A288452 -id:A288452 - cp's OEIS frontend

A177204 Pseudoperfect totient numbers (A288452) which are not powers of primes (A000961).

15, 33, 35, 39, 51, 55, 65, 69, 77, 85, 87, 111, 115, 119, 123, 141, 143, 153, 155, 159, 161, 175, 177, 183, 187, 201, 205, 209, 213, 219, 221, 235, 245, 247, 249, 253, 255, 265, 267, 287, 291, 295, 299, 303, 305, 309, 319, 321, 323, 325, 327, 329, 339, 341, 363, 371, 377, 391, 393, 403, 407, 411, 413, 415

Offset: 1

Amiram Eldar and Robert G. Wilson v, Jul 02 2017

A proper subset of A288452. All odd primes and all powers of 3 are pseudoperfect totient numbers. First term which has k factors counting multiplicity >1: 15, 153, 875, 4375, 62451, etc. First term which has k distinct factors >1: 15, 255, 21505, etc.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A288452, A000961.

fQ[n_] := Module[{tots = Rest@ NestWhileList[ EulerPhi, n, # > 1 &]}, MemberQ[Total /@ Subsets[tots, Length[ tots]], n]]; Select[ Range[ 3, 440, 2], fQ@# && !PrimePowerQ@# &]

A288453 Weird totient numbers: totient abundant numbers (A286265) that are not pseudoperfect totient numbers (A288452).

Original entry on oeis.org

91, 95, 133, 145, 185, 203, 215, 217, 259, 275, 301, 335, 343, 355, 365, 395, 427, 469, 497, 545, 551, 553, 575, 635, 637, 649, 655, 703, 725, 755, 763, 767, 785, 815, 817, 833, 865, 889, 893, 905, 917, 931, 949, 955, 973, 985, 995, 1007, 1027, 1057, 1073

Offset: 1

Amiram Eldar, Jun 09 2017

nonn

Analogous to A006037 (weird numbers) as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).

			The set of iterated phi of 91 is {72, 24, 8, 4, 2, 1} and none of its subsets sums to 91.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000396, A006037, A092693, A082897, A286265.

pseudoPerfectTotQ[n_] := Module[{tots = Most[Rest[FixedPointList[EulerPhi@# &, n]]]}, MemberQ[Total /@ Subsets[tots, Length[tots]], n]];
totAbundantQ[n_] := Plus @@ FixedPointList[EulerPhi@# &, n] > 2*n + 1;
weirdTotient[n_] := totAbundantQ[n] && ! pseudoPerfectTotQ[n];
Select[Range[1100], weirdTotient]

A335121 Admirable totient numbers: numbers that are equal to the sum of their iterated phi, with one of them taken with a minus sign.

Original entry on oeis.org

5, 7, 33, 35, 55, 87, 95, 175, 201, 215, 219, 245, 531, 747, 927, 939, 1047, 1295, 1463, 1473, 1551, 1855, 2015, 2103, 2421, 2431, 2547, 2619, 2631, 2765, 3535, 4833, 5067, 5215, 7655, 7743, 7851, 10503, 11127, 11307, 13055, 13707, 16247, 16593, 17805, 18471

Offset: 1

Amiram Eldar, May 24 2020

nonn

Analogous to A111592 (admirable numbers) as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).

			5 is a term since the values of the iterated phi of 5 are 4, 2 and 1 and 5 = 4 + 2 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..171 (terms below 10^9)

Subsequence of A286265.

Cf. A000010, A053478, A082897, A111592, A288452.

admTotQ[n_] := Module[{s = Most @ Rest @ FixedPointList[EulerPhi, n]}, (ab = Plus @@ s - n) > 0 && EvenQ[ab] && ab/2 < n && MemberQ[s, ab/2]]; Select[Range[8000], admTotQ]

cp's OEIS Frontend

A177204 Pseudoperfect totient numbers (A288452) which are not powers of primes (A000961).

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A288453 Weird totient numbers: totient abundant numbers (A286265) that are not pseudoperfect totient numbers (A288452).

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A335121 Admirable totient numbers: numbers that are equal to the sum of their iterated phi, with one of them taken with a minus sign.

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Mathematica