A062354 -id:A062354 - cp's OEIS frontend

A067683 Numbers k such that phi(k)*sigma(k) + 1 is prime.

1, 8, 9, 10, 12, 15, 20, 28, 30, 32, 35, 36, 42, 45, 50, 54, 58, 60, 70, 80, 92, 93, 95, 100, 110, 114, 122, 123, 124, 125, 130, 132, 142, 143, 145, 152, 155, 162, 165, 168, 169, 171, 172, 174, 175, 176, 178, 180, 183, 185, 186, 195, 198, 200, 204, 209, 212, 216

Offset: 1

Benoit Cloitre, Feb 04 2002

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

Cf. A000010 (phi), A000203 (sigma), A062354.

Select[Range[216], PrimeQ[EulerPhi[#]*DivisorSigma[1, #] + 1] &] (* Amiram Eldar, Apr 20 2025 *)

isok(k) = {my(f = factor(k)); isprime(eulerphi(k) * sigma(k) + 1);} \\ Amiram Eldar, Apr 20 2025

A069546 a(n) = Sum_{d|n} sigma(n*d).

Original entry on oeis.org

1, 10, 17, 53, 37, 170, 65, 236, 174, 370, 145, 901, 197, 650, 629, 987, 325, 1740, 401, 1961, 1105, 1450, 577, 4012, 968, 1970, 1618, 3445, 901, 6290, 1025, 4026, 2465, 3250, 2405, 9222, 1445, 4010, 3349, 8732, 1765, 11050, 1937, 7685, 6438, 5770

Offset: 1

Vladeta Jovovic, Apr 17 2002

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

Cf. A061391, A062354.

Cf. A002117, A013661.

[&+[DivisorSigma(1,n*d):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Sep 15 2019

Table[ Apply[ Plus, DivisorSigma[1, n*Divisors[n]]], {n, 1, 50}]
f[p_, e_] := (p^(e + 1)*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 28 2022 *)

a(n) = sumdiv(n, d, sigma(n*d)); \\ Michel Marcus, Sep 15 2019

Multiplicative with a(p^e) = (p^(e+1)*(p^(e+1)-1)-(p-1)*(e+1))/(p-1)^2.

Sum_{k=1..n} a(k) ~ c * n^3, where c = ((zeta(2)*zeta(3)^2)/3) * Product_{p prime} (1 + 1/p^2 - 1/p^4 - 1/p^5) = 1.09461730308... . - Amiram Eldar, Oct 28 2022

Edited and extended by Robert G. Wilson v, Apr 22 2002

A070943 Commuting elements: number of ordered pairs g, h in the group GL(2,Z_n) such that gh = hg.

Original entry on oeis.org

1, 18, 384, 1344, 11520, 6912, 96768, 92160, 303264, 207360, 1584000, 516096, 4402944, 1741824, 4423680, 6094848, 22560768, 5458752, 44323200, 15482880, 37158912, 28512000, 141064704, 35389440, 186000000, 79252992, 226748160, 130056192, 572947200, 79626240

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 12 2003

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A000010, A000203, A000252, A062354.

a[n_] := n^4*DivisorSigma[1, n]*EulerPhi[n]*Product[(1-1/p^2)*(1-1/p), {p, FactorInteger[n][[All, 1]]}]; a[1]=1; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, May 02 2013, after Eric M. Schmidt *)

def A070943(n) : return Integer(n^4 * sigma(n) * euler_phi(n) * prod((1-1/p^2)*(1-1/p) for (p,m) in factor(n))) # Eric M. Schmidt, May 02 2013

a(n) = A000252(n) * A062354(n).
a(n) = n^4 * Product_{p prime, p|n} (1-1/p^2)*(1-1/p) * sigma(n)*phi(n).
From Amiram Eldar, Oct 30 2022: (Start)
Multiplicative with a(p^e) = (p^(e+1)-1) * (p-1)^2 * (p+1) * p^(5*e-4).
Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 - 1/p^2 - 2/p^3 + 3/p^4 - 1/p^5) = 0.07103214283... . (End)

More terms from Benoit Cloitre, Sep 13 2003

More terms from Eric M. Schmidt, May 02 2013

A085646 Sum of the entries in the character table of the group GL(2,Z_n).

Original entry on oeis.org

1, 5, 24, 52, 120, 120, 336, 496, 654, 600, 1320, 1248, 2184, 1680, 2880

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 11 2003

Cf. A000252, A062354, A053191.

For an odd prime p : a(p) = p*(p^2-1).

A115911 Numbers k such that phi(k)*sigma(k) is a triangular number.

Original entry on oeis.org

1, 2, 9, 11, 23, 118, 373, 556, 1332, 2420, 3081, 5251, 7642, 12671, 116836, 127627, 135861, 172676, 198912, 365408, 421902, 426710, 901273, 921736, 954068, 1001396, 1003333, 1006567, 1077452, 1161754, 1162514, 1364225, 1632361

Offset: 1

Giovanni Resta, Feb 06 2006

nonn

			phi(1003333)*sigma(1003333) = 1003888529280 = T(1416960).

Amiram Eldar, Table of n, a(n) for n = 1..160

Cf. A062354, A069066.

Select[Range[10^4], IntegerQ @ Sqrt[8 * EulerPhi[#] * DivisorSigma[1, #] + 1] &] (* Amiram Eldar, Sep 16 2019 *)

isok(n) = ispolygonal(eulerphi(n)*sigma(n), 3); \\ Michel Marcus, Jan 09 2014

A271182 a(n) = prime(n)^(2*n) - prime(n)^(n-1).

Original entry on oeis.org

3, 78, 15600, 5764458, 25937409960, 23298084751188, 168377826535263360, 288441413566727295942, 3244150909895169974315088, 176994576151109738690640664532, 645590698195138072217104753157760, 43335257111193343900187118461545288548

Offset: 1

Wesley Ivan Hurt, Apr 07 2016

Cf. A000010, A000040, A000203, A062354, A062457.

[NthPrime(n)^(2*n)-NthPrime(n)^(n-1) : n in [1..12]];

A271182:=n->ithprime(n)^(2*n)-ithprime(n)^(n-1): seq(A271182(n), n=1..15);

Table[Prime[n]^(2*n) - Prime[n]^(n - 1), {n, 12}]

a(n) = prime(n)^(2*n) - prime(n)^(n-1); \\ Altug Alkan, Apr 07 2016

a(n) = sigma(prime(n)^n) * phi(prime(n)^n) = A062354(A062457(n)).

A309153 a(n) = A000203(n)*A001227(n).

Original entry on oeis.org

1, 3, 8, 7, 12, 24, 16, 15, 39, 36, 24, 56, 28, 48, 96, 31, 36, 117, 40, 84, 128, 72, 48, 120, 93, 84, 160, 112, 60, 288, 64, 63, 192, 108, 192, 273, 76, 120, 224, 180, 84, 384, 88, 168, 468, 144, 96, 248, 171, 279, 288, 196, 108, 480, 288, 240, 320, 180, 120, 672, 124, 192, 624, 127, 336, 576, 136, 252, 384

Offset: 1

Omar E. Pol, Jul 14 2019

A001227(n) is denoted by Delta_0(n) in Glaisher 1907.

a(n) = A000203(n) iff n is a power of 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A000079, A000203, A001227, A062354, A062355, A064840.

Cf. A001620, A002117, A244115, A306016

Array[DivisorSum[#, 1 &, OddQ] DivisorSigma[1, #] &, 69] (* Michael De Vlieger, Nov 22 2019 *)
f[p_, e_] := (e+1)*(p^(e+1)-1)/(p-1); f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 01 2022 *)

a(n) = sigma(n)*delta(n).
Multiplicative with a(2^e) = 2^(e+1) - 1 and a(p^e) = (e+1)*(p^(e+1)-1)/(p-1) for p > 2. - Amiram Eldar, Nov 01 2022
From Amiram Eldar, Dec 04 2023: (Start)
Dirichlet g.f.: (4^s - 3*2^s + 2)/(4^s - 2) * (zeta(s)*zeta(s-1))^2/zeta(2*s-1).
Sum_{k=1..n} a(k) ~ (Pi^4/(168*zeta(3))) * n^2 * (log(n) + 2*gamma - 1/2 + 22*log(2)/21 + 2*zeta'(2)/zeta(2) - 2*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)

A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

Original entry on oeis.org

1, 6, 12, 30, 30, 72, 56, 138, 123, 180, 132, 360, 182, 336, 360, 602, 306, 738, 380, 900, 672, 792, 552, 1656, 795, 1092, 1176, 1680, 870, 2160, 992, 2538, 1584, 1836, 1680, 3690, 1406, 2280, 2184, 4140, 1722, 4032, 1892, 3960, 3690, 3312, 2256, 7224, 2835, 4770, 3672, 5460

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A000385, A034761, A038040, A062354, A062952, A183699, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[1, n/d], {d, Divisors[n]}], {n, 52}]
Table[Sum[DivisorSigma[1, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 52}]
f[p_, e_] := (p^(2*e + 3) - (e + 1)*(p^2 - 1)*p^e - p)/((p - 1)^2*(p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 12 2022 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*sigma(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} sigma(gcd(n,k)) * sigma(n/gcd(n,k)).
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p^e) = (p^(2*e+3) - (e+1)*(p^2-1)*p^e - p)/((p-1)^2*(p+1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/3) * A183699 * A330523 = 0.581007... . (End)

A341638 a(n) = Sum_{d|n} phi(d) * sigma(d) * tau(n/d).

Original entry on oeis.org

1, 5, 10, 23, 26, 50, 50, 101, 97, 130, 122, 230, 170, 250, 260, 427, 290, 485, 362, 598, 500, 610, 530, 1010, 671, 850, 904, 1150, 842, 1300, 962, 1761, 1220, 1450, 1300, 2231, 1370, 1810, 1700, 2626, 1682, 2500, 1850, 2806, 2522, 2650, 2210, 4270, 2493, 3355, 2900, 3910, 2810, 4520

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Inverse Moebius transform of A062952.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A007430, A062354, A062952, A191831.

Cf. A002117, A013661, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 54}]
Table[Sum[DivisorSigma[0, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 54}]
f[p_, e_] := (p^(2*e + 4) - p^(e + 3) - 2*p^(e + 2) - p^(e + 1) + (e + 1)*p^3 - (e - 1)*p + 1)/(p^2 - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 26 2023 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*numdiv(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} tau(gcd(n,k)) * sigma(n/gcd(n,k)).
a(n) = Sum_{d|n} A062952(d).
a(n) = Sum_{k=1..n} tau(n/gcd(n,k))*sigma(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 09 2021
From Amiram Eldar, Jan 26 2023: (Start)
Multiplicative with a(p^e) = (p^(2*e+4) - p^(e+3) - 2*p^(e+2) - p^(e+1) + (e+1)*p^3 - (e-1)*p + 1)/(p^2-1)^2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)^2/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A013661 * A002117^2 * A330523 / 3 = 0.424578... . (End)

A386624 a(n) = Sum_{d|n} sigma(d) * phi(d) * mu(n/d).

Original entry on oeis.org

1, 2, 7, 11, 23, 14, 47, 46, 70, 46, 119, 77, 167, 94, 161, 188, 287, 140, 359, 253, 329, 238, 527, 322, 596, 334, 642, 517, 839, 322, 959, 760, 833, 574, 1081, 770, 1367, 718, 1169, 1058, 1679, 658, 1847, 1309, 1610, 1054, 2207, 1316, 2346, 1192, 2009, 1837, 2807, 1284, 2737, 2162, 2513, 1678, 3479, 1771, 3719, 1918, 3290, 3056, 3841, 1666, 4487, 3157, 3689

Offset: 1

Wesley Ivan Hurt, Jul 27 2025

Möbius transform of sigma(n) * phi(n) = A062354(n).

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

Cf. A000010 (phi), A000203 (sigma), A062354, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 100}]
f[p_, e_] := p^(2*e) - p^(e-1) - If[e > 1, p^(2*e-2) - p^(e-2), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 27 2025 *)

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i, 2];  p^(2*e) - p^(e - 1) - if(e > 1, p^(2*e - 2) - p^(e - 2), 1));} \\ Amiram Eldar, Jul 27 2025

a(n) = Sum_{d|n} A062354(d) * mu(n/d).
From Amiram Eldar, Jul 27 2025: (Start)
Multiplicative with a(p) = p^2 - 2, and a(p^e) = p^(2*e) - p^(2*e-2) - p^(e-1) + p^(e-2) for e >= 2.
Dirichlet g.f.: (zeta(s-2) * zeta(s-1) / zeta(s)) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s + 1/p^(2*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)/(3*zeta(3))) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523*zeta(2)/(3*zeta(3)) = 0.24444595409976589792... . (End)

cp's OEIS Frontend

A067683 Numbers k such that phi(k)*sigma(k) + 1 is prime.

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A069546 a(n) = Sum_{d|n} sigma(n*d).

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Magma

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A070943 Commuting elements: number of ordered pairs g, h in the group GL(2,Z_n) such that gh = hg.

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A085646 Sum of the entries in the character table of the group GL(2,Z_n).

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A115911 Numbers k such that phi(k)*sigma(k) is a triangular number.

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A271182 a(n) = prime(n)^(2*n) - prime(n)^(n-1).

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Maple

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A309153 a(n) = A000203(n)*A001227(n).

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A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

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