A369464 -id:A369464 - 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).

A098700 Numbers n such that x' = n has no integer solution, where x' is the arithmetic derivative of x.

Original entry on oeis.org

2, 3, 11, 17, 23, 29, 35, 37, 47, 53, 57, 65, 67, 79, 83, 89, 93, 97, 107, 117, 125, 127, 137, 145, 149, 157, 163, 173, 177, 179, 189, 197, 205, 207, 209, 217, 219, 223, 233, 237, 245, 257, 261, 277, 289, 303, 305, 307, 317, 323, 325, 337, 345, 353, 367, 373

Offset: 1

Robert G. Wilson v, Sep 21 2004

nonn

If x' = n has solutions, they occur for x <= (n/2)^2. - T. D. Noe, Oct 12 2004
The prime and composite terms are in A189483 and A189554, respectively.
A099302(a(n)) = 0. - Reinhard Zumkeller, Mar 18 2014

T. D. Noe, Table of n, a(n) for n = 1..5000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003. (See p. 7.)

Cf. A003415 (arithmetic derivative of n), A099302 (number of solutions to x' = n), A099303 (greatest x such that x' = n), A098699 (least x such that x' = n).
Cf. A239433 (complement), A002620.
Subsequence of A369464.

a098700 n = a098700_list !! (n-1)
a098700_list = filter
   (\z -> all (/= z) $ map a003415 [1 .. a002620 z]) [2..]
-- Reinhard Zumkeller, Mar 18 2014

a[1] = 0; a[n_] := Block[{f = Transpose[ FactorInteger[ n]]}, If[ PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; b = Table[ -1, {500}]; b[[1]] = 1; Do[c = a[n]; If[c < 500 && b[[c + 1]] == 0, b[[c + 1]] = n], {n, 10^6}]; Select[ Range[500], b[[ # ]] == 0 &]
dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; d1=Table[dn[n], {n, 40000}]; Select[Range[400], 0==Count[d1, # ]&]

list(lim)=my(v=List()); lim\=1; forfactored(n=1, lim^2, my(f=n[2],t); listput(v, n[1]*sum(i=1, #f~, f[i, 2]/f[i, 1]))); setminus([1..lim], Set(v)); \\ Charles R Greathouse IV, Oct 21 2021

from itertools import count, islice
from sympy import factorint
def A098700_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda m:sum((m*e//p for p,e in factorint(m).items())) != n,range(1,(n**2>>1)+1))),count(max(startvalue,2)))
A098700_list = list(islice(A098700_gen(),30)) # Chai Wah Wu, Sep 12 2022

Corrected and extended by T. D. Noe, Oct 12 2004

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)).

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}.

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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A098700 Numbers n such that x' = n has no integer solution, where x' is the arithmetic derivative of x.

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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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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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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.

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

A369251 Numbers that have 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.