cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 26 results. Next

A029939 a(n) = Sum_{d|n} phi(d)^2.

Original entry on oeis.org

1, 2, 5, 6, 17, 10, 37, 22, 41, 34, 101, 30, 145, 74, 85, 86, 257, 82, 325, 102, 185, 202, 485, 110, 417, 290, 365, 222, 785, 170, 901, 342, 505, 514, 629, 246, 1297, 650, 725, 374, 1601, 370, 1765, 606, 697, 970, 2117, 430, 1801, 834, 1285, 870, 2705, 730, 1717, 814, 1625
Offset: 1

Views

Author

Keywords

Comments

Equals the inverse Mobius transform (A051731) of A127473. - Gary W. Adamson, Aug 20 2008
Number of (i,j) in {1,2,...,n}^2 such that gcd(n,i) = gcd(n,j). - Benoit Cloitre, Dec 31 2020

Crossrefs

Programs

  • Maple
    with(numtheory): A029939 := proc(n) local i,j; j := 0; for i in divisors(n) do j := j+phi(i)^2; od; j; end;
    # alternative
    N:= 1000: # to get a(1)..a(N)
    A:= Vector(N,1):
    for d from 2 to N do
      pd:= numtheory:-phi(d)^2;
      md:= [seq(i,i=d..N,d)];
      A[md]:= map(`+`,A[md],pd);
    od:
    seq(A[i],i=1..N); # Robert Israel, May 30 2016
  • Mathematica
    Table[Total[EulerPhi[Divisors[n]]^2],{n,60}] (* Harvey P. Dale, Feb 04 2017 *)
    f[p_, e_] := (p^(2*e)*(p-1)+2)/(p+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^2); \\ Michel Marcus, Jan 17 2017

Formula

Multiplicative with a(p^e) = (p^(2*e)*(p-1)+2)/(p+1). - Vladeta Jovovic, Nov 19 2001
G.f.: Sum_{k>=1} phi(k)^2*x^k/(1 - x^k), where phi(k) is the Euler totient function (A000010). - Ilya Gutkovskiy, Jan 16 2017
a(n) = Sum_{k=1..n} phi(n/gcd(n, k)). - Ridouane Oudra, Nov 28 2019
Sum_{k>=1} 1/a(k) = 2.3943802654751092440350752246012273573942903149891228695146514601814537713... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/(3*zeta(2))) * Product_{p prime} (1 - 1/(p*(p+1))) = A253905 * A065463 / 3 = 0.171593... . - Amiram Eldar, Oct 25 2022

A057434 a(n) = Sum_{k=1..n} phi(k)^2.

Original entry on oeis.org

1, 2, 6, 10, 26, 30, 66, 82, 118, 134, 234, 250, 394, 430, 494, 558, 814, 850, 1174, 1238, 1382, 1482, 1966, 2030, 2430, 2574, 2898, 3042, 3826, 3890, 4790, 5046, 5446, 5702, 6278, 6422, 7718, 8042, 8618, 8874, 10474, 10618, 12382, 12782
Offset: 1

Views

Author

N. J. A. Sloane, Sep 08 2000

Keywords

Comments

Partial sums of A127473. - R. J. Mathar, Sep 29 2008

Crossrefs

Programs

  • Mathematica
    FoldList[Plus, 1, EulerPhi[Range[2, 50]]^2] (* Ivan Neretin, May 30 2015 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)^2); \\ Michel Marcus, Dec 20 2015

Formula

We can derive an asymptotic formula from a general formula given in the reference, namely: a(n) = C*n^3 + O(log(x)^(4/3)log(log(x))^(8/3)) where C = (1/3)/zeta(2)^2*Product_{p prime}(1+1/(p-1)/(p+1)^2) = 0.142749835225698(...). - Benoit Cloitre, Dec 22 2015
a(n) ~ c * n^3 / 3, where c = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.4282495056770944402187657075818235461212985133559361440319... - Vaclav Kotesovec, Dec 18 2019

A361148 a(n) = phi(n)^4.

Original entry on oeis.org

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
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 02 2023

Keywords

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).

Crossrefs

Programs

  • Mathematica
    Table[EulerPhi[n]^4, {n, 1, 50}]
  • PARI
    a(n) = eulerphi(n)^4;
    
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 + X - 4*p*X + 6*p^2*X - 4*p^3*X) / (1 - p^4*X))[n], ", "))

Formula

Multiplicative with a(p^e) = (p-1)^4 * p^(4*e-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

A109695 Decimal expansion of Sum_{n>=1} 1/phi(n)^2.

Original entry on oeis.org

3, 3, 9, 0, 6, 4, 2, 0, 0, 5, 5, 7, 2, 5, 0, 3, 9, 1, 6, 1, 4, 2, 5, 9, 5, 6, 6, 3, 0, 0, 2, 6, 3, 0, 7, 9, 3, 7, 4, 0, 5, 3, 7, 3, 8, 1, 2, 1, 4, 4, 7, 1, 6, 9, 1, 1, 8, 0, 7, 3, 9, 8, 1, 5, 6, 8, 5, 7, 3, 8, 1, 3, 1, 1, 1, 7, 7, 6, 3, 3, 2, 1, 3, 6, 5, 0, 4, 1, 0, 2, 4, 4, 4, 9, 5, 2, 3, 7, 4, 2, 9, 8, 2, 5, 7
Offset: 1

Views

Author

Keywords

Comments

The logarithm of the value can be expanded in a series Sum_{j>=2} c(j)*P(j) = P(2) + 2*P(3) + (7/2)*P(4) + ... where P(.) is the prime zeta function. The partial sums of the series are a slowly oscillating function of the upper limit of j, from which the bracketing interval [3.390642005572503655..., 3.390642005572504756...] for the constant can be computed. - R. J. Mathar, Feb 03 2009
Sum_{n>=1} 1/phi(n)^k is convergent iff k > 1 (reference Monier). - Bernard Schott, Dec 13 2020

Examples

			3.39064200557250391614259566300263079374053738121447169118...
		

References

  • Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.21, pp. 281 and 294.

Crossrefs

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; f[p_] := (1 + p^2/((p - 1)^2*(p^2 - 1))); Do[cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 120]]], {m, 100, 1000, 100}] (* Vaclav Kotesovec, Jun 25 2020 *)
  • PARI
    my(N=1000000000); prodeuler(p=2,N,1.+p^2/((p-1)^2*(p^2-1)))*(1+1/(N*log(N)))
    
  • PARI
    prodeulerrat(1 + p^2/((p-1)^2*(p^2-1))) \\ Amiram Eldar, Mar 15 2021

Formula

Equals Product_p Sum_{k>=0} 1/phi(p^k)^2 = Product_p (1 + p^2/((p-1)^2*(p^2-1))).
Equals Sum{n>=1} 1/A127473(n). - Amiram Eldar, Mar 15 2021

Extensions

Four more digits from R. J. Mathar, Feb 03 2009, 25 more Dec 18 2010
More digits from Vaclav Kotesovec, Jun 25 2020

A356533 a(n) = sigma_2(n)^2.

Original entry on oeis.org

1, 25, 100, 441, 676, 2500, 2500, 7225, 8281, 16900, 14884, 44100, 28900, 62500, 67600, 116281, 84100, 207025, 131044, 298116, 250000, 372100, 280900, 722500, 423801, 722500, 672400, 1102500, 708964, 1690000, 925444, 1863225, 1488400, 2102500, 1690000, 3651921
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 11 2022

Keywords

Crossrefs

Cf. A001157, A127473, A035116, A072861, A356535 (partial sums).

Programs

  • Mathematica
    Table[DivisorSigma[2, n]^2, {n, 1, 40}]
  • PARI
    a(n) = sigma(n, 2)^2; \\ Michel Marcus, Aug 11 2022

Formula

Dirichlet g.f.: zeta(s) * zeta(s-2)^2 * zeta(s-4) / zeta(2*s-4).
Multiplicative with a(p^e) = ((p^(2*e+2)-1)/(p^2-1))^2. - Amiram Eldar, Aug 11 2022
a(n) = A001157(n)^2. - R. J. Mathar, Aug 18 2022

A356534 a(n) = sigma_3(n)^2.

Original entry on oeis.org

1, 81, 784, 5329, 15876, 63504, 118336, 342225, 573049, 1285956, 1774224, 4177936, 4831204, 9585216, 12446784, 21911761, 24147396, 46416969, 47059600, 84603204, 92775424, 143712144, 148060224, 268304400, 248094001, 391327524, 417793600, 630612544, 594872100
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 11 2022

Keywords

Crossrefs

Cf. A001158, A127473, A035116, A072861, A356536 (partial sums).

Programs

  • Mathematica
    Table[DivisorSigma[3, n]^2, {n, 1, 40}]
  • PARI
    a(n) = sigma(n, 3)^2; \\ Michel Marcus, Aug 11 2022

Formula

Dirichlet g.f.: zeta(s) * zeta(s-3)^2 * zeta(s-6) / zeta(2*s-6).
Multiplicative with a(p^e) = ((p^(3*e+3)-1)/(p^3-1))^2. - Amiram Eldar, Aug 11 2022

A077101 a(n) = A051612(n)*A065387(n) = sigma(n)^2-phi(n)^2, where A051612(n) = sigma(n) - phi(n) and A065387(n) = sigma(n) + phi(n).

Original entry on oeis.org

0, 8, 12, 45, 20, 140, 28, 209, 133, 308, 44, 768, 52, 540, 512, 897, 68, 1485, 76, 1700, 880, 1196, 92, 3536, 561, 1620, 1276, 2992, 116, 5120, 124, 3713, 1904, 2660, 1728, 8137, 148, 3276, 2560, 7844, 164, 9072, 172, 6656, 5508, 4700, 188, 15120, 1485
Offset: 1

Views

Author

Labos Elemer, Nov 06 2002

Keywords

Comments

If n is prime, then a(n) = 4n.

Crossrefs

Programs

Formula

a(n) = A077099(n) * A077100(n). - Antti Karttunen, May 26 2017
From Amiram Eldar, Dec 04 2023: (Start)
a(n) = A072861(n) - A127473(n).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 5*zeta(3)/2 - Product_{p prime}(1 - (2*p-1)/p^3) = (5/2)*A002117 - A065464 = 2.576892... . (End)

Extensions

Edited by Dean Hickerson, Nov 07 2002

A082953 a(n) = A000252(n) / A070732(n).

Original entry on oeis.org

1, 2, 4, 8, 16, 8, 36, 32, 36, 32, 100, 32, 144, 72, 64, 128, 256, 72, 324, 128, 144, 200, 484, 128, 400, 288, 324, 288, 784, 128, 900, 512, 400, 512, 576, 288, 1296, 648, 576, 512, 1600, 288, 1764, 800, 576, 968, 2116
Offset: 1

Views

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), May 26 2003

Keywords

Comments

From Jianing Song, Apr 20 2019: (Start)
a(n) is the number of split complex numbers z = x + yj in a reduced system modulo n where x, y are integers, j^2 = 1; number of solutions to gcd(x^2 - y^2, n)=1 with x, y in [0, n-1].
a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - 1) with discriminant d = 4, where Z_n is the ring of integers modulo n. (End)

Crossrefs

Similar sequences: A127473 (size of (Z_n[x]/(x^2 - x))*, d = 1), 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).

Programs

  • Maple
    A082953 := proc(n) numtheory[phi](n)*numtheory[phi](2*n) ; end proc:
    seq(A082953(n),n=1..100) ; # R. J. Mathar, Jan 07 2011
  • Mathematica
    Array[Times @@ Map[EulerPhi, {#, 2 #}] &, 47] (* Michael De Vlieger, Apr 21 2019 *)
  • PARI
    a(n) = eulerphi(n)*eulerphi(2*n); \\ Michel Marcus, Jun 04 2025

Formula

a(n) = phi(n)*phi(2*n) = A000010(n)*A062570(n). - Vladeta Jovovic, May 02 2005
Multiplicative with a(2^e) = 2^(2e-1) and a(p^e) = (p-1)^2*p^(2e-2) for p > 2. - R. J. Mathar, Apr 14 2011
a(n) = phi(n)^2 if n odd; 2*phi(n)^2 if n even, where phi(n) = A000010(n). - Jianing Song, Apr 20 2019
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/5) * Product_{p prime} (1 - (2*p-1)/p^3) = (2/5) * A065464 = 0.171299... . - Amiram Eldar, Oct 30 2022
a(n) = gcd(n,2)*phi(n)^2 = A040001(n)*A127473(n). - Ridouane Oudra, Jun 04 2025

A356535 a(n) = Sum_{k=1..n} sigma_2(k)^2.

Original entry on oeis.org

1, 26, 126, 567, 1243, 3743, 6243, 13468, 21749, 38649, 53533, 97633, 126533, 189033, 256633, 372914, 457014, 664039, 795083, 1093199, 1343199, 1715299, 1996199, 2718699, 3142500, 3865000, 4537400, 5639900, 6348864, 8038864, 8964308, 10827533, 12315933, 14418433
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 11 2022

Keywords

Comments

Partial sums of A356533.

Crossrefs

Programs

  • Mathematica
    Table[Sum[DivisorSigma[2, k]^2, {k, 1, n}], {n, 1, 40}]
  • PARI
    a(n) = sum(k=1, n, sigma(k, 2)^2); \\ Michel Marcus, Aug 11 2022

Formula

a(n) ~ 189 * zeta(3)^2 * zeta(5) * n^5 / Pi^6.

A356536 a(n) = Sum_{k=1..n} sigma_3(k)^2.

Original entry on oeis.org

1, 82, 866, 6195, 22071, 85575, 203911, 546136, 1119185, 2405141, 4179365, 8357301, 13188505, 22773721, 35220505, 57132266, 81279662, 127696631, 174756231, 259359435, 352134859, 495847003, 643907227, 912211627, 1160305628, 1551633152, 1969426752, 2600039296
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 11 2022

Keywords

Comments

Partial sums of A356534.
In general, for m>0, Sum_{k=1..n} sigma_m(k)^2 ~ zeta(2*m+1) * zeta(m+1)^2 * n^(2*m+1) / ((2*m+1) * zeta(2*m+2)).

Crossrefs

Programs

  • Mathematica
    Table[Sum[DivisorSigma[3, k]^2, {k, 1, n}], {n, 1, 40}]
    Accumulate[DivisorSigma[3,Range[40]]^2] (* This program is much more efficient than the first program above. *) (* Harvey P. Dale, Feb 27 2023 *)
  • PARI
    a(n) = sum(k=1, n, sigma(k, 3)^2); \\ Michel Marcus, Aug 11 2022

Formula

a(n) ~ zeta(7) * n^7 / 6.
Showing 1-10 of 26 results. Next