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.

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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).

A369245 Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r. (Definition implies that p=3 and q > 3).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1
Offset: 0

Views

Author

Antti Karttunen, Jan 22 2024

Keywords

Comments

Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum of the form 3*(p+q) + p*q, where p and q are odd primes.
Question: Will there be an eventual growth spurt for this sequence? Even though all solutions must be multiples of 3 (but not of 9), because A006862(n) == 1 (mod 3), for n > 1, and the solutions belong to a set listed by A369461.
Similar sequence A369242 grows more vigorously because A033312(n) == -1 (mod 3) for n >= 3, thus allowing non-multiples of 3 as solutions. See comments in A369252.

Examples

			a(4) = 1 as there exists a natural number 399 = 3 * 7 * 19, whose arithmetic derivative (indicated with 399', see A003415) is computed as ((3*7) + (3*19) + (7*19)) = 211 = 1 + prime(4)# = A006862(4), and because 399 is the unique term in A046316 that satisfies the condition.
a(17) >= 1 because there exists (at least one) solution k = 4903038892893242229501 = 3 * 17 * 96138017507710631951 with A003415(k) = 1+A002110(17).
For other cases, see examples in A369246.

Crossrefs

Cf. A002110, A003415, A006862, A046316, A369054, A369246 (the solutions), A369252, A369461.
Cf. also A116979, A369000, A369239 for similar counts, also A369241, A369242 and A369247.

Programs

  • PARI
    \\ Needs also program from A369054.
A002110(n) = prod(i=1,n,prime(i));
A369245(n) = A369054(A002110(n)+1);
  • PARI
    \\ Optimized version of above, employs the fact that solutions must all be multiples of 3. Outputs also terms for A369246.
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("b369246_by_search_order_to.txt", n, " ", 3*p*q))); p = nextprime(1+p)));
A002110(n) = prod(i=1,n,prime(i));
A369245(n) = search_for_3k1_cases(A002110(n)+1);

Formula

a(n) = A369054(A006862(n)).

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)).

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.

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

Views

Author

Antti Karttunen, Jan 24 2024

Keywords

Comments

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.

Links

Crossrefs

Partial sums of A369461.
Cf. A369054, A369057, A369252, A369450, 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])));
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];

Formula

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 (p*q + p*r + 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

Views

Author

Antti Karttunen, Jan 24 2024

Keywords

Comments

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).

Links

Crossrefs

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

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])));
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];

Formula

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 (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)).

A369057 Total number of representations of natural numbers in range 1 .. 4n-1 as sums of the form 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, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 7, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 14, 15, 15, 16, 17, 17, 17, 19, 21, 22, 22, 22, 24, 24, 24, 24, 25, 26, 28, 30, 31, 32, 32, 33, 35, 35, 35, 35, 38, 38, 39, 39, 41, 42, 44, 44, 47, 48, 49, 50, 50, 50, 51, 52, 52, 54, 54, 54, 59, 61, 61, 61, 63, 64, 65, 65, 67
Offset: 1

Views

Author

Antti Karttunen, Jan 21 2024

Keywords

Comments

Terms a(10^n), for n=1..7 are: 2, 82, 1819, 34220, 628914, 11855507, 233030075, which gives a(n)/n ratios: 0.2, 0.82, 1.82, 3.42, 6.29, 11.86, 23.30, etc, Question: does the ratio just keep on growing?

Links

Crossrefs

Partial sums of A369055.
Cf. A369054.

Programs

  • PARI
    \\ Needs also program from A369055:
A369057list(up_to) = { my(v=vector(up_to)); s = 0; for(n=1,up_to,s+=A369055(n); v[n] = s); (v); };
v369057 = A369057list(up_to);
A369057(n) = v369057[n];

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);

A369242 Number of representations of 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, 2, 1, 3, 11, 24, 53, 176, 339, 1510, 2573
Offset: 0

Views

Author

Antti Karttunen, Jan 21 2024

Keywords

Crossrefs

Cf. A033312, A369054.
Cf. also A369241, A369245.

Programs

Formula

a(n) = A369054(A033312(n)).
For n >= 4, a(n) = A369055(n!/4).
Previous Showing 11-20 of 25 results. Next

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