A358714 -id:A358714

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A127473

a(n) = phi(n)^2.

+10
23

1, 1, 4, 4, 16, 4, 36, 16, 36, 16, 100, 16, 144, 36, 64, 64, 256, 36, 324, 64, 144, 100, 484, 64, 400, 144, 324, 144, 784, 64, 900, 256, 400, 256, 576, 144, 1296, 324, 576, 256, 1600, 144, 1764, 400, 576, 484, 2116, 256, 1764, 400, 1024, 576, 2704, 324, 1600 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,3`
	COMMENTS	`Number of maps of the form j \|--> mj + d with gcd(m, n) = 1 and gcd(d, n) = 1 from [1, 2, ..., n] to itself. - Joerg Arndt, Aug 29 2014` `Right border of A127474.` `Equals the Mobius transform (A054525) of A029939. - Gary W. Adamson, Aug 20 2008` `From Jianing Song, Apr 14 2019: (Start)` `a(n) is the number of solutions to gcd(xy, n) = 1 with x, y in [0, n-1].` `Let Z_n be the ring of integers modulo n, then a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - x) (or equivalently, Z_n[x]/(x^2 + x)) with discriminant d = 1 (that is, a(n) is the size of the group G(n) = (Z_n[x]/(x^2 - x))). Actually, G(n) is isomorphic to (Z_n)* X (Z_n)*. (End)`
	LINKS	`Jens Kruse Andersen, Table of n, a(n) for n = 1..10000` `N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).`
	FORMULA	`a(n) = A000010(n)^2.` `Multiplicative with a(p^e) = (p-1)^2p^(2e-2), e >= 1. Dirichlet g.f. zeta(s-2)Product_{primes p} (1 - 2/p^(s-1) + 1/p^s). - R. J. Mathar, Apr 04 2011` `Sum_{k>=1} 1/a(k) = A109695. - Vaclav Kotesovec, Sep 20 2020` `Sum_{k>=1} (-1)^k/a(k) = (1/7) * A109695. - Amiram Eldar, Nov 11 2020` `Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime}(1 - (2*p-1)/p^3) = A065464 / 3 = 0.142749... . - Amiram Eldar, Oct 25 2022`
	EXAMPLE	`a(5) = 16 since phi(5) = 4.`
	MAPLE	`A127473 := proc(n) numtheory[phi](n)^2 ; end proc:` `seq(A127473(n), n=1..40) ; # R. J. Mathar, Apr 04 2011`
	MATHEMATICA	`Table[EulerPhi[n]^2, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)`
	PROG	`(Magma) [(EulerPhi(n))^2: n in [1..180]]; // Vincenzo Librandi, Apr 04 2011` `(PARI) a(n) = eulerphi(n)^2; \\ Michel Marcus, Oct 16 2018`
	CROSSREFS	`Cf. A000010, A057434, A065464, A109695, A127474, A358714, A361148.` `Similar sequences: A082953 (size of (Z_n[x]/(x^2 - 1)), d = 4), A002618 ((Z_n[x]/(x^2)), d = 0), A079458 ((Z_n[x]/(x^2 + 1)), d = -4), A319445 ((Z_n[x]/(x^2 - x + 1)) or (Z_n[x]/(x^2 + x + 1))*, d = -3).`
	KEYWORD	`nonn,mult`
	AUTHOR	`Gary W. Adamson, Jan 15 2007`
	STATUS	`approved`

A361148

a(n) = phi(n)^4.

+10
5

1, 1, 16, 16, 256, 16, 1296, 256, 1296, 256, 10000, 256, 20736, 1296, 4096, 4096, 65536, 1296, 104976, 4096, 20736, 10000, 234256, 4096, 160000, 20736, 104976, 20736, 614656, 4096, 810000, 65536, 160000, 65536, 331776, 20736, 1679616, 104976, 331776, 65536, 2560000 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,3`
	COMMENTS	`In general, for k>=1, Sum_{m=1..n} phi(m)^k ~ c(k) * n^(k+1) / (k+1).` `Table of the first twenty constants c(k):` `c1 = 0.6079271018540266286632767792583658334261526480334792930736...` `c2 = 0.4282495056770944402187657075818235461212985133559361440319...` `c3 = 0.3371878737915899719616928161521582449491541277581639388802...` `c4 = 0.2862564715115608911732883400866386479560747005250468681615...` `c5 = 0.2550316684059564308661179534476184539887434047229867871927...` `c6 = 0.2342690874743831026992085481001750961630443094403694748409...` `c7 = 0.2194845388428573186801010214226853865762414525869501954550...` `c8 = 0.2083553180392308846240883587603960475166426933863125773262...` `c9 = 0.1996016550942289223053750541784521301740825495040856984950...` `c10 = 0.1924764951305819663569723926235916851341834741671794581256...` `c11 = 0.1865198318046079731059147989571847359151227252097897755685...` `c12 = 0.1814343147960482243026212589426877406632573154701351352790...` `c13 = 0.1770192204728143035012153190352692532613146649385520287635...` `c14 = 0.1731338036872585521607716180505314246174563305338731073703...` `c15 = 0.1696760784770144194638735708052066949428247152918280392147...` `c16 = 0.1665700322333281768929516390245288052095235102037486400080...` `c17 = 0.1637576294807392765019551841269187995536332906534705685240...` `c18 = 0.1611936368897236567526886186599877745065426644021588804182...` `c19 = 0.1588421683609925408830108209202958349394621277940566066627...` `c20 = 0.1566743130878534775247182243921577941535243896576096188342...` `c1 = A059956 = 6/Pi^2, c2 = A065464.` `Conjecture: c(k)*log(k) converges to a constant (around 0.534).`
	LINKS	`Amiram Eldar, Table of n, a(n) for n = 1..10000` `Vaclav Kotesovec, Plot of c(k)*log(k), for k = 1..350`
	FORMULA	`Multiplicative with a(p^e) = (p-1)^4 * p^(4e-4).` `Dirichlet g.f.: zeta(s-4) Product_{primes p} (1 + 1/p^s - 4/p^(s-1) + 6/p^(s-2) - 4/p^(s-3)).` `Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = Product_{primes p} (1 - 4/p^2 + 6/p^3 - 4/p^4 + 1/p^5) = 0.286256471511560891173288340086638647956074700525046868161...` `Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^4/((p-1)^4*(p^4-1))) = 2.20815077889083518654... . - Amiram Eldar, Sep 01 2023`
	MATHEMATICA	`Table[EulerPhi[n]^4, {n, 1, 50}]`
	PROG	`(PARI) a(n) = eulerphi(n)^4;` `(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X - 4pX + 6p^2X - 4p^3X) / (1 - p^4*X))[n], ", "))`
	CROSSREFS	`Cf. A000010, A002088, A127473, A057434, A358714, A361132, A361179.` `Cf. A059956, A065464.`
	KEYWORD	`nonn,easy,mult`
	AUTHOR	`Vaclav Kotesovec, Mar 02 2023`
	STATUS	`approved`

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