A308658 - OEIS

A308658

Square array read by downward antidiagonals: A(n, k) is the number of primes between the n-th and (n+k)-th perfect powers with exponent > 1, k > 0.

%I #67 Nov 06 2024 04:35:00

%S 2,4,2,4,2,0,6,4,2,2,9,7,5,5,3,9,7,5,5,3,0,11,9,7,7,5,2,2,11,9,7,7,5,

%T 2,2,0,15,13,11,11,9,6,6,4,4,18,16,14,14,12,9,9,7,7,3,22,20,18,18,16,

%U 13,13,11,11,7,4,25,23,21,21,19,16,16,14,14,10,7

%N Square array read by downward antidiagonals: A(n, k) is the number of primes between the n-th and (n+k)-th perfect powers with exponent > 1, k > 0.

%C The Redmond-Sun conjecture implies that A(n, 1) is 0 for only finitely many values of n and A(n, k) > 0 for all n and k when k > 1.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Redmond%E2%80%93Sun_conjecture">Redmond-Sun conjecture</a>

%F A(n, k) = A000720(A001597(n+k)) - A000720(A001597(n)), k > 0.

%F A(A274605(n), 1) = 0.

%F A(n,k) = Sum_{j=n..n+k-1} A(j,1) = A(n,k-1) + A(n+k-1,1) for k > 1. - _Pontus von Brömssen_, Nov 05 2024

%e The array starts as follows:

%e k = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

%e --------------------------------------------------------------------------

%e n= 1| 2, 4, 4, 6, 9, 9, 11, 11, 15, 18, 22, 25, 30, 30, 31, 34, 39

%e n= 2| 2, 2, 4, 7, 7, 9, 9, 13, 16, 20, 23, 28, 28, 29, 32, 37, 42

%e n= 3| 0, 2, 5, 5, 7, 7, 11, 14, 18, 21, 26, 26, 27, 30, 35, 40, 43

%e n= 4| 2, 5, 5, 7, 7, 11, 14, 18, 21, 26, 26, 27, 30, 35, 40, 43, 44

%e n= 5| 3, 3, 5, 5, 9, 12, 16, 19, 24, 24, 25, 28, 33, 38, 41, 42, 47

%e n= 6| 0, 2, 2, 6, 9, 13, 16, 21, 21, 22, 25, 30, 35, 38, 39, 44, 45

%e n= 7| 2, 2, 6, 9, 13, 16, 21, 21, 22, 25, 30, 35, 38, 39, 44, 45, 52

%e n= 8| 0, 4, 7, 11, 14, 19, 19, 20, 23, 28, 33, 36, 37, 42, 43, 50, 55

%e n= 9| 4, 7, 11, 14, 19, 19, 20, 23, 28, 33, 36, 37, 42, 43, 50, 55, 57

%e n=10| 3, 7, 10, 15, 15, 16, 19, 24, 29, 32, 33, 38, 39, 46, 51, 53, 57

%e n=11| 4, 7, 12, 12, 13, 16, 21, 26, 29, 30, 35, 36, 43, 48, 50, 54, 60

%e n=12| 3, 8, 8, 9, 12, 17, 22, 25, 26, 31, 32, 39, 44, 46, 50, 56, 63

%e n=13| 5, 5, 6, 9, 14, 19, 22, 23, 28, 29, 36, 41, 43, 47, 53, 60, 67

%e n=14| 0, 1, 4, 9, 14, 17, 18, 23, 24, 31, 36, 38, 42, 48, 55, 62, 67

%e n=15| 1, 4, 9, 14, 17, 18, 23, 24, 31, 36, 38, 42, 48, 55, 62, 67, 69

%e n=16| 3, 8, 13, 16, 17, 22, 23, 30, 35, 37, 41, 47, 54, 61, 66, 68, 74

%e n=17| 5, 10, 13, 14, 19, 20, 27, 32, 34, 38, 44, 51, 58, 63, 65, 71, 80

%e n=18| 5, 8, 9, 14, 15, 22, 27, 29, 33, 39, 46, 53, 58, 60, 66, 75, 83

%e n=19| 3, 4, 9, 10, 17, 22, 24, 28, 34, 41, 48, 53, 55, 61, 70, 78, 85

%e n=20| 1, 6, 7, 14, 19, 21, 25, 31, 38, 45, 50, 52, 58, 67, 75, 82, 90

%e .

%e For instance let n = k = 6, then

%e A(n, k) = A000720(A001597(n+k)) - A000720(A001597(n))

%e = A000720(A001597(12)) - A000720(A001597(6))

%e = A000720(81) - A000720(25) = 22 - 9 = 13.

%o (PARI) power(n) = if(n==1, return(1)); my(i=1); for(k=2, oo, if(ispower(k), i++); if(i==n, return(k)))

%o array(n, k) = for(x=1, n, for(y=x+1, x+k, print1(primepi(power(y))-primepi(power(x)), ", ")); print(""))

%o array(10, 20) \\ Print initial 10 rows and 20 columns of array

%o (SageMath)

%o perfpower = [0]+[k for k in srange(1, 300) if k.is_perfect_power()]

%o primepi = [0]+[prime_pi(k) for k in srange(1, 300)]

%o def A308658(n, k): return primepi[perfpower[n+k]] - primepi[perfpower[n]]

%o for n in (1..10): print([A308658(n, k) for k in (1..10)]) # _Peter Luschny_, Nov 18 2019

%Y Cf. A000720, A001597, A080769 (column 1), A274605.

%K nonn,tabl

%O 1,1

%A _Felix Fröhlich_, Nov 16 2019