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 1-10 of 25 results. Next

A369064 Record values in A369054.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 10, 12, 13, 15, 24, 25, 31, 33, 34, 35, 40, 43, 44, 45, 46, 49, 51, 54, 57, 58, 63, 65, 67, 68, 69, 71, 76, 78, 79, 80, 81, 83, 84, 85, 87, 95, 99, 103, 105, 106, 109, 120, 125, 132, 136, 152, 153, 157, 159, 162, 166, 171, 178, 180, 181, 186, 198, 213, 217, 219, 226, 228, 229, 231, 238, 246, 261, 263, 264, 270, 296, 302, 310
Offset: 1

Views

Author

Antti Karttunen, Jan 21 2024

Keywords

Crossrefs

Also record values in A369055.
Cf. A369054, A369063.

Formula

a(n) = A369054(A369063(n)).

A369063 Positions of records in A369054.

Original entry on oeis.org

0, 27, 75, 231, 311, 551, 1031, 1991, 2231, 3191, 5591, 14951, 17111, 35591, 39191, 49271, 50951, 76151, 103991, 107159, 114791, 117911, 119831, 166871, 180311, 201431, 255191, 329111, 366791, 391751, 400391, 435431, 465911, 514751, 516071, 576551, 587831, 599231, 712631, 789311, 890711, 890951, 948191, 1130951, 1579751
Offset: 1

Views

Author

Antti Karttunen, Jan 21 2024

Keywords

Comments

Where the number of ways to express a natural number as a sum (p*q + p*r + q*r) with three (not necessarily distinct) odd primes p, q and r attains record maximum.

Crossrefs

Cf. A369054, A369064 (record values).

A046316 Numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.

Original entry on oeis.org

27, 45, 63, 75, 99, 105, 117, 125, 147, 153, 165, 171, 175, 195, 207, 231, 245, 255, 261, 273, 275, 279, 285, 325, 333, 343, 345, 357, 363, 369, 385, 387, 399, 423, 425, 429, 435, 455, 465, 475, 477, 483, 507, 531, 539, 549, 555, 561, 575, 595, 603, 605
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Links

Crossrefs

A369979 sorted into ascending order.
Subsequence of A014612 and of A046340.
Cf. A255646 (final digits), A369054, A369058 (characteristic function), A369252 [= A003415(a(n))].
Subsequences: A046389, A046373, A046405, A075814, A338469, A338556, A338557, A369246.

Programs

  • Haskell
    a046316 n = a046316_list !! (n-1)
a046316_list = filter ((== 3) . a001222) [1, 3 ..]
-- Reinhard Zumkeller, May 05 2015
  • PARI
    list(lim)=my(v=List(),pq); forprime(p=3,lim\9, forprime(q=3,min(lim\3\p,p), pq=p*q; forprime(r=3,lim\pq, listput(v, pq*r)))); Set(v) \\ Charles R Greathouse IV, Aug 23 2017
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A046316(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b+1 for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

Extensions

Definition clarified by N. J. A. Sloane, Dec 19 2017

A099302 Number of integer solutions to x' = n, where x' is the arithmetic derivative of x.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 3, 0, 2, 1, 2, 2, 3, 0, 4, 1, 3, 1, 2, 0, 3, 2, 4, 1, 4, 0, 4, 0, 2, 2, 3, 1, 4, 1, 4, 2, 4, 0, 6, 1, 4, 1, 3, 0, 5, 2, 4, 0, 4, 1, 7, 2, 3, 1, 5, 0, 6, 0, 3, 1, 5, 2, 7, 1, 5, 3, 5, 1, 7, 0, 6, 2, 5, 0, 8, 1, 5, 2, 4, 0, 9, 3, 6, 0, 5, 1, 8, 0, 3, 1, 6, 2, 8, 2, 5, 1, 6
Offset: 2

Views

Author

T. D. Noe, Oct 12 2004, Apr 24 2011

Keywords

Comments

This is the i(n) function in the paper by Ufnarovski and Ahlander. Note that a(1) is infinite because all primes satisfy x' = 1. The plot shows the great difference in the number of solutions for even and odd n. Also compare sequence A189558, which gives the least number have n solutions, and A189560, which gives the least such odd number.
It appears that there are a finite number of even numbers having a given number of solutions. This conjecture is explored in A189561 and A189562.

References

Links

Crossrefs

Cf. A002620, A003415 (arithmetic derivative of n), A099303 (greatest x such that x' = n), A098699 (least x such that x' = n), A098700 (n such that x' = n has no integer solution), A239433 (n such that x' = n has at least one solution).
Cf. A002375 (a lower bound for even n), A369054 (a lower bound for n of the form 4m+3).

Programs

  • Haskell
    a099302 n = length $ filter (== n) $ map a003415 [1 .. a002620 n]
-- Reinhard Zumkeller, Mar 18 2014
  • Mathematica
    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}]; Table[Count[d1, n], {n, 2, 400}]
  • PARI
    up_to = 100000; \\ A002620(10^5) = 2500000000
A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A099302list(up_to) = { my(d,c,v=vector(up_to)); for(i=1, A002620(up_to), d = A003415(i); if(d>1 && d<=up_to, v[d]++)); (v); };
v099302 = A099302list(up_to);
A099302(n) = v099302[n]; \\ Antti Karttunen, Jan 21 2024
  • Python
    from sympy import factorint
def A099302(n): return sum(1 for m in range(1,(n**2>>2)+1) if sum((m*e//p for p,e in factorint(m).items())) == n) # Chai Wah Wu, Sep 12 2022

Formula

a(A098700(n)) = 0; a(A239433(n)) > 0. - Reinhard Zumkeller, Mar 18 2014
From Antti Karttunen, Jan 21 2024: (Start)
a(n) = Sum_{i=1..A002620(n)} [A003415(i)==n], where [ ] is the Iverson bracket.
a(2n) >= A002375(n), a(2n+1) >= A369054(2n+1).
(End)

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.

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.

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

Views

Author

Antti Karttunen, Jan 22 2024

Keywords

Links

Crossrefs

Intersection of A017557 and A369056, multiples of 3 in the latter.
Cf. A369054, A369055.

Programs

A369251 Numbers that have at least one representation as a sum (p*q + p*r + 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

Views

Author

Antti Karttunen, Jan 22 2024

Keywords

Comments

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

Links

Crossrefs

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).

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])));

Formula

{k | A369054(k) > 0}.

A369461 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, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 1, 1, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 3, 0, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 2, 0, 0, 2, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 2, 1, 0, 1, 1, 0, 0, 0, 4, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0
Offset: 1

Views

Author

Antti Karttunen, Jan 23 2024

Keywords

Comments

The sequence seems to contain an infinite number of zeros. See A369451 for the cumulative sum, and comments there.
Question: Are there any sections of this sequence, with parameters k >= 2, 0 <= i < k, for which a((k*n)-i) = 0 for all n >= 1? - Antti Karttunen, Nov 20 2024

Links

Crossrefs

Trisection of A369055.
Cf. A017605, A369054, A369451 (partial sums), A369460, 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])));
A369461(n) = A369054((12*n)-5);

Formula

a(n) = A369054(A017605(n-1)) = A369054((12*n)-5).
a(n) = A369055((3*n)-1).
Showing 1-10 of 25 results. Next

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