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.

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

Offset: 1

Author

Felix Fröhlich, Nov 16 2019

Keywords

• nonn
• tabl

Comments

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.

Examples

			The array starts as follows:
  k = 1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17
     --------------------------------------------------------------------------
n= 1| 2,  4,  4,  6,  9,  9, 11, 11, 15,  18,  22,  25,  30,  30,  31,  34,  39
n= 2| 2,  2,  4,  7,  7,  9,  9, 13, 16,  20,  23,  28,  28,  29,  32,  37,  42
n= 3| 0,  2,  5,  5,  7,  7, 11, 14, 18,  21,  26,  26,  27,  30,  35,  40,  43
n= 4| 2,  5,  5,  7,  7, 11, 14, 18, 21,  26,  26,  27,  30,  35,  40,  43,  44
n= 5| 3,  3,  5,  5,  9, 12, 16, 19, 24,  24,  25,  28,  33,  38,  41,  42,  47
n= 6| 0,  2,  2,  6,  9, 13, 16, 21, 21,  22,  25,  30,  35,  38,  39,  44,  45
n= 7| 2,  2,  6,  9, 13, 16, 21, 21, 22,  25,  30,  35,  38,  39,  44,  45,  52
n= 8| 0,  4,  7, 11, 14, 19, 19, 20, 23,  28,  33,  36,  37,  42,  43,  50,  55
n= 9| 4,  7, 11, 14, 19, 19, 20, 23, 28,  33,  36,  37,  42,  43,  50,  55,  57
n=10| 3,  7, 10, 15, 15, 16, 19, 24, 29,  32,  33,  38,  39,  46,  51,  53,  57
n=11| 4,  7, 12, 12, 13, 16, 21, 26, 29,  30,  35,  36,  43,  48,  50,  54,  60
n=12| 3,  8,  8,  9, 12, 17, 22, 25, 26,  31,  32,  39,  44,  46,  50,  56,  63
n=13| 5,  5,  6,  9, 14, 19, 22, 23, 28,  29,  36,  41,  43,  47,  53,  60,  67
n=14| 0,  1,  4,  9, 14, 17, 18, 23, 24,  31,  36,  38,  42,  48,  55,  62,  67
n=15| 1,  4,  9, 14, 17, 18, 23, 24, 31,  36,  38,  42,  48,  55,  62,  67,  69
n=16| 3,  8, 13, 16, 17, 22, 23, 30, 35,  37,  41,  47,  54,  61,  66,  68,  74
n=17| 5, 10, 13, 14, 19, 20, 27, 32, 34,  38,  44,  51,  58,  63,  65,  71,  80
n=18| 5,  8,  9, 14, 15, 22, 27, 29, 33,  39,  46,  53,  58,  60,  66,  75,  83
n=19| 3,  4,  9, 10, 17, 22, 24, 28, 34,  41,  48,  53,  55,  61,  70,  78,  85
n=20| 1,  6,  7, 14, 19, 21, 25, 31, 38,  45,  50,  52,  58,  67,  75,  82,  90
.
For instance let n = k = 6, then
A(n, k) = A000720(A001597(n+k)) - A000720(A001597(n))
= A000720(A001597(12)) - A000720(A001597(6))
= A000720(81) - A000720(25) = 22 - 9 = 13.
		

Links

• Wikipedia, Redmond-Sun conjecture

Crossrefs

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

Programs

PARI

power(n) = if(n==1, return(1)); my(i=1); for(k=2, oo, if(ispower(k), i++); if(i==n, return(k)))
array(n, k) = for(x=1, n, for(y=x+1, x+k, print1(primepi(power(y))-primepi(power(x)), ", ")); print(""))
array(10, 20) \\ Print initial 10 rows and 20 columns of array

SageMath

perfpower = [0]+[k for k in srange(1, 300) if k.is_perfect_power()]
primepi   = [0]+[prime_pi(k) for k in srange(1, 300)]
def A308658(n, k): return primepi[perfpower[n+k]] - primepi[perfpower[n]]
for n in (1..10): print([A308658(n, k) for k in (1..10)]) # Peter Luschny, Nov 18 2019

Formula

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

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

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