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 1-3 of 3 results.

A327862 Numbers whose arithmetic derivative is of the form 4k+2, cf. A003415.

Original entry on oeis.org

9, 21, 25, 33, 49, 57, 65, 69, 77, 85, 93, 121, 129, 133, 135, 141, 145, 161, 169, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265, 289, 301, 305, 309, 315, 321, 329, 341, 351, 361, 365, 375, 377, 381, 393, 413, 417, 437, 445, 453, 459, 469, 473, 481, 485, 489, 493, 495, 497, 501, 505, 517, 529, 533, 537
Offset: 1

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Author

Antti Karttunen, Sep 30 2019

Keywords

Comments

All terms are odd because the terms A068719 are either multiples of 4 or odd numbers.
Odd numbers k for which A064989(k) is one of the terms of A358762. - Antti Karttunen, Nov 30 2022
The second arithmetic derivative (A068346) of these numbers is odd. See A235991. - Antti Karttunen, Feb 06 2024

Links

Crossrefs

Setwise difference A235992 \ A327864.
Setwise difference A046337 \ A360110.
Union of A369661 (k' has an even number of prime factors) and A369662 (k' has an odd number of prime factors).
Subsequences: A001248 (from its second term onward), A108181, A327978, A366890 (when sorted into ascending order), A368696, A368697.
Cf. A003415, A064989, A068346, A068719, A327863, A327865, A353495 (characteristic function).
Cf. also A235991, A358762, A358772, A358774.

Programs

  • PARI
    A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
isA327862(n) = (2==(A003415(n)%4));
k=1; n=0; while(k<105, if(isA327862(n), print1(n, ", "); k++); n++);

A369661 Numbers k whose arithmetic derivative k' is of the form 4m+2, and k' has an even number of prime factors.

Original entry on oeis.org

9, 21, 25, 33, 49, 57, 69, 85, 93, 121, 129, 133, 145, 169, 177, 205, 213, 217, 237, 249, 253, 265, 289, 309, 329, 361, 375, 393, 417, 445, 459, 469, 473, 489, 493, 505, 517, 529, 533, 553, 565, 573, 581, 597, 629, 633, 669, 685, 697, 713, 753, 781, 783, 793, 813, 817, 819, 841, 865, 869, 875, 889, 913, 933, 949, 961
Offset: 1

Views

Author

Antti Karttunen, Feb 06 2024

Keywords

Comments

Equally, numbers k whose arithmetic derivative k' is congruent to 2 modulo 4 and A276085(k') is congruent to 3 modulo 4.
Numbers k such that A003415(k) is in A369966.
For all n >= 1, A003415((1/2)*A003415(a(n))) is odd.

Links

Crossrefs

Setwise difference A327862 \ A369662.
Cf. A003415, A369660 (characteristic function), A369966.
Subsequences: A108181, A369663 (terms of the form 4m+3).

Programs

A369664 Numbers k of the form 4m+1, whose arithmetic derivative k' is of the form 4u+2, and k' has an odd number of prime factors.

Original entry on oeis.org

65, 77, 141, 161, 185, 201, 209, 221, 301, 305, 321, 341, 365, 377, 381, 413, 437, 453, 481, 485, 497, 501, 537, 545, 589, 649, 681, 689, 717, 721, 729, 737, 745, 749, 785, 789, 849, 893, 901, 905, 917, 921, 989, 1037, 1073, 1081, 1101, 1121, 1133, 1141, 1157, 1165, 1169, 1189, 1205, 1253, 1261, 1293, 1313, 1317
Offset: 1

Views

Author

Antti Karttunen, Feb 06 2024

Keywords

Links

Crossrefs

Intersection of A016813 and A369662.
Subsequence of A369666.
Cf. A003415, A351228.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA369664(n) = if(1!=(n%4), 0, my(d=A003415(n)); (2==(d%4) && (bigomega(d)%2)));
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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