A062977 Difference between largest and smallest positive exponent in prime factorization of n; a(1) = 0 by convention.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 1, 0, 0, 0, 0, 2, 0

Offset: 1

Author

Henry Bottomley, Jul 24 2001

Keywords

• nonn
• easy

Examples

			a(24) = 2 since 24 = 2^3*3^1 and max(3,1) - min(3,1) = 3 - 1 = 2;
a(25) = 0 since 25 = 5^2 and max(2) - min(2) = 2 - 2 = 0.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 4000 terms from Harry J. Smith)
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A072774 (positions of zeros), A059404 (of nonzeros).

Cf. A001222, A033150, A051903, A051904, A071178, A108951, A325226.

Programs

Mathematica

dlsp[n_]:=Module[{xp=FactorInteger[n][[All,2]]},Max[xp]-Min[xp]]; Join[ {0},Array[ dlsp,120]] (* Harvey P. Dale, Jan 28 2021 *)

PARI

{ for (n=1, 4000, if (n<2, M=m=0, f=factor(n)~; M=m=f[2, 1]; for (i=2, length(f), M=max(M, f[2, i]); m=min(m, f[2, i]))); write("b062977.txt", n, " ", M - m) ) } \\ Harry J. Smith, Aug 14 2009

PARI

A062977(n) = if((1==n),0,n=(factor(n)[, 2]); vecmax(n)-vecmin(n)); \\ Antti Karttunen, Nov 17 2019

Formula

a(n) = A051903(n) - A051904(n).

a(A108951(n)) = A325226(n) = A001222(n) - A071178(n). - Antti Karttunen, Nov 17 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A033150 - 1 = 0.705211... . - Amiram Eldar, Jan 05 2024