A046868 -id:A046868 - cp's OEIS frontend

A327434 Prime(n)^2 - prime(n-1)*prime(n+1) for n in A046868.

4, 30, 42, 128, 98, 90, 36, 248, 158, 150, 182, 420, 210, 222, 1326, 560, 1212, 36, 350, 36, 728, 1548, 402, 144, 1832, 462, 968, 1064, 36, 36, 1088, 578, 570, 3126, 630, 2732, 2796, 782, 36, 782, 1620, 3372, 3468, 902, 1860, 930, 2012, 1980, 2028, 5234, 6600

Offset: 1

Chai Wah Wu, Sep 10 2019

nonn

Positive terms in A056221.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

CF. A046868, A056221.

A046869 Good primes (version 1): prime(n)^2 > prime(n-1)*prime(n+1).

Original entry on oeis.org

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 157, 163, 173, 179, 191, 197, 211, 223, 227, 239, 251, 257, 263, 269, 277, 281, 307, 311, 331, 347, 367, 373, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499, 521, 541

Offset: 1

N. J. A. Sloane

nonn

Also called geometrically strong primes. - Amarnath Murthy, Mar 08 2002
The idea can be extended by defining a geometrically strong prime of order k to be a prime that is greater than the geometric mean of r neighbors on both sides for all r = 1 to k but not for r = k+1. Similar generalizations can be applied to the sequence A051634. - Amarnath Murthy, Mar 08 2002
It appears that a(n) ~ 2*prime(n). - Thomas Ordowski, Jul 25 2012
Conjecture: primes p(n) such that 2*p(n) >= p(n-1) + p(n+1). - Thomas Ordowski, Jul 25 2012
Probably {3,7,23} U {good primes} = {primes p(n) > 2/(1/p(n-1) + 1/p(n+1))}. - Thomas Ordowski, Jul 27 2012
Except for A001359(1), A001359 is a subsequence. - Chai Wah Wu, Sep 10 2019

			37 is a member as 37^2 = 1369 > 31*41 = 1271.

R. K. Guy, Unsolved Problems in Number Theory, Section A14.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A001359, A006562, A028388, A046868, A051634, A051635, A068828.

[NthPrime(n): n in [2..100] | NthPrime(n)^2 gt NthPrime(n-1)*NthPrime(n+1)]; // Bruno Berselli, Oct 23 2012

with(numtheory): a := [ ]: P := [ ]: M := 300: for i from 2 to M do if p(i)^2>p(i-1)*p(i+1) then a := [ op(a),i ]; P := [ op(P),p(i) ]; fi; od: a; P;

Do[ If[ Prime[n]^2 > Prime[n - 1]*Prime[n + 1], Print[ Prime[n] ] ], {n, 2, 100} ]
Transpose[Select[Partition[Prime[Range[300]],3,1],#[[2]]^2>#[[1]]#[[3]]&]][[2]] (* Harvey P. Dale, May 13 2012 *)
Select[Prime[Range[2, 100]], #^2 > NextPrime[#]*NextPrime[#, -1] &] (* Jayanta Basu, Jun 29 2013 *)

forprime(n=o=p=3,999,o+0<(o=p)^2/(p=n) & print1(o", "))
isA046869(p)={ isprime(p) & p^2>precprime(p-1)*nextprime(p+1) } \\ M. F. Hasler, Jun 15 2011

Corrected and extended by Robert G. Wilson v, Dec 06 2000

Edited by N. J. A. Sloane at the suggestion of Giovanni Resta, Aug 20 2007

A056221 Image of primes (A000040) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.

Original entry on oeis.org

-1, 4, -6, 30, -18, 42, -30, -22, 128, -112, 98, 90, -78, -70, 36, 248, -232, 158, 150, -280, 182, -142, -130, 420, 210, -198, 222, -210, -1074, 1326, -238, 560, -1092, 1212, -592, 36, 350, -310, 36, 728, -1428, 1548, -378, 402, -1966, 144, 1832, 462, -450, -442

Offset: 1

N. J. A. Sloane, Aug 06 2000

sign

a(n) > 0 if and only if n+1 is in A046868. a(n) < 0 if and only if n+1 is in A233671. - Chai Wah Wu, Sep 10 2019

Stefano Spezia, Table of n, a(n) for n = 1..10000
L. Panaitopol, On the sequence p(n)^2=p(n-1)*p(n+1), J. Inequal. Pure Appl. Math., Volume 3, Issue 4, Article 53, 2002.

Cf. A000040, A046868, A056222, A233671, A342567.

A056221 := proc(n)
        ithprime(n+1)^2-ithprime(n)*ithprime(n+2) ;
end proc:
seq(A056221(n),n=1..10) ; # R. J. Mathar, Dec 10 2011

a[n_]:=Prime[n+1]^2-Prime[n]Prime[n+2]; Array[a,50] (* Stefano Spezia, Jul 15 2024 *)

a(n) = determinant of matrix
| prime(n+1) prime(n)|
| prime(n+2) prime(n+1)|. - Zak Seidov, Jul 23 2008, indices corrected by Gary Detlefs, Dec 09 2011
a(n) = 2*A342567(n+1) for n >= 2. - Hugo Pfoertner, Jun 20 2021

A233671 Numbers k such that prime(k)^2 < prime(k-1)*prime(k+1).

Original entry on oeis.org

2, 4, 6, 8, 9, 11, 14, 15, 18, 21, 23, 24, 27, 29, 30, 32, 34, 36, 39, 42, 44, 46, 50, 51, 53, 58, 61, 62, 65, 66, 68, 70, 71, 72, 76, 77, 79, 80, 82, 84, 86, 87, 90, 91, 94, 96, 97, 99, 101, 105, 106, 110, 114, 117, 118, 121, 123, 124, 125, 127, 132, 135

Offset: 1

Clark Kimberling, Dec 14 2013

If 1 is appended to A046868, the resulting sequence is the complement of A233671. Does A233671 have asymptotic density 1/2? Does every positive integer occur infinitely many times in the difference sequence of A233671?

			a(1) = 2 because 3^2 < 2*5.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A046868, A000040.

Select[Range[2, 200], Prime[#]^2 < Prime[# - 1]*Prime[# + 1] &]
PrimePi[#]&/@Select[Partition[Prime[Range[200]],3,1],#[[2]]^2<(#[[1]] #[[3]])&][[All,2]] (* Harvey P. Dale, Dec 09 2021 *)

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A327434 Prime(n)^2 - prime(n-1)*prime(n+1) for n in A046868.

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A046869 Good primes (version 1): prime(n)^2 > prime(n-1)*prime(n+1).

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A056221 Image of primes (A000040) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.

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A233671 Numbers k such that prime(k)^2 < prime(k-1)*prime(k+1).

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