A256471 -id:A256471 - cp's OEIS frontend

A256470 a(n) = A256469(n) - A256468(n).

0, 1, 2, 3, 1, 6, 1, 3, 3, 0, 3, 2, -2, 0, 4, 6, 0, 4, 4, 0, 2, 3, 10, 13, -1, -2, 3, 4, 4, 34, 5, 3, 5, 17, 6, 2, 8, -2, -6, 3, -4, -4, -3, -2, -1, 9, 25, -2, -6, -4, 12, 4, 6, 9, -6, 18, 1, -2, -11, 7, -8, 27, -10, 3, -1, 12, 11, 13, -3, 5, 3, 5, -13, -8, 10, 16, -4, 14, 3, 12, -3, 23, 5, 4, 6, -8, 19, -13, 1, 0

Offset: 1

Antti Karttunen, Mar 30 2015

sign

a(n) = Difference between the number of primes occurring in range [prime(n)*prime(n+1), prime(n+1)^2] and the number of primes occurring in range [prime(n)^2, prime(n)*prime(n+1)].

In other words, a(n) tells how many more primes there are in the latter part of the range prime(n)^2 .. prime(n+1)^2 (after the geometric mean of its limits), than in its first part (before the geometric mean of its limits).

Antti Karttunen, Table of n, a(n) for n = 1..6541
A. Karttunen, Sequences A256470 and A050216 compared with OEIS Plot2-script (See also the ratio-plots linked in A256468.)

Positions of zeros: A256471. Cf. also A256472, A256473.
Positions of nonnegative terms: A256474, negative terms: A256475.
Positions of strictly positive terms: A256476, terms less than or equal to zero: A256477.
Cf. A256449, A256468, A256469.

(define (A256470 n) (- (A256469 n) (A256468 n)))

a(n) = A256469(n) - A256468(n).

a(n) = 3 - A256449(n).

A256472 Primes p for which there are exactly as many primes in the range [p^2, pnextprime(p)] as there are in the range [pnextprime(p), nextprime(p)^2], where nextprime(p) gives the next prime after prime p.

Original entry on oeis.org

2, 29, 43, 59, 71, 463, 601, 881, 1049, 1103, 1171, 1451, 2311, 2591, 2689, 2693, 2777, 3083, 3089, 3917, 8543, 12889, 13007, 13799, 13873, 13877, 15497, 18457, 19477, 20369, 23017, 25073, 26641, 29179, 32801, 33757, 35327, 41647, 43987, 46279, 47041, 49211, 51577, 63113

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

			For p=2, we have in the range [2*2, 2*3] just one prime {5}, and also in the latter range [2*3, 3*3] just one prime {7}, thus 2 is included in the sequence.

Subsequence of A256484.

Cf. A000040, A256471, A256473.

Select[Prime@ Range@ 500, Count[Range[#^2, # NextPrime[#]], ?PrimeQ] == Count[Range[# NextPrime[#], NextPrime[#]^2], ?PrimeQ] &] (* Michael De Vlieger, Mar 30 2015 *)

(define (A256472 n) (A000040 (A256471 n)))

a(n) = A000040(A256471(n)).

A256474 Numbers n for which there are at least as many primes in the range [prime(n)prime(n+1), prime(n+1)^2] as in the range [prime(n)^2, prime(n)prime(n+1)].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 46, 47, 51, 52, 53, 54, 56, 57, 60, 62, 64, 66, 67, 68, 70, 71, 72, 75, 76, 78, 79, 80, 82, 83, 84, 85, 87, 89, 90, 91, 93, 94, 95, 97, 99, 100

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

Positions of nonnegative terms in A256470.

Antti Karttunen, Table of n, a(n) for n = 1..3457

Complement: A256475.
Union of A256471 and A256476.
Cf. A256484 (corresponding primes).
Cf. A256470.

Select[Range@100, Count[Range[Prime[#] Prime[# + 1], Prime[# + 1]^2], ?PrimeQ] >= Count[Range[Prime[#]^2, Prime[#] Prime[# + 1]], ?PrimeQ] &] (* Michael De Vlieger, Mar 30 2015 *)

A256475 Numbers n for which there are more primes in range [prime(n)^2, prime(n)prime(n+1)] than in range [prime(n)prime(n+1), prime(n+1)^2].

Original entry on oeis.org

13, 25, 26, 38, 39, 41, 42, 43, 44, 45, 48, 49, 50, 55, 58, 59, 61, 63, 65, 69, 73, 74, 77, 81, 86, 88, 92, 96, 98, 101, 103, 106, 107, 108, 109, 116, 117, 120, 121, 122, 124, 125, 128, 141, 142, 143, 145, 146, 148, 149, 151, 155, 158, 159, 166, 169, 172, 173, 177, 179, 181, 182, 183, 190, 191, 194, 195, 196, 197, 206

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

Positions of negative terms in A256470.

Equally: Numbers n for which there are less primes in range [prime(n)*prime(n+1), prime(n+1)^2] than in range [prime(n)^2, prime(n)*prime(n+1)].

Antti Karttunen, Table of n, a(n) for n = 1..3084

Complement: A256474.
Setwise difference of A256477 and A256471.
Cf. A256485 (corresponding primes).
Cf. A256470.

Select[Range@210, Count[Range[Prime[#]^2, Prime[#] Prime[# + 1]], ?PrimeQ] > Count[Range[Prime[#] Prime[# + 1], Prime[# + 1]^2], ?PrimeQ] &] (* Michael De Vlieger, Mar 30 2015 *)

A256473 Odd primes p for which there are exactly as many primes in the range [prevprime(p)^2, prevprime(p)p] as there are in the range [prevprime(p)p, p^2], where prevprime(p) gives the previous prime before prime p.

Original entry on oeis.org

3, 31, 47, 61, 73, 467, 607, 883, 1051, 1109, 1181, 1453, 2333, 2593, 2693, 2699, 2789, 3089, 3109, 3919, 8563, 12893, 13009, 13807, 13877, 13879, 15511, 18461, 19483, 20389, 23021, 25087, 26647, 29191, 32803, 33767, 35339, 41651, 43991, 46301, 47051, 49223, 51581, 63127

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

			For p=3, we have in the range [2*2, 2*3] just one prime {5}, and also in the latter range [2*3, 3*3] just one prime {7}, thus 3 is included in the sequence.

Cf. A000040, A065091, A256471, A256472.

Select[Prime@ Range[2, 500], Count[Range[NextPrime[#, -1]^2, # NextPrime[#, -1]], ?PrimeQ] == Count[Range[# NextPrime[#, -1], #^2], ?PrimeQ] &] (* Michael De Vlieger, Mar 30 2015 *)

(define (A256473 n) (A000040 (+ 1 (A256471 n))))

a(n) = A065091(A256471(n)) = A000040(1+A256471(n)).

A256476 Numbers n for which there are more primes in range [prime(n)prime(n+1), prime(n+1)^2] than in range [prime(n)^2, prime(n)prime(n+1)].

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 16, 18, 19, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 46, 47, 51, 52, 53, 54, 56, 57, 60, 62, 64, 66, 67, 68, 70, 71, 72, 75, 76, 78, 79, 80, 82, 83, 84, 85, 87, 89, 91, 93, 94, 95, 97, 99, 100, 102, 104, 105, 111, 112, 113, 114, 115, 118, 119, 123

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

Positions where A256470 is strictly positive.

Antti Karttunen, Table of n, a(n) for n = 1..3413

Complement: A256477.
Setwise difference of A256474 and A256471.
Cf. A256470.

Select[Range@125, Count[Range[Prime[#] Prime[# + 1], Prime[# + 1]^2], ?PrimeQ] > Count[Range[Prime[#]^2, Prime[#] Prime[# + 1]], ?PrimeQ] &] (* Michael De Vlieger, Mar 30 2015 *)

A256477 Numbers n for which the number of primes in the range [prime(n)prime(n+1), prime(n+1)^2] is less than or equal to the number of primes in the range [prime(n)^2, prime(n)prime(n+1)].

Original entry on oeis.org

1, 10, 13, 14, 17, 20, 25, 26, 38, 39, 41, 42, 43, 44, 45, 48, 49, 50, 55, 58, 59, 61, 63, 65, 69, 73, 74, 77, 81, 86, 88, 90, 92, 96, 98, 101, 103, 106, 107, 108, 109, 110, 116, 117, 120, 121, 122, 124, 125, 128, 141, 142, 143, 145, 146, 148, 149, 151, 152, 155, 158, 159, 166, 169

Offset: 1

Antti Karttunen, Mar 30 2015

nonn

Positions where A256470 is zero or negative.

Antti Karttunen, Table of n, a(n) for n = 1..3128

Complement: A256476.
Union of A256471 and A256475.
Cf. A256470.

Select[Range@ 170, Count[Range[Prime[#] Prime[# + 1], Prime[# + 1]^2], ?PrimeQ] <= Count[Range[Prime[#]^2, Prime[#] Prime[# + 1]], ?PrimeQ] &] (* Michael De Vlieger, Mar 30 2015 *)

cp's OEIS Frontend

A256470 a(n) = A256469(n) - A256468(n).

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A256472 Primes p for which there are exactly as many primes in the range [p^2, p*nextprime(p)] as there are in the range [p*nextprime(p), nextprime(p)^2], where nextprime(p) gives the next prime after prime p.

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A256474 Numbers n for which there are at least as many primes in the range [prime(n)*prime(n+1), prime(n+1)^2] as in the range [prime(n)^2, prime(n)*prime(n+1)].

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A256475 Numbers n for which there are more primes in range [prime(n)^2, prime(n)*prime(n+1)] than in range [prime(n)*prime(n+1), prime(n+1)^2].

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A256473 Odd primes p for which there are exactly as many primes in the range [prevprime(p)^2, prevprime(p)*p] as there are in the range [prevprime(p)*p, p^2], where prevprime(p) gives the previous prime before prime p.

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A256476 Numbers n for which there are more primes in range [prime(n)*prime(n+1), prime(n+1)^2] than in range [prime(n)^2, prime(n)*prime(n+1)].

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A256477 Numbers n for which the number of primes in the range [prime(n)*prime(n+1), prime(n+1)^2] is less than or equal to the number of primes in the range [prime(n)^2, prime(n)*prime(n+1)].

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A256472 Primes p for which there are exactly as many primes in the range [p^2, pnextprime(p)] as there are in the range [pnextprime(p), nextprime(p)^2], where nextprime(p) gives the next prime after prime p.

A256474 Numbers n for which there are at least as many primes in the range [prime(n)prime(n+1), prime(n+1)^2] as in the range [prime(n)^2, prime(n)prime(n+1)].

A256475 Numbers n for which there are more primes in range [prime(n)^2, prime(n)prime(n+1)] than in range [prime(n)prime(n+1), prime(n+1)^2].

A256473 Odd primes p for which there are exactly as many primes in the range [prevprime(p)^2, prevprime(p)p] as there are in the range [prevprime(p)p, p^2], where prevprime(p) gives the previous prime before prime p.

A256476 Numbers n for which there are more primes in range [prime(n)prime(n+1), prime(n+1)^2] than in range [prime(n)^2, prime(n)prime(n+1)].

A256477 Numbers n for which the number of primes in the range [prime(n)prime(n+1), prime(n+1)^2] is less than or equal to the number of primes in the range [prime(n)^2, prime(n)prime(n+1)].