, 331, 449, 1307, 1381, 1777, 2039, 2719, 3041, 3617, 3739, 4243, 4643, 4919, 6263, 7079, 9049, 9629, 12799, 16931, 18257, 20483, 21059, 24061, 32441, 32803, 34267, 38069, 42071, 46703, 49331, 53411, 57173, 64063, 68699, 74587, 75683, 105691, 110291

Offset: 1

K. D. Bajpai, Sep 05 2013

nonn

Conjecture: the sequence is infinite.

			a(6)= 449 which is a prime. 21^2+pi(21) = 441 + 8 = 449.

K. D. Bajpai, Table of n, a(n) for n = 1..2000

Cf. A076757

with(numtheory): KD:= proc() local a,b;  a:= n^2+pi(n); if isprime(a) then RETURN(a): fi; end: seq(KD(),n=1..1000);

cp's OEIS Frontend

A202605 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Fibonacci self-fusion matrix (A202453).

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A077510 Numbers k such that k + pi(k) is a prime.

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PARI

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A186098 Primes removed by sieve generating A095117.

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Mathematica

PARI

A228865 Primes of the form n^2 + pi(n).

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