A189762 -id:A189762 - cp's OEIS frontend

A099303 Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.

0, 0, 4, 6, 9, 10, 15, 14, 25, 0, 35, 22, 49, 26, 55, 0, 77, 34, 91, 38, 121, 0, 143, 46, 169, 27, 187, 0, 221, 58, 247, 62, 289, 0, 323, 0, 361, 74, 391, 42, 437, 82, 403, 86, 529, 0, 551, 94, 589, 63, 667, 0, 713, 106, 703, 0, 841, 70, 899, 118, 961, 122, 943, 0, 1073, 0

Offset: 2

T. D. Noe, Oct 12 2004

nonn

This is the largest member of the set I(n) in the paper by Ufnarovski and Ahlander. They show that a(n) <= (n/2)^2.

Because this sequence is quite different for even and odd n, it is bisected into A102084 and A189762. The upper bound for odd n appears to be (n/3)^(3/2), which is attained when n = 3p^2 for primes p>5. - T. D. Noe, Apr 27 2011

See A003415

T. D. Noe, Table of n, a(n) for n=2..1000

Cf. A003415 (arithmetic derivative of n), A099302 (number of solutions to x' = n), A098699 (least x such that x' = n), A098700 (n such that x' = n has no integer solution).

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[x=Max[Flatten[Position[d1, n]]]; If[x>-Infinity, x, 0], {n, 2, 400}]

from sympy import factorint
def A099303(n):
    for m in range(n**2>>2,0,-1):
        if sum((m*e//p for p,e in factorint(m).items())) == n:
            return m
    return 0 # Chai Wah Wu, Sep 12 2022

A102084 a(1) = 0; for n>0, write 2n=p+q (p, q prime), pq maximal; then a(n)=pq (see A073046).

Original entry on oeis.org

0, 4, 9, 15, 25, 35, 49, 55, 77, 91, 121, 143, 169, 187, 221, 247, 289, 323, 361, 391, 437, 403, 529, 551, 589, 667, 713, 703, 841, 899, 961, 943, 1073, 1147, 1189, 1271, 1369, 1363, 1517, 1591, 1681, 1763, 1849, 1927, 2021, 1891, 2209, 2279, 2257, 2491

Offset: 1

Michael Taktikos, Feb 16 2005

nonn

For n>1, largest semiprime whose sum of prime factors = 2n. Assumes the Goldbach conjecture is true. Also the largest semiprime <= n^2.

Also the greatest integer x such that x' = 2*n, or 0 if there is no such x, where x' is the arithmetic derivative (A003415). Bisection of A099303. The only even number without an anti-derivative is 2. All terms are <= n^2, with equality only when n is prime. In fact a(n) = n^2 - k^2, where k is the least number such that both n-k and n+k are prime; k = A047160(n). It appears that the anti-derivatives of even numbers are overwhelmingly semiprimes of the form n^2 - k^2. For example, 1000 has 28 anti-derivatives, all of this form. Sequence A189763 lists the even numbers that have anti-derivatives not of this form. - T. D. Noe, Apr 27 2011

			n=13: 2n = 26; 26 = 23 + 3 = 19 + 7 = 13 + 13; 13*13 = maximal => p*q = 13*13 = 169.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A073046, A003415, A047160, A099303, A189762.

f[n_] := Block[{pf = FactorInteger[n]}, If[Plus @@ Last /@ pf == 2, If[ Length[pf] == 2, Plus @@ First /@ pf, 2pf[[1, 1]]], 0]]; t = Table[0, {51}]; Do[a = f[n]; If[ EvenQ[a] && 0 < a < 104, t[[a/2]] = n], {n, 2540}]; t (* Robert G. Wilson v, Jun 14 2005 *)
Table[k = 0; While[k < n && (! PrimeQ[n - k] || ! PrimeQ[n + k]), k++]; If[k == n, 0, (n - k)*(n + k)], {n, 100}] (* T. D. Noe, Apr 27 2011 *)

a(n) = n^2 - A047160(n)^2. - Jason Kimberley, Jun 26 2012

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A189763 Even numbers n such that solutions to x' = n are not all semiprimes, where x' denotes the arithmetic derivative (A003415).

Original entry on oeis.org

12, 16, 24, 32, 44, 48, 56, 60, 68, 72, 80, 92, 96, 108, 112, 120, 124, 128, 140, 152, 156, 162, 164, 168, 176, 188, 192, 212, 216, 220, 236, 240, 244, 248, 252, 272, 284, 288, 296, 316, 318, 320, 324, 332, 336, 350, 356, 360, 368, 378, 380, 384, 392, 408

Offset: 1

T. D. Noe, Apr 27 2011

nonn

It appears that the anti-derivatives of even numbers n are overwhelmingly semiprimes of the form (n/2)^2 - k^2. For example, 1000 has 28 anti-derivatives, all semiprimes.

Cf. A003415, A189762, A189764.

cp's OEIS Frontend

A099303 Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.

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A102084 a(1) = 0; for n>0, write 2n=p+q (p, q prime), pq maximal; then a(n)=pq (see A073046).

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A189763 Even numbers n such that solutions to x' = n are not all semiprimes, where x' denotes the arithmetic derivative (A003415).

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cp's OEIS Frontend

A099303 Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

A102084 a(1) = 0; for n>0, write 2n=p+q (p, q prime), p*q maximal; then a(n)=p*q (see A073046).

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Author

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Programs

Mathematica

Formula

Extensions

A189763 Even numbers n such that solutions to x' = n are not all semiprimes, where x' denotes the arithmetic derivative (A003415).

Views

Author

Keywords

Comments

Crossrefs

A102084 a(1) = 0; for n>0, write 2n=p+q (p, q prime), pq maximal; then a(n)=pq (see A073046).