A098700 -id:A098700 - cp's OEIS frontend

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

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.

A189555 Numbers n such that x' = n has two solutions, where x' is the arithmetic derivative (A003415) of x.

Original entry on oeis.org

10, 12, 14, 18, 20, 21, 28, 31, 38, 39, 45, 55, 61, 71, 81, 87, 101, 103, 111, 119, 123, 129, 131, 147, 183, 185, 199, 211, 213, 215, 241, 243, 255, 269, 291, 297, 299, 327, 339, 343, 351, 355, 359, 361, 363, 381, 395, 399, 401, 411, 421, 433, 439, 471, 493

Offset: 1

T. D. Noe, Apr 24 2011

nonn

Ufnarovski and Ahlander conjecture that this sequence, and any such sequence that has numbers n such that x' = n has k solutions, is infinite. See A098700 and A189481 for the k=0 and 1 cases. It appears that the only even terms here are 10, 12, 14, 18, 20, 28, and 38. The prime terms are in A189556.

See A003415.

T. D. Noe, Table of n, a(n) for n = 1..1000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Cf. A003415, A098700 (no solution), A099302, A189481 (1 solution).

n such that A099302(n) = 2.

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

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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A189555 Numbers n such that x' = n has two solutions, 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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