A048691 -id:A048691 - cp's OEIS frontend

A360909 Multiplicative with a(p^e) = 3*e + 2.

1, 5, 5, 8, 5, 25, 5, 11, 8, 25, 5, 40, 5, 25, 25, 14, 5, 40, 5, 40, 25, 25, 5, 55, 8, 25, 11, 40, 5, 125, 5, 17, 25, 25, 25, 64, 5, 25, 25, 55, 5, 125, 5, 40, 40, 25, 5, 70, 8, 40, 25, 40, 5, 55, 25, 55, 25, 25, 5, 200, 5, 25, 40, 20, 25, 125, 5, 40, 25, 125

Offset: 1

Vaclav Kotesovec, Feb 25 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), this sequence (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

a[n_] := Times @@ ((3*Last[#] + 2) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1+3*X-X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 3/p^s - 1/p^(2*s)).
Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)), (with a product that converges for s=1).
Sum_{k=1..n} a(k) ~ c * n * log(n)^4 / 24, where c = Product_{primes p} (1 - 7/p^2 + 11/p^3 - 6/p^4 + 1/p^5) = 0.091414252314317101861531055690354339957600046..., more precise (but very complicated) asymptotics can be obtained (in Mathematica notation) as Residue[Zeta[s]^5 * f[s] * n^s / s, {s, 1}], where f[s] = Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)).

A360997 Multiplicative with a(p^e) = e + 3.

Original entry on oeis.org

1, 4, 4, 5, 4, 16, 4, 6, 5, 16, 4, 20, 4, 16, 16, 7, 4, 20, 4, 20, 16, 16, 4, 24, 5, 16, 6, 20, 4, 64, 4, 8, 16, 16, 16, 25, 4, 16, 16, 24, 4, 64, 4, 20, 20, 16, 4, 28, 5, 20, 16, 20, 4, 24, 16, 24, 16, 16, 4, 80, 4, 16, 20, 9, 16, 64, 4, 20, 16, 64, 4, 30, 4, 16

Offset: 1

Vaclav Kotesovec, Feb 28 2023

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

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), this sequence (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

Cf. A000005, A007425, A007426, A074816.

g[p_, e_] := e+3; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1+2*X-2*X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: Product_{primes p} (1 + (4*p^s - 3)/(p^s - 1)^2).
Dirichlet g.f.: zeta(s)^4 * Product_{primes p} (1 - 5/p^(2*s) + 6/p^(3*s) - 2/p^(4*s)).
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A361264(n)).
a(n) = A074816(n)*A007426(n)/A007425(n). (End)

A059908 a(n) = |{m : multiplicative order of n mod m = 3}|.

Original entry on oeis.org

0, 1, 2, 4, 3, 2, 8, 2, 12, 5, 12, 2, 12, 2, 4, 20, 5, 6, 10, 2, 6, 14, 12, 2, 40, 9, 4, 6, 18, 10, 16, 6, 6, 8, 12, 12, 39, 2, 12, 8, 8, 6, 16, 6, 18, 26, 12, 6, 50, 3, 18, 8, 18, 2, 32, 12, 8, 20, 4, 6, 60, 2, 12, 26, 21, 4, 64, 10, 6, 8, 8, 6, 20, 14, 4, 12, 6, 4, 64, 2, 70, 7, 12, 6, 24

Offset: 1

Vladeta Jovovic, Feb 08 2001

The multiplicative order of a mod m, gcd(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

			a(2) = |{7}| = 1, a(3) = |{13,26}| = 2, a(4) = |{7,9,21,63}| = 4, a(5) = |{31,62,124}| = 3, a(6) = |{43,215}| = 2, a(7) = |{9,18,19,38,57,114,171,342}| = 8,...

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A059907, A059909-A059911, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

Row n=3 of A212957. - Alois P. Heinz, Oct 24 2012

Table[DivisorSigma[0,n^3-1]-DivisorSigma[0,n-1],{n,90}] (* Harvey P. Dale, Feb 03 2015 *)

a(n) = if(n == 1, 0, numdiv(n^3-1) - numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^3-1)-tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

A061701 Smallest number m such that GCD of d(m^2) and d(m) is 2n+1 where d(m) is the number of divisors of m.

Original entry on oeis.org

1, 12, 4608, 1728, 1260, 509607936, 2985984, 144, 56358560858112, 5159780352, 302400, 6232805962420322304, 207360000, 887040, 201226394483583074212773888, 15407021574586368, 248832, 2286144000, 26623333280885243904, 522547200, 8430527379596857675529996470321152

Offset: 0

Labos Elemer, Jun 18 2001

nonn

a(n) exists for every n. In other words, every positive odd integer k is equal to the GCD of d(m^2) and d(m) for some m. To see this, let m = 2^(k^2 - 1) * 3^((k-1)/2). Then d(m) = k^2 * (k+1)/2 and d(m^2) = (2 k^2 - 1) * k. Both of these are divisible by k and (8k-4) d(m) - (2k+1) d(m^2) = k, so the GCD is k. - Dean Hickerson, Jun 23 2001

All the terms are in A025487 because A061680(m) = gcd(d(m^2), d(m)) depends only on the prime signature of m. - Amiram Eldar, Nov 26 2023

			For n = 7, GCD[d(20736),d(144)] = GCD[45,15] = 15 = 2*7+1.

Amiram Eldar, Table of n, a(n) for n = 0..28

Cf. A000005, A000290, A025487, A048691, A061680.

a(n) = Min[m : GCD[d(m^2), d(m)] = 2n+1].

More terms from David Wasserman, Jun 20 2002

a(12)-a(13) corrected and a(17)-a(20) added by Amiram Eldar, Nov 26 2023

A117677 a(n) = number of divisors of n^2 (excluding 1 and n^2).

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 1, 5, 3, 7, 1, 13, 1, 7, 7, 7, 1, 13, 1, 13, 7, 7, 1, 19, 3, 7, 5, 13, 1, 25, 1, 9, 7, 7, 7, 23, 1, 7, 7, 19, 1, 25, 1, 13, 13, 7, 1, 25, 3, 13, 7, 13, 1, 19, 7, 19, 7, 7, 1, 43, 1, 7, 13, 11, 7, 25, 1, 13, 7, 25, 1, 33, 1, 7, 13, 13, 7, 25, 1, 25, 7, 7, 1, 43, 7, 7, 7, 19, 1, 43, 7

Offset: 1

Mark Taggart (mt2612f(AT)aol.com), Apr 12 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Equals A048691(n) - 2 for n>1. Cf. A117679, A117010, A070824.

Join[{0},DivisorSigma[0,Range[2,100]^2]-2] (* Harvey P. Dale, Oct 17 2016 *)

a(n) = A070824(n^2). - Ray Chandler, Nov 08 2018

A166721 Squares for which no smaller square has the same number of divisors.

Original entry on oeis.org

1, 4, 16, 36, 64, 144, 576, 900, 1024, 1296, 3600, 4096, 5184, 9216, 14400, 32400, 36864, 44100, 46656, 65536, 82944, 129600, 176400, 230400, 262144, 331776, 589824, 705600, 746496, 810000, 921600, 1166400, 1587600, 2073600, 2359296, 2822400, 2985984, 3240000

Offset: 1

Alexander Isaev (i2357(AT)mail.ru), Oct 20 2009

From Jon E. Schoenfield, Mar 03 2018: (Start)
Numbers k^2 such there is no positive m < k such that A000005(m^2) = A000005(k^2).
Square terms in A007416. (End)

			The positive squares begin 1, 4, 9, 16, 25, 36, 49, 64, ..., and their corresponding numbers of divisors are 1, 3, 3, 5, 3, 9, 3, 7, ...; thus, a(1)=1, a(2)=4, 9 is not a term (it has the same number of divisors as does 4; the same is true of 25, 49, etc.), a(3)=16, a(4)=36, a(5)=64, ... - _Jon E. Schoenfield_, Mar 03 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..300 from Alois P. Heinz)

Cf. A000005, A005179, A007416, A025487, A048691, A136404, A166722.

 Sort[Module[{nn=2000,tbl},tbl=Table[{n^2,DivisorSigma[0,n^2]},{n,nn}];Table[ SelectFirst[ tbl,#[[2]]==k&],{k,nn}]][[All,1]]/."NotFound"->Nothing] (* Harvey P. Dale, Jun 06 2022 *)

lista(nn) = {v = []; for (n=1, nn, d = numdiv(n^2); if (! vecsearch(v, d), print1(n^2, ", "); v = Set(concat(v, d))););} \\ Michel Marcus, Mar 04 2018

Proper definition and substantial editing by Jon E. Schoenfield, Mar 03 2018

A319132 a(n) = Sum_{d|n} Sum_{j|d} mu(j)^2*j, where mu = Möbius function (A008683).

Original entry on oeis.org

1, 4, 5, 7, 7, 20, 9, 10, 9, 28, 13, 35, 15, 36, 35, 13, 19, 36, 21, 49, 45, 52, 25, 50, 13, 60, 13, 63, 31, 140, 33, 16, 65, 76, 63, 63, 39, 84, 75, 70, 43, 180, 45, 91, 63, 100, 49, 65, 17, 52, 95, 105, 55, 52, 91, 90, 105, 124, 61, 245, 63, 132, 81, 19, 105, 260, 69, 133, 125, 252

Offset: 1

Ilya Gutkovskiy, Sep 11 2018

Inverse Möbius transform of A048250.

Metin Sariyar, Table of n, a(n) for n = 1..16000
N. J. A. Sloane, Transforms.

Cf. A007429, A008683, A048250, A048691, A055615, A072691, A319131.

[&+[&+[MoebiusMu(j)^2*j:j in Divisors(d)]:d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019

[&+[MoebiusMu(d)^2*d*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019

with(numtheory): seq(add(mobius(d)^2*d*tau(n/d), d in divisors(n)), n=1..70); # Ridouane Oudra, Nov 13 2019

Table[Sum[Sum[MoebiusMu[j]^2 j, {j, Divisors[d]}], {d, Divisors[n]}], {n, 70}]
nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, # &, SquareFreeQ[#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 70; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(DivisorSum[k, # &, SquareFreeQ[#] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := (p + 1)*e + 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

a(n) = sumdiv(n, d, moebius(d)^2*d*numdiv(n/d)); \\ Michel Marcus, Nov 13 2019; corrected Jun 13 2022

G.f.: Sum_{k>=1} A048250(k)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(A048250(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
a(p^k) = (p + 1)*k + 1, where p is a prime.
a(n) = Sum_{d|n} mu(d)^2*d*tau(n/d). - Ridouane Oudra, Nov 13 2019
Multiplicative with a(p^e) = (p+1)*e+1. - Amiram Eldar, Oct 25 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 = 0.822467... (A072691). - Amiram Eldar, Nov 13 2022
Dirichlet g.f.: zeta(s)^2*zeta(s-1)/zeta(2*s-2). - Amiram Eldar, Jan 03 2023

A348455 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= number of divisors of n^2) is the number of ways to tile an n X n chessboard with rook-connected polyominoes of area d_k, where d_k is the k-th divisor of n^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 10, 1, 1, 70, 117, 36, 1, 1, 4006, 1, 1, 6728, 80092, 178939, 451206, 442791, 264500, 80518, 1, 1, 158753814, 1

Offset: 1

N. J. A. Sloane, Oct 27 2021

This is an essentially identical triangle to A348453, except that the data in each row has effectively been reversed. Rather than copying everything here, please refer to A348453 for further information.

			Triangle begins:
1,
1, 2, 1,
1, 10, 1,
1, 70, 117, 36, 1,
1, 4006, 1,
1, 6728, 80092, 178939, 451206, 442791, 264500, 80518, 1,
1, 158753814, 1,
1, ?, ?, 187497290034, ?, 7157114189, 1,
...

Cf. A348452, A348453, A348454, A348456.

Cf. A048691 (row lengths).

A059909 a(n) = |{m : multiplicative order of n mod m = 4}|.

Original entry on oeis.org

0, 2, 6, 4, 12, 4, 26, 18, 14, 6, 24, 12, 64, 8, 16, 8, 66, 20, 36, 8, 64, 24, 76, 6, 28, 12, 64, 24, 48, 12, 100, 40, 50, 48, 36, 8, 96, 40, 28, 8, 104, 12, 208, 36, 24, 36, 200, 18, 48, 36, 36, 24, 128, 8, 152, 16, 172, 24, 48, 12, 48, 36, 56, 72, 40, 8, 128, 56, 48, 40

Offset: 1

Vladeta Jovovic, Feb 08 2001

The multiplicative order of a mod m, gcd(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

			a(2) = |{5, 15}| = 2, a(3) = |{5, 10, 16, 20, 40, 80}| = 6, a(4) = |{17, 51, 85, 255}| = 4, a(5) = |{13, 16, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624}| = 12, ...

Cf. A059907, A059908, A059910-A059911, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

Table[DivisorSigma[0,n^4-1]-DivisorSigma[0,n^2-1],{n,70}] (* Harvey P. Dale, Nov 30 2011 *)

a(n) = if(n == 1, 0, numdiv(n^4-1) - numdiv(n^2-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^4-1)-tau(n^2-1), where tau(n) = number of divisors of n A000005. More generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

A059910 a(n) = |{m : multiplicative order of n mod m = 5}|.

Original entry on oeis.org

0, 1, 4, 6, 9, 4, 4, 6, 20, 9, 8, 2, 6, 6, 12, 44, 5, 6, 18, 14, 12, 4, 4, 2, 56, 13, 20, 4, 6, 2, 40, 6, 18, 12, 12, 44, 63, 6, 28, 4, 16, 14, 8, 2, 18, 12, 28, 14, 70, 3, 42, 12, 42, 6, 24, 8, 56, 44, 60, 6, 60, 2, 4, 90, 21, 20, 24, 2, 18, 60, 88, 6, 12, 2, 28, 26, 6, 28, 8, 14, 170

Offset: 1

Vladeta Jovovic, Feb 08 2001

The multiplicative order of a mod m, gcd(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

Cf. A059907-A059909, A059911, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

a[n_] := Subtract @@ DivisorSigma[0, {n^5-1, n-1}]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025 *)

a(n) = if(n == 1, 0, numdiv(n^5-1) - numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^5-1)-tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

cp's OEIS Frontend

A360909 Multiplicative with a(p^e) = 3*e + 2.

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A360997 Multiplicative with a(p^e) = e + 3.

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A059908 a(n) = |{m : multiplicative order of n mod m = 3}|.

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A061701 Smallest number m such that GCD of d(m^2) and d(m) is 2n+1 where d(m) is the number of divisors of m.

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A117677 a(n) = number of divisors of n^2 (excluding 1 and n^2).

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A166721 Squares for which no smaller square has the same number of divisors.

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A319132 a(n) = Sum_{d|n} Sum_{j|d} mu(j)^2*j, where mu = Möbius function (A008683).

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A348455 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= number of divisors of n^2) is the number of ways to tile an n X n chessboard with rook-connected polyominoes of area d_k, where d_k is the k-th divisor of n^2.

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