A189554 -id:A189554 - 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

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));

cp's OEIS Frontend

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

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