cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A369450 Partial sums of A369460, where A369460(n) = number of representations of 12n-9 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 5, 6, 6, 7, 8, 9, 11, 11, 11, 12, 14, 14, 14, 17, 17, 19, 20, 20, 21, 21, 23, 25, 25, 26, 28, 29, 29, 29, 30, 31, 33, 34, 35, 36, 38, 40, 42, 43, 43, 44, 45, 46, 46, 48, 48, 50, 53, 53, 55, 58, 58, 59, 59, 61, 62, 63, 63, 65, 66, 66, 67, 68, 68, 71, 72, 74, 75, 75, 75, 78, 80, 81, 82, 84, 84, 85
Offset: 1

Views

Author

Antti Karttunen, Jan 24 2024

Keywords

Comments

In the case of the numbers of the form 12m+3 (i.e., multiples of 3 among the numbers of the form 4m+3) any such representation must either have p = q = 3, or p == q == r == +1 (mod 3), or -1 (mod 3) for all three primes (see the table given in comments of A369252), therefore the cumulative sum here has an intermediate growth among a(n), A369451(n) and A369452(n).

Links

Crossrefs

Partial sums of A369460.
Cf. A369054, A369057, A369252, A369451, A369452.

Programs

  • PARI
    up_to = 1024; \\ 2*(10^4);
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])));
A369460(n) = A369054((12*n)-9);
A369450list(up_to) = { my(v=vector(up_to)); s = 0; for(n=1,up_to,s+=A369460(n); v[n] = s); (v); };
v369450 = A369450list(up_to);
A369450(n) = v369450[n];

Formula

a(1) = A369460(1), for n > 1, a(n) = A369460(n) + a(n-1).
a(n) = A369057(3*n) - (A369451(n) + A369452(n)).

A369054 Number of representations of n as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

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
Offset: 0

Views

Author

Antti Karttunen, Jan 20 2024

Keywords

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.

Examples

			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.

Links

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))].

Programs

  • 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])));

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).

A369055 Number of representations of 4n-1 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 2, 2, 1, 0, 0, 2, 0, 0, 0, 1, 1, 2, 2, 1, 1, 0, 1, 2, 0, 0, 0, 3, 0, 1, 0, 2, 1, 2, 0, 3, 1, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 5, 2, 0, 0, 2, 1, 1, 0, 2, 0, 1, 1, 2, 2, 0, 2, 1, 0, 2, 0, 3, 1, 0, 0, 4, 1, 0, 1
Offset: 1

Views

Author

Antti Karttunen, Jan 20 2024

Keywords

Comments

Number of solutions to 4n-1 = x', where x' is the arithmetic derivative of x (A003415), and x is a product of three odd primes, A046316.
The number of 0's in range [1..10^n], for n=1..7 are: 8, 46, 288, 2348, 21330, 206355, 2079925, etc.
Goldbach's conjecture can be expressed by claiming that each even number > 4 is an arithmetic derivative of an odd semiprime, as (p*q)' = p+q, where p and q are odd primes. One way to extend Goldbach's conjecture to three primes involves applying the arithmetic derivative to all possible products of three odd primes (A046316) as: (p*q*r)' = (p*q) + (p*r) + (q*r), and asking, "Onto which subset of natural numbers does this map surjectively?" Clearly, the above formula can only produce numbers of the form 4m+3, and furthermore, an analysis at A369252 shows that the trisections of this sequence have quite different expected values, being on average the highest in the trisection A369462, which gives the number of representations for the numbers of the form 12m+11. This motivates a new kind of Goldbach-3 conjecture: "All numbers of the form 12*m-1, with m large enough, have at least one representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r." Furthermore, empirical data for sequence A369463 suggests that "large enough" in this case might well be 4224080, as 1+(12*4224079) = 50688949 = A369463(285), with the next term of A369463 so far unknown. Similar conjectures can be envisaged for the arithmetic derivatives of products of four or more primes. - Antti Karttunen, Jan 25 2024

Examples

			a(7) = 1 because 4*7 - 1 = 27, which can be represented as a sum of the form (p*q) + (p*r) + (q*r), with all three primes p, q and r = 3.
a(19) = 2 because 4*19 - 1 = 75, which can be represented as a sum of the form (p*q) + (p*r) + (q*r) in two ways, with p=3, q=3 and r=11, or with p = q = r = 5.
a(9999995) = 0 because (4*9999995)-1 = 39999979, which cannot be expressed as a sum (p*q) + (p*r) + (q*r) for any three odd primes p, q and r, whether distinct or not.

Links

Crossrefs

Cf. A002620, A003415, A046316, A369054, A369056, A369057 (partial sums), A369058, A369064 (record values), A369241, A369245, A369248, A369460, A369461, A369462, A369463.
Cf. A369460, A369461, A369462 (trisections), A369450, A369451, A369452 (and their partial sums).
Cf. also A351029, A369239.

Programs

  • PARI
    \\ We iterate over weakly increasing triplets of odd primes:
A369055list(up_to) = { my(v = [3,3,3], ip = #v, d, u = vector(up_to), lim = -1+(4*up_to)); while(1, d = ((v[1]*v[2]) + (v[1]*v[3]) + (v[2]*v[3])); if(d > lim, ip--, ip = #v; u[(d+1)/4]++); if(!ip, return(u)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])); };
v369055 = A369055list(100001);
A369055(n) = v369055[n];

Formula

a(n) = A369054(4*n-1).
a(n) = Sum_{i=1..A002620(4*n-1)} A369058(i)*[A003415(i)==4*n-1], where [ ] is the Iverson bracket.

A369252 Arithmetic derivative applied to the numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.

Original entry on oeis.org

27, 39, 51, 55, 75, 71, 87, 75, 91, 111, 103, 123, 95, 119, 147, 131, 119, 151, 183, 151, 135, 195, 167, 155, 231, 147, 199, 191, 187, 255, 167, 267, 211, 291, 195, 215, 247, 191, 263, 215, 327, 251, 247, 363, 203, 375, 311, 271, 255, 239, 411, 231, 311, 343, 299, 231, 435, 359, 331, 447, 311, 263, 391, 483, 263
Offset: 1

Views

Author

Antti Karttunen, Jan 22 2024

Keywords

Comments

The table showing the possible modulo 3 combinations for p, q, r and the sum ((p*q) + (p*r) + (q*r)):
| p | q | r | sum ((p*q) + (p*r) + (q*r)) (mod 3)
--+------+------+------+----------------------------------------
| 0 | 0 | 0 | 0, p=q=r=3, sum is 27.
--+------+------+------+----------------------------------------
| 0 | 0 | +/-1 | 0, p=q=3, r > 3.
--+------+------+------+----------------------------------------
| 0 | +1 | +1 | +1
--+------+------+------+----------------------------------------
| 0 | -1 | -1 | +1
--+------+------+------+----------------------------------------
| 0 | -1 | +1 | -1
--+------+------+------+----------------------------------------
| 0 | +1 | -1 | -1
--+------+------+------+----------------------------------------
| +1 | +1 | +1 | 0
--+------+------+------+----------------------------------------
| -1 | -1 | -1 | 0
--+------+------+------+----------------------------------------
| -1 | +1 | +1 | -1, regardless of the order, thus x3.
--+------+------+------+----------------------------------------
| +1 | -1 | -1 | -1, regardless of the order, thus x3.
--+------+------+------+----------------------------------------
Notably a(n) is a multiple of 3 only when A046316(n) is either a multiple of 9, or all primes p, q and r are either == +1 (mod 3) or all are == -1 (mod 3), and the case a(n) == +1 (mod 3) is only possible when A046316(n) is a multiple of 3, but not of 9, and furthermore, it is required that r == q (mod 3). See how these combinations affects sequences like A369241, A369245, A369450, A369451, A369452.
For n=1..9 the number of terms of the form 3k, 3k+1 and 3k+2 in range [1..10^n-1] are:
6, 2, 1,
39, 22, 38,
291, 209, 499,
2527, 1884, 5588,
23527, 17020, 59452,
227297, 156240, 616462,
2232681, 1453030, 6314288,
22119496, 13661893, 64218610,
220098425, 129624002, 650277572.
It seems that 3k+2 terms are slowly gaining at the expense of 3k+1 terms when n grows, while the density of the multiples of 3 might converge towards a limit.

Links

Crossrefs

Cf. A369251 (same sequence sorted into ascending order, with duplicates removed).
Cf. A369464 (numbers that do not occur in this sequence).
Cf. A003415, A046316, A369054, A369058, A369248.
Cf. also the trisections of A369055: A369460, A369461, A369462 and their partial sums A369450, A369451, A369452, also A369241, A369245.
Only terms of A004767 occur here.

Formula

a(n) = A003415(A046316(n)).

A369461 Number of representations of 12n-5 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 1, 1, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 3, 0, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 2, 0, 0, 2, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 2, 1, 0, 1, 1, 0, 0, 0, 4, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0
Offset: 1

Views

Author

Antti Karttunen, Jan 23 2024

Keywords

Comments

The sequence seems to contain an infinite number of zeros. See A369451 for the cumulative sum, and comments there.
Question: Are there any sections of this sequence, with parameters k >= 2, 0 <= i < k, for which a((k*n)-i) = 0 for all n >= 1? - Antti Karttunen, Nov 20 2024

Links

Crossrefs

Trisection of A369055.
Cf. A017605, A369054, A369451 (partial sums), A369460, A369462.

Programs

  • PARI
    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])));
A369461(n) = A369054((12*n)-5);

Formula

a(n) = A369054(A017605(n-1)) = A369054((12*n)-5).
a(n) = A369055((3*n)-1).

A369462 Number of representations of 12n-1 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 1, 2, 0, 1, 1, 3, 1, 1, 2, 5, 0, 1, 0, 2, 2, 2, 1, 4, 1, 3, 0, 3, 1, 2, 2, 3, 0, 2, 1, 8, 1, 1, 1, 4, 2, 2, 3, 3, 0, 4, 0, 4, 1, 1, 4, 3, 1, 3, 1, 6, 2, 3, 0, 5, 3, 1, 2, 6, 2, 6, 2, 2, 0, 1, 1, 5, 1, 2, 1, 10, 1, 3, 1, 3, 4, 2, 1, 6, 3, 6, 1, 4, 1, 3, 1, 5, 2, 3, 0
Offset: 1

Views

Author

Antti Karttunen, Jan 23 2024

Keywords

Comments

See A369452 for the cumulative sum, and comments there.
Question: Is there only a finite number of 0's in this sequence? See discussion at A369055 and see A369463 for empirical data.

Links

Crossrefs

Trisection of A369055.
Cf. A017653, A369054, A369252, A369452 (partial sums), A369460, A369461, A369463 (= (12*i)-1, where i are the indices of zeros in this sequence).

Programs

  • PARI
    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])));
A369462(n) = A369054((12*n)-1);

Formula

a(n) = A369054(A017653(n-1)) = A369054(12*n - 1).
a(n) = A369055(3*n).
Showing 1-6 of 6 results.

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