A369054 - OEIS

A369054

Number of representations of n as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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

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OFFSET

0,76

COMMENTS

Number of solutions to n = x', where x' is the arithmetic derivative of x (A003415), and x is a product of three odd primes (not all necessarily distinct, A046316).

See the conjecture in A369055.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..50000

Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

FORMULA

a(n) = Sum_{i=1..A002620(n)} A369058(i)*[A003415(i)==n], where [ ] is the Iverson bracket.

For n >= 2, a(n) <= A099302(n).

EXAMPLE

a(27) = 1 as 27 can be expressed in exactly one way in the form (p*q + p*r + q*r), with p, q, r all being 3 in this case, as 27 = (3*3 + 3*3 + 3*3).

a(311) = 5 as 311 = (3*5 + 3*37 + 5*37) = (3*7 + 3*29 + 7*29) = (3*13 + 3*17 + 13*17) = (5*7 + 5*23 + 7*23) = (7*11 + 7*13 + 11*13). Expressed in the terms of arithmetic derivatives, of the A099302(311) = 8 antiderivatives of 311 [366, 430, 494, 555, 609, 663, 805, 1001], only the last five are products of three odd primes: 555 = 3*5*37, 609 = 3*7*29, 663 = 3*13*17, 805 = 5*7*23, 1001 = 7 * 11 * 13.

PROG

(PARI)

\\ Use this for building up a list up to a certain n. We iterate over weakly increasing triplets of odd primes:

A369054list(up_to) = { my(v = [3, 3, 3], ip = #v, d, u = vector(up_to)); while(1, d = ((v[1]*v[2]) + (v[1]*v[3]) + (v[2]*v[3])); if(d > up_to, ip--, ip = #v; u[d]++); if(!ip, return(u)); v[ip] = nextprime(1+v[ip]); for(i=1+ip, #v, v[i]=v[i-1])); };

v369054 = A369054list(100001);

A369054(n) = if(!n, n, v369054[n]);

(PARI)

\\ Use this for computing the value of arbitrary n. We iterate over weakly increasing pairs of odd primes:

A369054(n) = if(3!=(n%4), 0, my(v = [3, 3], ip = #v, r, c=0); while(1, r = (n-(v[1]*v[2])) / (v[1]+v[2]); if(r < v[2], ip--, ip = #v; if(1==denominator(r) && isprime(r), c++)); if(!ip, return(c)); v[ip] = nextprime(1+v[ip]); for(i=1+ip, #v, v[i]=v[i-1])));

CROSSREFS

Cf. A369055 [quadrisection, a(4n-1)], and its trisections A369460 [= a((12*n)-9)], A369461 [= a((12*n)-5)], A369462 [= a((12*n)-1)].

Cf. A002620, A003415, A046316, A099302, A369056, A369058, A369252.

Cf. A369251 (positions of terms > 0), A369464 (positions of 0's).

Cf. A369063 (positions of records), A369064 (values of records).

Cf. A369241 [= a(2^n - 1)], A369242 [= a(n!-1)], A369245 [= a(A006862(n))], A369247 [= a(3*A057588(n))].

Sequence in context: A027345 A328959 A086080 * A281477 A070139 A116860

Adjacent sequences: A369051 A369052 A369053 * A369055 A369056 A369057

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 20 2024

STATUS

approved