A369056 -id:A369056 - cp's OEIS frontend

A369054 Number of representations of n as a sum (pq + pr + 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

Offset: 0

Antti Karttunen, Jan 20 2024

nonn

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.

			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.

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.

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))].

\\ 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]);

\\ 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])));

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 (pq + pr + 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

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.

A369248 Numbers of the form 12m+3 for which there is no representation as a sum (pq + pr + qr) with three odd primes p <= q <= r.

Original entry on oeis.org

3, 15, 63, 99, 159, 171, 207, 219, 243, 279, 303, 339, 387, 399, 531, 579, 603, 639, 675, 699, 747, 783, 819, 879, 891, 963, 1059, 1107, 1143, 1179, 1215, 1227, 1299, 1323, 1359, 1467, 1527, 1563, 1611, 1659, 1731, 1779, 1791, 1803, 1899, 1923, 1971, 1983, 2007, 2019, 2115, 2235, 2319, 2403, 2427, 2487, 2499, 2547

Offset: 1

Antti Karttunen, Jan 22 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10516

Intersection of A017557 and A369056, multiples of 3 in the latter.

Cf. A369054, A369055.

isA369248(n) = ((3==(n%12)) && !A369054(n)); \\ Needs also code from A369054.

A369251 Numbers that have at least one representation as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

27, 39, 51, 55, 71, 75, 87, 91, 95, 103, 111, 119, 123, 131, 135, 147, 151, 155, 167, 183, 187, 191, 195, 199, 203, 211, 215, 231, 239, 247, 251, 255, 263, 267, 271, 275, 287, 291, 299, 311, 315, 327, 331, 335, 343, 351, 355, 359, 363, 371, 375, 383, 391, 395, 407, 411, 419, 423, 431, 435, 439, 447, 451, 455, 459

Offset: 1

Antti Karttunen, Jan 22 2024

nonn

By necessity all terms are of the form 4m+3 (in A004767).

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

Complement of A369464.
Sequence A369252 sorted into ascending order, with duplicates removed.
Setwise difference A004767 \ A369056.
Subsequence of A239433.
Cf. A369054, A369055.
Cf. A369250 (primes in this sequence).

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])));

{k | A369054(k) > 0}.

A369464 Numbers for which there is no representation as a sum (pq + pr + 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

Antti Karttunen, Jan 24 2024

nonn

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

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);

A369249 Primes of the form 4m+3 for which there is no representation as a sum (pq + pr + qr) with three odd primes p <= q <= r.

Original entry on oeis.org

3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 79, 83, 107, 127, 139, 163, 179, 223, 227, 283, 307, 347, 367, 379, 443, 463, 499, 523, 547, 571, 619, 643, 659, 683, 787, 827, 883, 907, 947, 967, 1039, 1087, 1123, 1171, 1259, 1327, 1423, 1459, 1483, 1523, 1567, 1579, 1627, 1699, 1723, 1747, 1759, 1787, 1987, 1999, 2083, 2143

Offset: 1

Antti Karttunen, Jan 22 2024

nonn

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

Intersection of A002145 and A369056.
Setwise difference A002145 \ A369250.
Cf. A369054.
Cf. also A189483.

isA369249(n) = ((3==(n%4)) && isprime(n) && !A369054(n)); \\ Needs also program from A369054.

A369250 Primes for which there is at least one representation as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

71, 103, 131, 151, 167, 191, 199, 211, 239, 251, 263, 271, 311, 331, 359, 383, 419, 431, 439, 467, 479, 487, 491, 503, 563, 587, 599, 607, 631, 647, 691, 719, 727, 739, 743, 751, 811, 823, 839, 859, 863, 887, 911, 919, 971, 983, 991, 1019, 1031, 1051, 1063, 1091, 1103, 1151, 1163, 1187, 1223, 1231, 1279, 1283, 1291

Offset: 1

Antti Karttunen, Jan 22 2024

nonn

All such primes are by necessity of the form 4m+3 (in A002145). See A369249 for those 4m+3 primes that do not have such a representation.

Also by necessity, in these cases the primes in the sum (p*q + p*r + q*r) must all be distinct, that is, we actually need p < q < r, otherwise the sum would not be a prime.

			71 is present as 71 = (3*5) + (3*7) + (5*7) = A003415(105).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Primes with such a representation vs. 4m+3 primes without such a representation (Plot2 comparison of the densities)

Primes in A369251.
Setwise difference A002145 \ A369249.
Subsequence of A189441.
Cf. A003415, A369054, A369056.

isA369250(n) = (isprime(n) && (A369054(n)>0)); \\ Needs also program from A369054.

A369463 Numbers of the form 12m-1 for which there is no representation as a sum (pq + pr + qr) 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

Antti Karttunen, Jan 23 2024

nonn

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).

Antti Karttunen, Table of n, a(n) for n = 1..285 (please update, also comments, if more terms are found!)

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

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));

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

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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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A369248 Numbers of the form 12*m+3 for which there is no representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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A369251 Numbers that have at least one representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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

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A369249 Primes of the form 4*m+3 for which there is no representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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A369250 Primes for which there is at least one representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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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.

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

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

A369248 Numbers of the form 12m+3 for which there is no representation as a sum (pq + pr + qr) with three odd primes p <= q <= r.

A369251 Numbers that have at least one representation as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

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

A369249 Primes of the form 4m+3 for which there is no representation as a sum (pq + pr + qr) with three odd primes p <= q <= r.

A369250 Primes for which there is at least one representation as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

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