A034448 -id:A034448 - cp's OEIS frontend

A348504 a(n) = sigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

1, 1, 1, 7, 1, 1, 1, 5, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 5, 31, 1, 10, 7, 1, 1, 1, 21, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 10, 1, 5, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 13, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 5, 1, 13, 1, 7, 1, 1, 1, 21, 1, 57

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 72 = 8*9, where a(72) = 13 != 5*13 = a(8) * a(9).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A005117 (positions of ones), A034448, A048146, A348503, A348505.

Differs from A344696 for the first time at n=72, where a(72) = 13, while A344696(72) = 65. Cf. also A348048.

f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := (sigma = Times @@ f2 @@@ (fct = FactorInteger[n])) / GCD[sigma, Times @@ f1 @@@ fct]; Array[a, 100] (* Amiram Eldar, Oct 29 2021 *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348504(n) = { my(u=sigma(n)); (u/gcd(u, A034448(n))); };

a(n) = A000203(n) / A348503(n) = A000203(n) / gcd(A000203(n), A034448(n)).

A348506 Numbers k such that sigma(k) is a multiple of usigma(k), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 118, 119

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

Conjectured to be the union of A005117 and A063880.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of ones in A348505.
Cf. A005117 and A063880.
Cf. A000203, A034448.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
isA348506(n) = !(sigma(n)%A034448(n));

A348732 a(n) = A003959(n) - A034448(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 18, 6, 0, 0, 16, 0, 0, 0, 64, 0, 18, 0, 24, 0, 0, 0, 72, 10, 0, 36, 32, 0, 0, 0, 210, 0, 0, 0, 94, 0, 0, 0, 108, 0, 0, 0, 48, 36, 0, 0, 256, 14, 30, 0, 56, 0, 108, 0, 144, 0, 0, 0, 96, 0, 0, 48, 664, 0, 0, 0, 72, 0, 0, 0, 342, 0, 0, 40, 80, 0, 0, 0, 384, 174, 0, 0, 128, 0, 0, 0, 216, 0, 108

Offset: 1

Antti Karttunen, Oct 31 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..21125

Cf. A003959, A005117 (positions of zeros), A034448, A034460, A048146, A348029, A348507.

f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348732(n) = (A003959(n)-A034448(n));

a(n) = A003959(n) - A034448(n).
a(n) = A348507(n) - A034460(n).
a(n) = A048146(n) + A348029(n).

A349000 a(n) = A323166(A276086(n)), where A323166(n) = gcd(n, usigma(n)), usigma (A034448) is multiplicative with a(p^e) = (p^e)+1, and A276086 gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

1, 1, 1, 6, 1, 6, 1, 2, 3, 6, 15, 90, 1, 2, 1, 6, 5, 30, 1, 2, 3, 6, 45, 90, 1, 2, 1, 6, 5, 30, 1, 2, 1, 6, 1, 6, 1, 2, 3, 6, 15, 90, 1, 2, 1, 6, 5, 30, 7, 14, 21, 42, 315, 630, 1, 2, 1, 6, 5, 30, 1, 2, 1, 6, 1, 6, 5, 10, 15, 30, 15, 90, 25, 50, 25, 150, 25, 150, 175, 350, 525, 1050, 7875, 15750, 25, 50, 25, 150, 125, 750

Offset: 0

Antti Karttunen, Nov 07 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A034448, A276086, A323166, A348996, A348997, A348999.

A349000(n) = { my(m1=1, m2=1, p=2, u); while(n, if(n%p, u = p^(n%p); m1 *= u; m2 *= (1+u)); n = n\p; p = nextprime(1+p)); gcd(m1,m2); };

a(n) = A323166(A276086(n)) = gcd(A276086(n), A348996(n)).

A379515 Numerators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 2, 11, 43, 53, 4, 37, 293, 329, 103, 113, 107, 809, 129, 809, 12913, 41119, 39691, 41833, 8081, 33395, 32443, 33871, 10973, 148361, 48275, 7149, 34861, 108119, 319937, 164941, 1761311, 112361, 662011, 5405483, 26502319, 516671461, 508357441, 3620857237, 3556192637

Offset: 1

Amiram Eldar, Dec 23 2024

			Fractions begin with 1, 2/3, 11/12, 43/60, 53/60, 4/5, 37/40, 293/360, 329/360, 103/120, 113/120, 107/120, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3. See p. 4, eq. (vi).
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.

Cf. A034448, A064609, A308041, A323482, A370898, A379513, A379516 (denominators).

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Numerator[Accumulate[Table[(-1)^(n+1)/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / usigma(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A034448(k)).

a(n)/A379516(n) = E * log(n) + F + O(log(n)^(5/3) * log(log(n))^(4/3) / n^u), where u > 0, E = A308041 * (2/(A323482 + 1/2) - 1) = 0.10259754363391420806..., and F is a constant.

A379516 Denominators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 3, 12, 60, 60, 5, 40, 360, 360, 120, 120, 120, 840, 140, 840, 14280, 42840, 42840, 42840, 8568, 34272, 34272, 34272, 11424, 148512, 49504, 7072, 35360, 106080, 318240, 159120, 1750320, 109395, 656370, 5250960, 26254800, 498841200, 498841200, 3491888400, 3491888400

Offset: 1

Amiram Eldar, Dec 23 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3. See p. 4, eq. (vi).
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.

Cf. A034448, A064609, A370898, A379514, A379515 (numerators).

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Denominator[Accumulate[Table[(-1)^(n+1)/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / usigma(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A034448(k)).

A063845 a(n) = sigma(usigma(n)), where usigma(n) is the sum of unitary divisors of n (A034448) and sigma(n) is the sum of the divisors (A000203).

Original entry on oeis.org

1, 4, 7, 6, 12, 28, 15, 13, 18, 39, 28, 42, 24, 60, 60, 18, 39, 72, 42, 72, 63, 91, 60, 91, 42, 96, 56, 90, 72, 195, 63, 48, 124, 120, 124, 93, 60, 168, 120, 120, 96, 252, 84, 168, 168, 195, 124, 126, 93, 168, 195, 144, 120, 224, 195, 195, 186

Offset: 1

N. J. A. Sloane, Oct 30 2001

nonn

Harry J. Smith, Table of n, a(n) for n=1..1000

usigma[n_] := DivisorSum[n, Boole[CoprimeQ[#, n/#]]*#& ]; a[n_] := DivisorSigma[1, usigma[n]]; Array[a, 60] (* Jean-François Alcover, Dec 02 2015 *)

usigma(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)) { for (n=1, 1000, write("b063845.txt", n, " ", sigma(usigma(n))) ) } \\ Harry J. Smith, Sep 01 2009

a(n) = A000203(A034448(n)). - Michel Marcus, Dec 02 2015

A064212 a(n) = sigma(n) + usigma(n) = A000203(n) + A034448(n).

Original entry on oeis.org

2, 6, 8, 12, 12, 24, 16, 24, 23, 36, 24, 48, 28, 48, 48, 48, 36, 69, 40, 72, 64, 72, 48, 96, 57, 84, 68, 96, 60, 144, 64, 96, 96, 108, 96, 141, 76, 120, 112, 144, 84, 192, 88, 144, 138, 144, 96, 192, 107, 171, 144, 168, 108, 204, 144, 192, 160

Offset: 1

N. J. A. Sloane, Oct 31 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000203, A034448, A048146.

Cf. A002117, A072691.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) + Times @@ f2 @@@ fct; a[1] = 2; Array[a, 100] (* Amiram Eldar, Apr 01 2024 *)

usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ for (n=1, 1000, write("b064212.txt", n, " ", sigma(n) + usigma(n)) ) } \\ Harry J. Smith, Sep 10 2009

Sum_{k=1..n} a(k) ~ (Pi^2/12)*(1 + 1/zeta(3)) * n^2. - Amiram Eldar, Apr 01 2024

A064992 a(n) = usigma(n+1) - usigma(n), where usigma(n) is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

2, 1, 1, 1, 6, -4, 1, 1, 8, -6, 8, -6, 10, 0, -7, 1, 12, -10, 10, 2, 4, -12, 12, -10, 16, -14, 12, -10, 42, -40, 1, 15, 6, -6, 2, -12, 22, -4, -2, -12, 54, -52, 16, 0, 12, -24, 20, -18, 28, -6, -2, -16, 30, -12, 0, 8, 10, -30, 60, -58, 34, -16, -15, 19, 60, -76, 22, 6, 48, -72, 18, -16, 40

Offset: 1

N. J. A. Sloane, Oct 31 2001

sign

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A034448.

usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) } { u=1; for (n=1, 1000, u1=usigma(n + 1); a=u1 - u; u=u1; write("b064992.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 02 2009

a(n) = A034448(n+1) - A034448(n).

A258101 Number x such that usigma(x) = (-1)sigma(x), where usigma(x) is the sum of unitary divisors of x (A034448) and (-1)sigma(x) is defined in A049060 .

Original entry on oeis.org

1, 4, 15867, 21357, 49887, 63468, 69875, 85428, 86387, 149875, 199548, 247475, 271607, 279500, 293944, 318681, 345548, 599500, 637659, 989900, 1086428, 1169091, 1274724, 1897875, 1913571, 2550636, 2665269, 2801880, 2855691

Offset: 1

Paolo P. Lava, May 20 2015

nonn

			usigma(1) = (-1)sigma(1) = 1;
usigma(4) =  (-1)sigma(4) = 5;
usigma(15867) = (-1)sigma(15867) = 18480; etc.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A034448, A049060, A258106.

with(numtheory): P:=proc(q) local a,b,d,k,n;
a:=0; for n from 1 to q do a:=divisors(n); d:=0; for k from 1 to nops(a)
do if gcd(a[k],n/a[k])=1 then d:=d+a[k]; fi; od; a:=ifactors(n)[2]; b:=1;
for k from 1 to nops(a) do b:=b*(-1+sum(a[k][1]^j,j=1..a[k][2])); od;
if b=d then print(n); fi; od; end: P(10^9);

aQ[n_] := Module[{f = FactorInteger[n]}, p = f[[;;,1]]; e = f[[;;,2]]; Times@@(p^e+1) == Times@@((p^(e+1)-2*p+1)/(p-1))]; Join[{1}, Select[Range[2, 200000 ], aQ]] (* Amiram Eldar, Jun 23 2019 *)

cp's OEIS Frontend

A348504 a(n) = sigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

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A348506 Numbers k such that sigma(k) is a multiple of usigma(k), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

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A348732 a(n) = A003959(n) - A034448(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

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A349000 a(n) = A323166(A276086(n)), where A323166(n) = gcd(n, usigma(n)), usigma (A034448) is multiplicative with a(p^e) = (p^e)+1, and A276086 gives the prime product form of primorial base expansion of n.

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A379515 Numerators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

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A379516 Denominators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

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A063845 a(n) = sigma(usigma(n)), where usigma(n) is the sum of unitary divisors of n (A034448) and sigma(n) is the sum of the divisors (A000203).

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A064212 a(n) = sigma(n) + usigma(n) = A000203(n) + A034448(n).

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