A144774 -id:A144774 - cp's OEIS frontend

A144775 Define f(n) = 2 * rad(n) if four divides n and rad(n) otherwise: then a(n) = 0 for composite n where f(n) is not less than n and otherwise equals the number of positive integers k less than n for which f(k) < f(n).

0, 1, 2, 0, 4, 0, 6, 3, 2, 0, 10, 0, 12, 0, 0, 4, 16, 8, 18, 0, 0, 0, 22, 13, 7, 0, 2, 0, 28, 0, 30, 5, 0, 0, 0, 16, 36, 0, 0, 24, 40, 0, 42, 0, 21, 0, 46, 16, 13, 15, 0, 0, 52, 11, 0, 35, 0, 0, 58, 0, 60, 0, 31, 5, 0, 0, 66, 0, 0, 0, 70, 20, 72, 0, 27, 0, 0, 0, 78, 32, 2, 0, 82, 0, 0, 0, 0, 55

Offset: 1

Reikku Kulon, Sep 21 2008

This sequence obtains a new maximum a(n) = n - 1 for prime n.

a(n) = 0 often, but not always, when n - 1 and n + 1 are twin primes.

			f(8) = 2 * rad(8) = 4. f(k) < 4 for 1, 2 and 3 (f(k) = k for 0 < k < 8); a(8) = 3.

Cf. A000040, A001097, A007947, A144100, A144774

A144776 Define f(n) = 1 if n is prime, 2 * rad(n) if four divides n and rad(n) otherwise: then a(n) = 0 for composite n where f(n) is not less than n and otherwise equals the number of positive integers k less than n for which f(k) < f(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 8, 0, 12, 0, 0, 0, 0, 0, 17, 14, 0, 10, 0, 0, 0, 0, 14, 0, 0, 0, 22, 0, 0, 0, 28, 0, 0, 0, 0, 29, 0, 0, 26, 25, 26, 0, 0, 0, 24, 0, 42, 0, 0, 0, 0, 0, 0, 41, 21, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 42, 0, 0, 0, 0, 46, 23, 0, 0, 0, 0, 0, 0, 64, 0, 58, 0, 0, 0, 0, 0

Offset: 1

Reikku Kulon, Sep 21 2008

For the given terms, nearly all n for which a(n) obtains a new maximum are multiples of eight. Only 18, 36 and 45 are not.

			f(8) = 4 and f(9) = 3. For 1, 2, 3, 5 and 7, f(k) = 1, so a(8) = a(9) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000040, A001097, A007947, A144100, A144774, A144775

A007947(n) = factorback(factorint(n)[, 1]);
f(n) = if(isprime(n),1,if(n%4,A007947(n),2*A007947(n)));
A144776(n) = if(n<2,0,my(x=f(n)); if(!isprime(n)&&(x>=n),0,sum(k=1,n-1,(f(k)Antti Karttunen, Jul 03 2018

cp's OEIS Frontend

A144775 Define f(n) = 2 * rad(n) if four divides n and rad(n) otherwise: then a(n) = 0 for composite n where f(n) is not less than n and otherwise equals the number of positive integers k less than n for which f(k) < f(n).

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