A369450 -id:A369450 - cp's OEIS frontend

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

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

Antti Karttunen, Jan 20 2024

nonn

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

			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.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.
Eric Weisstein's World of Mathematics, Goldbach Conjecture.
Index entries for sequences related to Goldbach conjecture

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.

\\ 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];

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 pqr 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

Antti Karttunen, Jan 22 2024

nonn

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.

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

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.

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

A369451 Partial sums of A369461, where A369461(n) = number of representations of 12n-5 as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 3, 5, 5, 5, 6, 7, 8, 8, 8, 10, 10, 11, 11, 11, 11, 11, 12, 14, 15, 15, 15, 18, 18, 18, 18, 20, 21, 21, 22, 23, 23, 24, 24, 25, 25, 25, 25, 26, 26, 27, 27, 30, 30, 30, 31, 32, 34, 34, 34, 36, 37, 38, 38, 39, 39, 39, 40, 42, 42, 42, 43, 46, 46, 46, 46, 47, 47, 47, 47, 47, 47, 49, 50, 52

Offset: 1

Antti Karttunen, Jan 24 2024

nonn

In the case of the numbers of the form 12m+7 (i.e., intersection of numbers of the form 3k+1 with the numbers of the form 4m+3) any such representation must have p = 3 and q > 3 (see the table given in comments of A369252), therefore the cumulative sum here grows slowest among A369450(n), a(n) and A369452(n). Notably, it seems that a(n) < n for all n.

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

Partial sums of A369461.

Cf. A369054, A369057, A369252, A369450, A369452.

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])));
A369461(n) = A369054((12*n)-5);
A369451list(up_to) = { my(v=vector(up_to)); s = 0; for(n=1,up_to,s+=A369461(n); v[n] = s); (v); };
v369451 = A369451list(up_to);
A369451(n) = v369451[n];

a(1) = A369461(1), for n > 1, a(n) = A369461(n) + a(n-1).

a(n) = A369057(3*n) - (A369450(n) + A369452(n)).

A369452 Partial sums of A369462, where A369462(n) = number of representations of 12n-1 as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 5, 6, 8, 8, 10, 11, 13, 13, 14, 15, 18, 19, 20, 22, 27, 27, 28, 28, 30, 32, 34, 35, 39, 40, 43, 43, 46, 47, 49, 51, 54, 54, 56, 57, 65, 66, 67, 68, 72, 74, 76, 79, 82, 82, 86, 86, 90, 91, 92, 96, 99, 100, 103, 104, 110, 112, 115, 115, 120, 123, 124, 126, 132, 134, 140, 142, 144, 144

Offset: 1

Antti Karttunen, Jan 24 2024

nonn

In the case of the numbers of the form 12m+11 (i.e., intersection of numbers of the form 3k+2 with the numbers of the form 4m+3) any such representation must be one of the four most common combinations that p, q and r may obtain mod-3-wise (see the table given in comments of A369252), therefore this sequence grows fastest among A369450(n), A369451(n) and a(n).

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

Cf. A369054, A369057, A369252, A369450, A369451, A369462.

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])));
A369462(n) = A369054((12*n)-1);
A369452list(up_to) = { my(v=vector(up_to)); s = 0; for(n=1,up_to,s+=A369462(n); v[n] = s); (v); };
v369452 = A369452list(up_to);
A369452(n) = v369452[n];

a(1) = A369462(1), for n > 1, a(n) = A369462(n) + a(n-1).

(n) = A369057(3*n) - (A369450(n) + A369451(n)).

A369460 Number of representations of 12n-9 as a sum (pq + pr + 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

Antti Karttunen, Jan 23 2024

nonn

See A369450 for the cumulative sum, and comments there.

Antti Karttunen, Table of n, a(n) for n = 1..100000

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.

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);

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

A378209 Antiderivatives of 334406399, numbers k for which A003415(k) = A024451(9) = A003415(A002110(9)).

Original entry on oeis.org

223092870, 975351895, 1527890095, 1885679383, 2189118743, 2329696457, 2338611863, 3485765789, 4586671213, 5453593183, 5472849253, 5674340053, 8071055747, 8931775397, 9332889127, 9453996491, 9601098443, 10293819917, 12717530039, 17343441881, 18636581773, 19498393573, 20167656703, 23244839627, 23515890737, 23556538969

Offset: 1

Antti Karttunen, Nov 20 2024

Apart from the initial term A002110(9), all other terms are products of three distinct odd primes, A046389. Compare to the comments in A369239.
Note that A024451(9) = 334406399 = 43 * 163 * 47711 == -1 (mod 12). Compare the sequences A369450, A369451 and A369452 to see why there is such a sudden peak in A377993 at n=9, when compared to the nearby terms before and after.
For all n=1..330, A327969(a(n)) <= 7 = A099307(a(n)), because, when we apply A003415 successively, we get: A003415(334406399) -> 9835475 [= A369651(9)] -> 4893565 -> 978718 -> 564671 (which is a prime) -> 1 -> 0.

Antti Karttunen, Table of n, a(n) for n = 1..330
Index entries for sequences related to primorial numbers

Row 9 of irregular triangle A377992.
Subsequence of A099308, and after the initial term, subsequence of A046389.
Cf. A002110, A003415, A024451, A099307, A327969, A369239, A369651.
Cf. A369450, A369451, A369452.

cp's OEIS Frontend

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

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

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A369451 Partial sums of A369461, where A369461(n) = number of representations of 12n-5 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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A369452 Partial sums of A369462, where A369462(n) = number of representations of 12n-1 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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A369460 Number of representations of 12n-9 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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A378209 Antiderivatives of 334406399, numbers k for which A003415(k) = A024451(9) = A003415(A002110(9)).

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

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

A369451 Partial sums of A369461, where A369461(n) = number of representations of 12n-5 as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

A369452 Partial sums of A369462, where A369462(n) = number of representations of 12n-1 as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

A369460 Number of representations of 12n-9 as a sum (pq + pr + q*r) with three odd primes p <= q <= r.