A322817 a(n) = A001222(A065642(n)) - A001222(n), where A065642(n) gives the next larger m that has same prime factors as n (ignoring multiplicity), and A001222 gives the number of prime factors, when counted with multiplicity.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 0, 1, 1, -1, 1, 2, 1, 1, 1, 1, 1, -1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 2, 1, 2, 1, 1, 1, 1, 1

Offset

1,50

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = A001222(A065642(n)) - A001222(n).

Example

For n = 2 = 2^1, the next larger number with only 2's as its prime factors is 4 = 2^2, thus a(2) = 1.
For n = 12 = 2^2 * 3^1, the next larger number with the same prime factors is 18 = 2^1 * 3^2, with the same value of A001222, thus a(12) = 0.
For n = 40 = 2^3 * 5^1, the next larger number with the same prime factors is 50 = 2^1 * 5^2. While 40 has 3+1 = 4 prime factors in total, 50 has 1+2 = 3, thus a(40) = 3 - 4 = -1.
For n = 50, the next larger number with the same prime factors is 80 = 2^4 * 5^1, thus a(50) = (4+1)-(2+1) = 2.

Prog

(PARI)
A007947(n) = factorback(factorint(n)[, 1]);
A065642(n) = { my(r=A007947(n)); if(1==n, n, n = n+r; while(A007947(n) <> r, n = n+r); n); };
A322817(n) = (bigomega(A065642(n)) - bigomega(n));

Crossrefs

Cf. A001222, A065642, A322818.

Sequence in context: A338639 A249351 A123706 * A194325 A300547 A025452

Adjacent sequences: A322814 A322815 A322816 * A322818 A322819 A322820

Keyword

sign

Author

Antti Karttunen, Dec 27 2018

Status

approved