A115905 -id:A115905 - cp's OEIS frontend

A115910 Numbers k such that phi(k)*k is a triangular number.

1, 3, 15, 84, 217, 255, 1710, 2967, 5363, 11242, 24066, 27370, 29697, 29953, 60977, 65535, 69324, 103257, 159761, 209305, 228333, 446516, 598559, 615410, 691410, 1231305, 1238358, 1365175, 1467732, 1841154, 1966220, 2081070, 2562370

Offset: 1

Giovanni Resta, Feb 06 2006

nonn

			phi(24066)*24066 = 164611440 = T(18144).

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A002618, A000217, A115905, A115909.

Select[Range[2.6*10^6],OddQ[Sqrt[8#*EulerPhi[#]+1]]&] (* Harvey P. Dale, Jul 11 2017 *)

isok(n) = ispolygonal(n*eulerphi(n), 3); \\ Michel Marcus, Jan 25 2014

A116040 n+phi(n)+phi(phi(n)) is a triangular number.

Original entry on oeis.org

1, 3, 7, 16, 22, 23, 24, 46, 50, 54, 57, 67, 77, 81, 109, 135, 164, 204, 273, 304, 311, 331, 391, 415, 438, 500, 532, 541, 671, 685, 690, 702, 783, 855, 1025, 1130, 1264, 1301, 1375, 1459, 1603, 1667, 1707, 1775, 1864, 1953, 2053, 2055, 2136, 2369

Offset: 1

Giovanni Resta, Feb 13 2006

nonn

			77+phi(77)+phi(phi(77)) = 153 = T(17).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A115905, A116041, A116042, A116043.

tnoQ[n_]:=Module[{eph=EulerPhi[n]},IntegerQ[(Sqrt[8(n+eph+ EulerPhi[ eph])+ 1 ]-1)/2]]; Select[Range[2500],tnoQ] (* Harvey P. Dale, Oct 01 2011 *)

A115906 Numbers k such that k + sigma(k) + phi(k) is a triangular number.

Original entry on oeis.org

1, 2, 5, 7, 9, 16, 144, 186, 410, 656, 680, 805, 963, 968, 1152, 1331, 1419, 1476, 1557, 1608, 2158, 2220, 2376, 2401, 2529, 2840, 3376, 3447, 3570, 3591, 4018, 5030, 5769, 5967, 6530, 6720, 6854, 6955, 7250, 7856, 8004, 8514, 8946, 9144

Offset: 1

Giovanni Resta, Feb 06 2006

nonn

			1331 + sigma(1331) + phi(1331) = 4005 = T(89).

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

Cf. A000010, A000203, A000217, A069066, A115904, A115905.

isok(n) = {my(f=factor(n)); ispolygonal(n + sigma(f) + eulerphi(f), 3);} \\ Amiram Eldar, Apr 06 2023

A116044 n+phi(n)+phi(phi(n)) is a palindrome.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 14, 23, 28, 45, 51, 55, 56, 63, 75, 79, 85, 95, 136, 138, 146, 161, 168, 169, 186, 189, 195, 201, 227, 231, 241, 247, 272, 274, 275, 276, 292, 297, 306, 322, 323, 336, 389, 398, 418, 423, 430, 444, 473, 504, 514, 543, 552, 555, 567, 574, 582

Offset: 1

Giovanni Resta, Feb 13 2006

			227+phi(227)+phi(phi(227)) = 565.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A115905.

filter:= proc(n) local t,x,L;
uses numtheory;
  t:= phi(n);
  x:= n + t + phi(t);
L:= convert(x,base,10);
andmap(i -> L[i]=L[-i], [$1..nops(L)/2]);
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 03 2020

cp's OEIS Frontend

A115910 Numbers k such that phi(k)*k is a triangular number.

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A116040 n+phi(n)+phi(phi(n)) is a triangular number.

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A115906 Numbers k such that k + sigma(k) + phi(k) is a triangular number.

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PARI

A116044 n+phi(n)+phi(phi(n)) is a palindrome.

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