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 1-7 of 7 results.

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)).

A369056 Numbers k of the form 4m+3 for which there is no representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 31, 35, 43, 47, 59, 63, 67, 79, 83, 99, 107, 115, 127, 139, 143, 159, 163, 171, 175, 179, 207, 219, 223, 227, 235, 243, 259, 279, 283, 295, 303, 307, 319, 323, 339, 347, 367, 379, 387, 399, 403, 415, 427, 443, 463, 499, 515, 523, 531, 547, 559, 571, 579, 595, 603, 619, 639, 643, 655, 659, 675
Offset: 1

Views

Author

Antti Karttunen, Jan 20 2024

Keywords

Comments

Numbers k in A004767 for which A369054(k) = 0.
Numbers k of the form 4m-1 such that they are not arithmetic derivative (A003415) of any term of A046316.
Question: Is it possible that this sequence might be finite (although very long)? See comments in A369055.

Links

Crossrefs

Setwise difference A004767 \ A369251.
Cf. A003415, A046316, A369054, A369055.
Subsequences: A369248 (terms that are multiples of 3), A369249 (primes in this sequence).
Cf. also A369250 (4m+3 primes missing from this sequence).

Programs

  • Maple
    N:= 1000: # for terms <= N
S:= {seq(i,i=3..N,4)}:
P:= select(isprime, [seq(i,i=3..N/3,2)]):
for i from 1 to nops(P) do
  p:= P[i];
  for j from i to nops(P) do
    q:= P[j];
    if 2*p*q + q^2 > N then break fi;
    for k from j to nops(P) do
      r:= P[k];
      v:= p*q + p*r + q*r;
      if v > N then break fi;
      S:= S minus {v};
od od od:
sort(convert(S,list)); # Robert Israel, Apr 17 2024
  • PARI
    isA369056(n) = ((3==(n%4)) && !A369054(n)); \\ Needs also program from A369054.

A369460 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, 1, 1, 0, 2, 1, 0, 1, 1, 1, 2, 0, 0, 1, 2, 0, 0, 3, 0, 2, 1, 0, 1, 0, 2, 2, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 0, 1, 1, 1, 0, 2, 0, 2, 3, 0, 2, 3, 0, 1, 0, 2, 1, 1, 0, 2, 1, 0, 1, 1, 0, 3, 1, 2, 1, 0, 0, 3, 2, 1, 1, 2, 0, 1, 3, 2, 1, 1, 2, 1, 0, 2, 2, 3, 0, 1, 2, 0, 4, 1, 0, 2, 1, 0, 0, 2, 2
Offset: 1

Views

Author

Antti Karttunen, Jan 23 2024

Keywords

Comments

See A369450 for the cumulative sum, and comments there.

Links

Crossrefs

Trisection of A369055.
Cf. A369054, A369248 (gives the positions of 0's in this sequence when nine is added and divided by 12), A369450 (partial sums), A369461, 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])));
A369460(n) = A369054((12*n)-9);

Formula

a(n) = A369055((3*n)-2).

A369241 Number of representations of 2^n - 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, 0, 0, 2, 1, 1, 0, 2, 1, 3, 0, 2, 2, 6, 0, 13, 1, 13, 0, 15, 0, 24, 1, 49, 4, 47, 0, 156, 6, 129, 0, 441, 1, 616
Offset: 0

Views

Author

Antti Karttunen, Jan 21 2024

Keywords

Comments

Any solutions for odd cases must have p = 3, with q and r > 3, because A000225(2n-1) == 1 (mod 3), while on even n, 2^n - 1 is a multiple of 3. This explains why the odd bisection grows much more sluggishly than the even bisection.
Question 2: Is there an infinite number of 0's in this sequence? See also comments in A369055.

Crossrefs

Cf. A000225, A369054, A369055, A369248.
Cf. also A369242, A369245.

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])));
search_for_3k1_cases(n) = if(3!=(n%4), 0, my(p = 5, q, c=0); while(1, q = (n-(3*p)) / (3+p); if(q < p, return(c), if(1==denominator(q) && isprime(q), c++; write("b369241_by_solutions_of_odd_bisection_to.txt", n, " ", 3*p*q))); p = nextprime(1+p)));
A369241(n) = if(n%2, search_for_3k1_cases((2^n)-1), A369054((2^n)-1));

Formula

a(n) = A369054(A000225(n)).
For n >= 2, a(n) = A369055(2^(n-2)).

A369464 Numbers for which there is no representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88
Offset: 1

Views

Author

Antti Karttunen, Jan 24 2024

Keywords

Crossrefs

Complement of A369251. Numbers not in A369252.
Union of A004773 and A369056.
Positions of 0's in A369054.
Cf. A098700, A369248, A369249, A369463 (subsequences).

Programs

  • PARI
    isA369251(n) = if(3!=(n%4),0, my(v = [3,3], ip = #v, r); 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), return(1))); if(!ip, return(0)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])));
isA369464(n) = !isA369251(n);

A369463 Numbers of the form 12*m-1 for which there is no representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

11, 23, 35, 47, 59, 83, 107, 143, 179, 227, 323, 347, 443, 515, 659, 683, 827, 947, 1259, 1523, 1763, 1787, 2075, 2267, 2675, 2963, 3023, 3203, 3275, 3347, 3467, 3635, 4523, 4643, 4859, 5003, 5147, 5747, 5819, 6395, 6803, 6827, 7235, 8003, 8123, 8171, 8747, 8963, 9323, 9659, 9827, 10367, 10427, 12347, 12923, 13187
Offset: 1

Views

Author

Antti Karttunen, Jan 23 2024

Keywords

Comments

Equal to (12*i)-1, where i are the positions of 0's in A369462.
Terms of the form 3k+2 in A369056. These seem to be much more rare than terms of A369248.
Question: Is this a finite sequence, with the last term a(285) = 50688947 = (12*4224079)-1? See conjecture in A369055.
If it exists, a(286) > 201326603 (= (12*(2^24))+11).

Links

Crossrefs

Intersection of A016789 and A369056 (and of A369464).
Subsequence of A017653.
Cf. A369054, A369055, A369462.
Cf. also A369248.

Programs

  • PARI
    isA369251(n) =  if(3!=(n%4),0, my(v = [3,3], ip = #v, r); 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), return(1))); if(!ip, return(0)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])));
isA369463(n) = ((11==(n%12)) && !isA369251(n));
Showing 1-7 of 7 results.

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