A121048 -id:A121048 - cp's OEIS frontend

A115896 Numbers k such that k + phi(k) is a palindrome.

1, 2, 3, 4, 5, 6, 17, 21, 61, 63, 71, 142, 157, 167, 183, 184, 190, 197, 201, 213, 215, 219, 237, 255, 263, 283, 284, 293, 305, 322, 325, 338, 359, 375, 379, 389, 395, 407, 412, 427, 445, 452, 458, 459, 460, 483, 535, 539, 549, 566, 568, 586, 595, 603, 941

Offset: 1

Giovanni Resta, Feb 06 2006

			183 + phi(183) = 183 + 120 = 303.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A115890, A115895, A115897.

Cf. A121048 (n+phi(n)).

ispali:= proc(n) local L; L:= convert(n,base,10); andmap(t -> L[t]=L[-t], [$1..nops(L)/2]) end proc:
select(t -> ispali(t+numtheory:-phi(t)), [$1..1000]); # Robert Israel, Sep 19 2022

nppQ[n_]:=Module[{idn=IntegerDigits[n+EulerPhi[n]]},idn==Reverse[idn]]; Select[Range[1000],nppQ] (* Harvey P. Dale, Aug 18 2013 *)

ispal(n) = my(d=digits(n)); d == Vecrev(d) \\ A002113
isok(k) = ispal(k+eulerphi(k)) \\ Alexandru Petrescu, Sep 19 2022

A114072 Numbers k such that phi(k) + k is a fourth power.

Original entry on oeis.org

12, 41, 172, 176, 192, 313, 385, 972, 1008, 1201, 2732, 2752, 2816, 3072, 6668, 7321, 14281, 15552, 16128, 25616, 29425, 41761, 43696, 43712, 44032, 45056, 49152, 78732, 78768, 79056, 81648, 84240, 97241, 106688, 123921, 139921

Offset: 1

Giovanni Resta, Feb 13 2006

nonn

			phi(41) + 41 = 81 = 3^4.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000010, A121048, subsequence of A114073.

Select[Range[140000],IntegerQ[Surd[#+EulerPhi[#],4]]&] (* Harvey P. Dale, Sep 27 2020 *)

isok(n) = ispower(eulerphi(n) + n, 4); \\ Michel Marcus, Jan 09 2014

A114073 Numbers k such that phi(k) + k is a square.

Original entry on oeis.org

5, 12, 13, 41, 44, 48, 61, 68, 108, 113, 125, 172, 176, 181, 192, 252, 268, 272, 313, 385, 421, 432, 452, 524, 613, 657, 684, 688, 704, 761, 768, 772, 825, 964, 972, 1008, 1013, 1072, 1088, 1201, 1301, 1332, 1412, 1728, 1729, 1741, 1808, 1861, 2092

Offset: 1

Giovanni Resta, Feb 13 2006

nonn

			phi(41) + 41 = 81 = 9^2.

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

Cf. A000010, A121048, supersequence of A114072.

Select[Range[2500], IntegerQ[Sqrt[EulerPhi[#] + #]] &] (* Amiram Eldar, Jan 18 2024 *)

isok(n) = issquare(eulerphi(n) + n); \\ Michel Marcus, Jan 09 2014

A309835 Numbers k such that k + phi(k) is a repunit.

Original entry on oeis.org

5798663, 5555564201311, 5555574497311, 5555593942711, 66815976110703, 69437045907973255623

Offset: 1

Giovanni Resta, Aug 19 2019

Also in the sequence is 555555555555555555555556288388841217550575591423513701223. - Robert Israel, Aug 20 2019

The number 5975946235638859341313216528710061511 is also in the sequence. - Daniel Suteu, Aug 22 2019

			5798663 is a terms since phi(5798663) = 5312448 and 5798663 + 5312448 = 11111111.

Subsequence of A116018.

Cf. A000010, A002275, A121048, A244444.

isok(k) = my(d=digits(k+eulerphi(k))); vecmin(d)==1 && vecmax(d)==1; \\ Daniel Suteu, Aug 22 2019

a(5) from Daniel Suteu confirmed by Max Alekseyev, Oct 25 2023

a(6) from Max Alekseyev, Nov 30 2023

A063794 Numbers k such that usigma(k) = k + phi(k).

Original entry on oeis.org

2, 28, 1368, 1700, 342000, 2061000, 2120832, 65834560, 89082000

Offset: 1

Jason Earls, Aug 18 2001

No other terms < 2700000000. - Jud McCranie, Nov 04 2001

Cf. A000010, A034448, A121048.

us(n) = sumdiv(n,d, if(gcd(d,n/d)==1,d));
for(n=1,10^8, if(us(n)==n+eulerphi(n),print1(n, ", ")))

More terms from Jud McCranie, Nov 04 2001

Offset corrected by Mohammed Yaseen, Jul 17 2023

A383044 Numbers m such that phi(m) + phi(m+phi(m)) = m where phi is the Euler totient function.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 20, 24, 28, 32, 40, 48, 56, 64, 70, 80, 94, 96, 112, 128, 140, 160, 188, 192, 224, 256, 280, 320, 376, 384, 448, 512, 560, 640, 752, 768, 896, 1024, 1120, 1280, 1504, 1536, 1792, 2048, 2240, 2560, 3008, 3072, 3584, 4096, 4480, 5120, 6016, 6144, 7168, 8192, 8960

Offset: 1

Michel Marcus, Apr 14 2025

nonn

Empirical observation: Let phi(m) + phi(m + phi(m)) = A*m / B, GCD(A,B) = 1. For some (A,B) like (1,1) - this sequence, (2,3), (4,5), (4,7), (5,7), (7,9), (14,9), (8,11), ..., there exists (finitely/infinitely many ?) solutions to phi(m) + phi(m + phi(m)) = A*m / B. Experimentally it looks like for m = 3*A033845(n) = 18*A003586(n), phi(m) + phi(m + phi(m)) = 7*m / 9. - Ctibor O. Zizka, Apr 25 2025

Stefan Steinerberger, On an iterated arithmetic function problem of Erdos and Graham, arXiv:2504.08023 [math.NT], 2025.

Cf. A000010, A121048.

q[m_] := Module[{phi = EulerPhi[m]}, phi + EulerPhi[m + phi] == m]; Select[Range[10000], q] (* Amiram Eldar, Apr 14 2025 *)

isok(m) = eulerphi(m) + eulerphi(m+eulerphi(m)) == m;

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A115896 Numbers k such that k + phi(k) is a palindrome.

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A114072 Numbers k such that phi(k) + k is a fourth power.

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A114073 Numbers k such that phi(k) + k is a square.

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A309835 Numbers k such that k + phi(k) is a repunit.

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A063794 Numbers k such that usigma(k) = k + phi(k).

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A383044 Numbers m such that phi(m) + phi(m+phi(m)) = m where phi is the Euler totient function.

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Mathematica

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