A033631 -id:A033631 - cp's OEIS frontend

A190503 Numbers k such that sigma(phi(k)) divides sigma(k).

1, 2, 6, 12, 14, 22, 24, 28, 44, 46, 48, 56, 68, 87, 88, 92, 94, 96, 112, 118, 166, 174, 176, 184, 188, 192, 214, 224, 236, 332, 334, 352, 358, 362, 368, 376, 384, 390, 410, 428, 448, 454, 472, 526, 664, 668, 694, 704, 716, 718, 736, 752, 766, 768, 856, 896

Offset: 1

T. D. Noe, May 11 2011

nonn

These numbers appear indirectly in A067740, which seeks the least k such that sigma(k)/sigma(phi(k)) = n. Most of these numbers are even. The odd terms (1, 87, 1257, 41559, 56679, ...) all appear to produce sigma(k)/sigma(phi(k)) = 1.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000010 (phi), A000203 (sigma), A062402, A067740.
Cf. A033631 (k such that sigma(k)/sigma(phi(k)) = 1).
Cf. A066831 (k such that sigma(k) divides sigma(phi(k))).

Select[Range[1000], IntegerQ[DivisorSigma[1,#]/DivisorSigma[1,EulerPhi[#]]] &]

is(k) = {my(f = factor(k), s = sigma(f), p = eulerphi(f)); !(s % sigma(p));} \\ Amiram Eldar, May 17 2024

A255199 Numbers k such that mu(k) = mu(phi(k)) where mu(k) is the Möbius function and phi(k) is Euler's totient function.

Original entry on oeis.org

1, 3, 8, 12, 14, 16, 20, 22, 24, 25, 27, 28, 31, 32, 36, 40, 43, 44, 45, 46, 48, 50, 52, 54, 56, 60, 63, 64, 67, 68, 71, 72, 75, 76, 79, 80, 81, 84, 88, 90, 92, 94, 96, 99, 100, 103, 104, 108, 112, 116, 117, 118, 120, 124, 125, 126, 128, 131, 132, 135, 136, 139

Offset: 1

Tom Edgar, Feb 16 2015

nonn

If k and phi(k) are both not squarefree then k is in the list.
A prime p is in the list if p - 1 is squarefree and bigomega(p - 1) = A001222(p - 1) is odd.
It follows that the subsequence of primes is A078330. - Bernard Schott, Apr 03 2021

			8 is in the list since mu(8) = 0 and mu(phi(8)) = mu(4) = 0.
7 is not in the list since mu(7) = -1 and mu(phi(7)) = mu(6) = 1.

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

Cf. A000010, A008683, A033631, A013929, A078330.

Select[Range[200], MoebiusMu[#] == MoebiusMu[EulerPhi[#]] &] (* Alonso del Arte, Feb 16 2015 *)

for(n=1, 140, if(moebius(n) == moebius(eulerphi(n)), print1(n,", "))) \\  Indranil Ghosh, Mar 11 2017

[n for n in [1..1000] if moebius(n)==moebius(euler_phi(n))]

A255219 Squarefree numbers k such that mu(k) = mu(phi(k)) where mu(k) is the Möbius function and phi(k) is Euler's totient function.

Original entry on oeis.org

1, 3, 14, 22, 31, 43, 46, 67, 71, 79, 94, 103, 118, 131, 139, 166, 191, 214, 223, 239, 283, 311, 334, 358, 367, 419, 422, 431, 439, 443, 454, 499, 526, 599, 607, 619, 643, 647, 659, 662, 683, 694, 718, 743, 766, 787, 823, 827, 907, 926, 934, 947, 958, 971, 1006

Offset: 1

Tom Edgar, Feb 17 2015

nonn

A prime p is a term in the sequence if p - 1 is squarefree and bigomega(p - 1) = A001222(p - 1) is odd (see A078330).

			31 is a term since mu(31) = -1 and mu(phi(31)) = mu(30) = -1.
7 is not a term since mu(7) = -1 and mu(phi(7)) = mu(6) = 1.
24 is not a term since mu(24) = 0 (i.e., 24 is not squarefree).

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

Cf. A000010, A001222, A005117, A008683, A033631, A078330, A255199.

Select[Range[1000], Abs[MoebiusMu[#] + MoebiusMu[EulerPhi[#]]] == 2 &] (* Alonso del Arte, Feb 17 2015 *)

for(n=1, 1006, if(abs(moebius(n) + moebius(eulerphi(n))) == 2, print1(n,", "))) \\ Indranil Ghosh, Mar 10 2017

[n for n in [1..1006] if moebius(n)==moebius(euler_phi(n)) if moebius(n)!=0]

A333338 Numbers k such that sigma_2(k) = sigma_2(phi(k)).

Original entry on oeis.org

1, 7, 11891, 130801, 273493, 1438811, 3008423, 6290339, 15826921, 33092653, 69193729, 144677797, 174096131, 364019183, 761131019, 1591455767, 1915057441, 3327589331, 4004211013, 8372441209, 17506013437, 21065631851, 36603482641, 44046321143, 76534554613, 92096853299

Offset: 1

Ivan N. Ianakiev, Mar 14 2020

nonn

The sequence is infinite since it contains all the numbers of the form 11^i*23^j*47 for i,j > 0. Up to 10^11 the only terms not of this form are 1 and 7. - Giovanni Resta, Mar 15 2020

			50 = 1^2 + 7^2 (sum of the squares of the divisors of 7) = 1^2 + 2^2 + 3^2 + 6^2 (sum of the squares of the divisors of 6 = phi(7)). So 7 is in the sequence.

Cf. A000010, A001157, A070418, A033631.

Select[Range[10!],DivisorSigma[2,#]==DivisorSigma[2,EulerPhi[#]]&]

isok(m) = sigma(m, 2) == sigma(eulerphi(m), 2); \\ Michel Marcus, Mar 15 2020

More terms from Giovanni Resta, Mar 15 2020

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A190503 Numbers k such that sigma(phi(k)) divides sigma(k).

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A255199 Numbers k such that mu(k) = mu(phi(k)) where mu(k) is the Möbius function and phi(k) is Euler's totient function.

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A255219 Squarefree numbers k such that mu(k) = mu(phi(k)) where mu(k) is the Möbius function and phi(k) is Euler's totient function.

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A333338 Numbers k such that sigma_2(k) = sigma_2(phi(k)).

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