A189483 -id:A189483 - cp's OEIS frontend

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

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

A189441 Primes that are the arithmetic derivative (A003415) of some number.

Original entry on oeis.org

5, 7, 13, 19, 31, 41, 43, 59, 61, 71, 73, 101, 103, 109, 113, 131, 139, 151, 167, 181, 191, 193, 199, 211, 227, 229, 239, 241, 251, 263, 269, 271, 281, 283, 293, 311, 313, 331, 347, 349, 359, 383, 401, 419, 421, 431, 433, 439, 461, 463, 467, 479, 487, 491

Offset: 1

T. D. Noe, Apr 22 2011

nonn

Complement of A189483 in the primes. Sequence A157037 has the numbers that produce these primes. It is possible for several numbers to produce the same prime. In fact, for each of these primes there is a tree of numbers whose (perhaps multiple) derivatives equal the prime.

Every upper twin prime is here because for such p, the derivative of 2(p-2) is p.

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

Cf. A003415, A099302, A157037.

Primes p such that A099302(p) > 0.

A351078 First noncomposite number reached when iterating the map x -> x', when starting from x = n, or 0 if no such number is ever reached. Here x' is the arithmetic derivative of x, A003415.

Original entry on oeis.org

0, 1, 2, 3, 0, 5, 5, 7, 0, 5, 7, 11, 0, 13, 5, 0, 0, 17, 7, 19, 0, 7, 13, 23, 0, 7, 0, 0, 0, 29, 31, 31, 0, 5, 19, 0, 0, 37, 7, 0, 0, 41, 41, 43, 0, 0, 7, 47, 0, 5, 0, 0, 0, 53, 0, 0, 0, 13, 31, 59, 0, 61, 5, 0, 0, 7, 61, 67, 0, 0, 59, 71, 0, 73, 0, 0, 0, 7, 71, 79, 0, 0, 43, 83, 0, 13, 0, 0, 0, 89, 0, 0, 0, 19, 5

Offset: 0

Antti Karttunen, Feb 11 2022

Primes of A189483 occur only once, on the corresponding indices, while A189441 may also occur in other positions.

There are interesting white "filament-like regions" in the scatter plot.

			For n = 15, if we iterate with A003415, we get a path 15 -> 8 -> 12 -> 16 -> 32 -> 80 -> 176 -> 368 -> ..., where the terms just keep on growing without ever reaching a prime or 1, therefore a(15) = 0.
For n = 18, its path down to zero, when iterating A003415 is: 18 -> 21 -> 10 -> 7 -> 1 -> 0, and the first noncomposite term on the path is prime 7, therefore a(18) = 7.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003415, A189441, A189483, A351079, A351259 [= a(A351255(n))].
Cf. A099309 (positions of zeros after the initial one at a(0)=0), A328115 (positions of 5's), A328117 (positions of 7's).
Cf. also A327968.

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A351078(n) = { while(n>1&&!isprime(n), n = A003415checked(n)); (n); };

For all n, a(4*n) = a(27*n) = a((p^p)*n) = a(A099309(n)) = 0.

a(p) = p for all primes p.

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.

A189554 Composite numbers n such that x' = n has no integer solution, where x' is the arithmetic derivative (A003415) of x.

Original entry on oeis.org

35, 57, 65, 93, 117, 125, 145, 177, 189, 205, 207, 209, 217, 219, 237, 245, 261, 289, 303, 305, 323, 325, 345, 377, 387, 393, 413, 415, 427, 429, 453, 473, 477, 485, 497, 513, 515, 517, 529, 531, 533, 537, 553, 561, 597, 605, 625, 629, 639, 657, 665, 681

Offset: 1

T. D. Noe, Apr 24 2011

nonn

These are the composite terms of A098700. The prime terms of A098700 are in A189483. Apparently all terms are odd.

See A003415.

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

Cf. A003415, A098700, A099302, A189483.

Composite k such that A099302(k) = 0.

A351095 Semiprimes that are not the arithmetic derivative (A003415) of any integer.

Original entry on oeis.org

35, 57, 65, 93, 145, 177, 205, 209, 217, 219, 237, 289, 303, 305, 323, 377, 393, 413, 415, 427, 453, 473, 485, 497, 515, 517, 529, 533, 537, 553, 597, 629, 681, 697, 699, 713, 749, 781, 785, 793, 817, 835, 849, 869, 895, 917, 933, 965, 989, 1037, 1057, 1059, 1077, 1081, 1133, 1137, 1145, 1149, 1159, 1169, 1227, 1243

Offset: 1

Antti Karttunen, Feb 03 2022

nonn

17^2 = 289 and 23^2 = 529 are the first squares present.

Complement of A351096 in A001358. Subsequence of A098700 and of A189554.

Cf. A003415, A189483.

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]));
isA351095(n) = if(2!=bigomega(n), 0, for(k=1,A002620(n),if(A003415(k)==n,return(0))); (1));

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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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A189441 Primes that are the arithmetic derivative (A003415) of some number.

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A351078 First noncomposite number reached when iterating the map x -> x', when starting from x = n, or 0 if no such number is ever reached. Here x' is the arithmetic derivative of x, A003415.

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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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A189554 Composite numbers n such that x' = n has no integer solution, where x' is the arithmetic derivative (A003415) of x.

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A351095 Semiprimes that are not the arithmetic derivative (A003415) of any integer.

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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.