A057820 First differences of sequence of consecutive prime powers (A000961).

1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 3, 4, 2, 6, 2, 2, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 6, 6, 4, 2, 4, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 2, 8, 5, 1, 6, 6, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 6, 6, 4, 2, 4, 6, 2, 6, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10

Offset: 1

Author

Labos Elemer, Nov 08 2000

Keywords

• nonn

Comments

a(n) = 1 iff A000961(n) = A006549(k) for some k. - Reinhard Zumkeller, Aug 25 2002

Also run lengths of distinct terms in A070198. - Reinhard Zumkeller, Mar 01 2012

Does this sequence contain all positive integers? - Gus Wiseman, Oct 09 2024

Examples

			Odd differences arise in pairs in neighborhoods of powers of 2, like {..,2039,2048,2053,..} gives {..,11,5,..}
		

Links

• Michael B. Porter, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000961, A036616, A001223.

For perfect-powers (A001597) we have A053289.

For non-perfect-powers (A007916) we have A375706.

Positions of ones are A375734.

Run-compression is A376308.

Run-lengths are A376309.

Sorted positions of first appearances are A376340.

The second (instead of first) differences are A376596, zeros A376597.

Prime-powers:

- terms: A000961 or A246655, complement A024619

- differences: A057820 (this), first appearances A376341

- runs: A373675, A373673, A373674, A174965

- anti-runs: A373576, A120430, A006549, A373671

Non-prime-powers:

- terms: A361102

- differences: A375708 (ones A375713)

- runs: A373678, A373676, A373677, A110969 (A373669, sorted A373670)

- anti-runs: A373679, A373575, A255346, A373672

Cf. A000015, A014210, A014963, A025475, A025528, A027833, A037201, A046933, A076259, A078147, A093555, A345531, A376310.

Programs

Haskell

a057820_list = zipWith (-) (tail a000961_list) a000961_list
-- Reinhard Zumkeller, Mar 01 2012

Maple

A057820 := proc(n)
        A000961(n+1)-A000961(n) ;
end proc: # R. J. Mathar, Sep 23 2016

Mathematica

Map[Length, Split[Table[Apply[LCM, Range[n]], {n, 1, 150}]]] (* Geoffrey Critzer, May 29 2015 *)
Join[{1},Differences[Select[Range[500],PrimePowerQ]]] (* Harvey P. Dale, Apr 21 2022 *)

PARI

isA000961(n) = (omega(n) == 1 || n == 1)
n_prev=1;for(n=2,500,if(isA000961(n),print(n-n_prev);n_prev=n)) \\ Michael B. Porter, Oct 30 2009

Python

from sympy import primepi, integer_nthroot
def A057820(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    r, k = m, f(m)+1
    while r != k: r, k = k, f(k)+1
    return r-m # Chai Wah Wu, Sep 12 2024

Formula

a(n) = A000961(n+1) - A000961(n).

Extensions

Offset corrected and b-file adjusted by Reinhard Zumkeller, Mar 03 2012