A034448 -id:A034448 - cp's OEIS frontend

A061765 a(n) = usigma(sigma(n)), where usigma(n) is the sum of unitary divisors of n (A034448) and sigma(n) is the sum of the divisors (A000203).

1, 4, 5, 8, 12, 20, 9, 24, 14, 30, 20, 40, 24, 36, 36, 32, 30, 56, 30, 96, 33, 50, 36, 120, 32, 96, 54, 72, 72, 90, 33, 80, 68, 84, 68, 112, 60, 120, 72, 180, 96, 132, 60, 160, 168, 90, 68, 160, 80, 128, 90, 150, 84, 216, 90, 216, 102, 180, 120, 288

Offset: 1

N. J. A. Sloane, Oct 30 2001

nonn

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000203, A034448.

a[n_] := Times @@ (1 + Power @@@ FactorInteger[DivisorSigma[1, n]]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 26 2022 *)

a(n) = A034448(A000203(n)). - Amiram Eldar, Aug 26 2022

A285906 Numbers n such that sigma(n)/usigma(n) > sigma(m)/usigma(m) for all m < n, where sigma(n) is the sum of divisors of n (A000203) and usigma(n) is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 72, 144, 216, 288, 432, 864, 1728, 2592, 3456, 3600, 5184, 7200, 10800, 21600, 43200, 64800, 86400, 108000, 129600, 216000, 259200, 324000, 432000, 518400, 529200, 648000, 1058400, 2116800, 3175200, 4233600, 5292000, 6350400, 10584000

Offset: 1

Amiram Eldar, Apr 28 2017

nonn

This sequence is infinite. The smallest values of n for which sigma(n)/usigma(n) > 2, 3 and 4 are a(7), a(19), and a(44).

			sigma(72)=195, usigma(72)=90, and their ratio 195/90=13/6 is higher than sigma(m)/usigma(m) for all m<72, thus 72 is in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..56

Cf. A000203, A034448.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
a={}; rmax=0; Do[r=DivisorSigma[1, n]/usigma[n]; If[r>rmax, AppendTo[a, n]; rmax=r], {n, 3000}]; a

A348944 a(n) = (1/2) * (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

1, 3, 4, 7, 6, 12, 8, 18, 13, 18, 12, 28, 14, 24, 24, 49, 18, 39, 20, 42, 32, 36, 24, 72, 31, 42, 46, 56, 30, 72, 32, 138, 48, 54, 48, 97, 38, 60, 56, 108, 42, 96, 44, 84, 78, 72, 48, 196, 57, 93, 72, 98, 54, 138, 72, 144, 80, 90, 60, 168, 62, 96, 104, 397, 84, 144, 68, 126, 96, 144, 72, 261, 74, 114, 124, 140, 96

Offset: 1

Antti Karttunen, Nov 05 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 36 = 2^2 * 3^2, where a(36) = 97 != 91 = 7*13 = a(4)*a(9).

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

Arithmetic mean of A003959 and A034448.
Cf. A348732, A348733, A348945.
Cf. also A325973.

f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[1] = 1; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f) / 2; Array[a, 100] (* Amiram Eldar, Nov 05 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
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));

A348945 a(n) = A348944(n) - sigma(n), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 0, 0, 75, 0, 0, 0, 6, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 18, 0, 24, 0, 0, 0, 0, 0, 0, 0, 270, 0, 0, 0, 0, 0, 0, 0, 66, 0, 0, 0, 0, 0, 0, 0, 108, 48, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 300, 0, 0, 0, 10

Offset: 1

Antti Karttunen, Nov 05 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A003959, A034448, A048146, A348029, A348944, A348946.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 0; a[n_] := (Times @@ f2 @@@ (f = FactorInteger[n]) + Times @@ f3 @@@ f) / 2 - Times @@ f1 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 05 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])); };
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
A348945(n) = (A348944(n)-sigma(n));

a(n) = A348944(n) - A000203(n) = ((1/2) * (A003959(n)+A034448(n))) - A000203(n).

a(n) = (1/2) * (A348029(n)-A048146(n)).

A055033 a(n) = usigma(usigma(n)), where usigma(n) is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 4, 5, 6, 12, 20, 9, 10, 18, 30, 20, 30, 24, 36, 36, 18, 30, 72, 30, 72, 33, 50, 36, 50, 42, 96, 40, 54, 72, 90, 33, 48, 68, 84, 68, 78, 60, 120, 72, 84, 96, 132, 60, 120, 120, 90, 68, 90, 78, 168, 90, 144, 84, 160, 90, 90, 102, 180, 120, 216, 96

Offset: 1

N. J. A. Sloane, Oct 30 2001

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A034448, A051027, A061765, A064012.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; a[n_] := usigma[usigma[n]]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
a(n) = usigma(usigma(n)); \\ Amiram Eldar, Jul 24 2024

a(n) = A034448(A034448(n)). - Amiram Eldar, Jul 24 2024

A254503 Möbius transform of A034448.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 4, 6, 10, 11, 6, 13, 14, 15, 8, 17, 12, 19, 10, 21, 22, 23, 12, 20, 26, 18, 14, 29, 30, 31, 16, 33, 34, 35, 12, 37, 38, 39, 20, 41, 42, 43, 22, 30, 46, 47, 24, 42, 40, 51, 26, 53, 36, 55, 28, 57, 58, 59, 30, 61, 62, 42, 32, 65, 66, 67, 34, 69, 70

Offset: 1

Álvar Ibeas, Jan 31 2015

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Cf. A000010 (totient), A001694 (powerful), A005117 (squarefree), A034448 (usigma), A057521 (powerful part), A055231 (unitary squarefree kernel).

Table[DivisorSum[n, MoebiusMu[#]^2*EulerPhi[n/#] &, CoprimeQ[n/#, #] &], {n, 70}] (* Michael De Vlieger, Jun 27 2018 *)
f[p_, e_] := (p - 1)*p^(e - 1); f[p_, 1] := p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((e=f[i, 2]) > 1, f[i, 1] = eulerphi(f[i, 1]^e); f[i, 2] = 1);); factorback(f);} \\ Michel Marcus, Feb 06 2015

a(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, moebius(d)^2 * eulerphi(n/d))); \\ Daniel Suteu, Jun 27 2018

a(n) = phi(A057521(n)) * A055231(n).
If n is squarefree, a(n) = n; if n is powerful, a(n) = phi(n).
Multiplicative with a(p) = p; a(p^e) = phi(p^e), for e > 1.
Dirichlet g.f.: zeta(s-1) / zeta(2s-1).
a(n) = Sum_{d|n, gcd(n/d, d) = 1} mu(d)^2 * phi(n/d). - Daniel Suteu, Jun 27 2018
Sum_{k=1..n} a(k) ~ n^2 / (2*zeta(3)). - Vaclav Kotesovec, Jan 11 2019

A302572 Unitary barely deficient numbers: unitary deficient numbers k such that usigma(k)/k > usigma(m)/m for all unitary deficient numbers m < k, where usigma(k) is the sum of the unitary divisors of k (A034448).

Original entry on oeis.org

1, 2, 10, 84, 110, 1155, 6490, 34320, 55335, 80652, 163212, 449295, 676390, 1360810, 1503370, 1788490, 3214090, 22627605, 32062485, 35604492, 103712410, 365690892, 615206030, 815634435

Offset: 1

Amiram Eldar, Apr 10 2018

			The values of usigma(k)/k are 1, 1.5, 1.8, 1.904..., 1.963..., 1.994...

Cf. A034448, A071927, A302570.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; seq = {}; r = 0; Do[s = usigma[n]/n; If[s < 2 && s > r, AppendTo[seq, n]; r = s], {n, 1, 1000000}]; seq

A323163 Greatest common divisor of product (1+(p^e)) and product p^(e-1), where p ranges over prime factors of n, with e corresponding exponent; a(n) = gcd(A034448(n), A003557(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 3, 10, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A034448, A322318, A323159.

Differs from A062760 for the first time at n=36, where a(36) = 2, while A062760(36) = 1.

A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323163(n) = gcd(A003557(n), A034448(n));

a(n) = gcd(A003557(n), A034448(n)).

A325812 Numbers k such that gcd(A034448(k)-k, k-A048146(k)) is equal to abs(k-A048146(k)).

Original entry on oeis.org

1, 6, 12, 28, 56, 60, 108, 120, 132, 168, 264, 280, 312, 408, 420, 440, 456, 496, 528, 540, 552, 696, 700, 728, 744, 756, 760, 888, 984, 992, 1032, 1128, 1140, 1188, 1272, 1404, 1416, 1456, 1464, 1608, 1704, 1710, 1752, 1836, 1896, 1992, 2052, 2136, 2328, 2424, 2472, 2484, 2568, 2616, 2646, 2712, 3048, 3132, 3144, 3288, 3336, 3344

Offset: 1

Antti Karttunen, May 23 2019

nonn

Numbers k for which A325813(k) is equal to abs(A325814(k)).
Numbers k such that A325814(k) is not zero (not in A064591) and divides A034460(k).
Conjecture: after the initial one all other terms are even. If this holds then there are no odd perfect numbers.

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences where any odd perfect numbers must occur

Cf. A034448, A034460, A048146, A064591, A325813, A325814, A325822.

Cf. A000396 (a subsequence).

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325813(n) = gcd(A034460(n), A325814(n));
isA325812(n) = (A325813(n)==abs(A325814(n)));
\\ Alternatively:
isA325812(n) = (A325814(n) && !(A034460(n)%A325814(n)));

A325963 Numbers n for which A034448(n)-n is equal to n-A048250(n).

Original entry on oeis.org

1, 4, 24, 240, 349440

Offset: 1

Antti Karttunen, Jun 04 2019

No other terms below 536870912 (2^29).

a(6) > 10^12, if it exists. - Giovanni Resta, Jun 07 2019

Cf. A034448, A048250, A034460, A325313.

Positions of zeros in A325977.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
isA325963(n) = ((A034448(n)-n) == (n-A048250(n)));

cp's OEIS Frontend

A061765 a(n) = usigma(sigma(n)), where usigma(n) is the sum of unitary divisors of n (A034448) and sigma(n) is the sum of the divisors (A000203).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A285906 Numbers n such that sigma(n)/usigma(n) > sigma(m)/usigma(m) for all m < n, where sigma(n) is the sum of divisors of n (A000203) and usigma(n) is the sum of unitary divisors of n (A034448).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A348944 a(n) = (1/2) * (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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A348945 a(n) = A348944(n) - sigma(n), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A055033 a(n) = usigma(usigma(n)), where usigma(n) is the sum of unitary divisors of n (A034448).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A254503 Möbius transform of A034448.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A302572 Unitary barely deficient numbers: unitary deficient numbers k such that usigma(k)/k > usigma(m)/m for all unitary deficient numbers m < k, where usigma(k) is the sum of the unitary divisors of k (A034448).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A323163 Greatest common divisor of product (1+(p^e)) and product p^(e-1), where p ranges over prime factors of n, with e corresponding exponent; a(n) = gcd(A034448(n), A003557(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A325812 Numbers k such that gcd(A034448(k)-k, k-A048146(k)) is equal to abs(k-A048146(k)).

Views