A034448 -id:A034448 - cp's OEIS frontend

A034460 a(n) = usigma(n) - n, where usigma(n) = sum of unitary divisors of n (A034448).

0, 1, 1, 1, 1, 6, 1, 1, 1, 8, 1, 8, 1, 10, 9, 1, 1, 12, 1, 10, 11, 14, 1, 12, 1, 16, 1, 12, 1, 42, 1, 1, 15, 20, 13, 14, 1, 22, 17, 14, 1, 54, 1, 16, 15, 26, 1, 20, 1, 28, 21, 18, 1, 30, 17, 16, 23, 32, 1, 60, 1, 34, 17, 1, 19, 78, 1, 22, 27, 74, 1, 18, 1, 40, 29, 24, 19, 90, 1, 22, 1, 44

Offset: 1

N. J. A. Sloane

			Unitary divisors of 12 are 1, 3, 4, 12. a(12) = 1 + 3 + 4 = 8.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Mathematics of Computation 83.288 (2014): 1903-1913; alternative link.

Cf. A034444, A034448, A077610, A306633.
Cf. A063936 (squares > 1).
Cf. A063919 (essentially the same sequence).

a034460 = sum . init . a077610_row  -- Reinhard Zumkeller, Aug 15 2012

A034460 := proc(n)
    A034448(n)-n ;
end proc:
seq(A034460(n),n=1..40) ; # R. J. Mathar, Nov 10 2014

usigma[n_] := Sum[ If[GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; a[n_] := usigma[n] - n; Table[ a[n], {n, 1, 82}] (* Jean-François Alcover, May 15 2012 *)
a[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; a[1] = 0; Array[a, 100] (* Amiram Eldar, Oct 03 2022 *)

a(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n \\ Charles R Greathouse IV, Aug 01 2016

a(n) = Sum_{k = 1..A034444(n)-1} A077610(n,k). - Reinhard Zumkeller, Aug 15 2012

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) - 1)/2 = 0.1842163888... . - Amiram Eldar, Feb 22 2024

A064125 Numbers k such that k and k+1 have the same sum of unitary divisors (A034448).

Original entry on oeis.org

14, 44, 55, 152, 957, 1334, 1400, 1634, 1652, 2204, 2232, 2295, 2685, 3195, 3451, 3956, 4256, 5547, 7191, 8216, 8495, 8636, 8907, 9144, 9503, 9844, 10152, 11515, 17255, 18423, 19491, 20145, 20155, 27404, 27643, 30247, 33998, 38180, 41265

Offset: 1

Jason Earls, Sep 10 2001

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 800 terms from Harry J. Smith)

usigma(n,s=1,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1], s=s*(1+fac[i,1]^fac[i,2])); return(s); j=[]; for(n=1,50000, if(usigma(n)==usigma(n+1),j=concat(j,n))); j

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) }
{ n=0; s=0; for (m=1, 10^9, us=usigma(m+1); if(s==us, write("b064125.txt", n++, " ", m); if (n==800, break)); s=us ) } \\ Harry J. Smith, Sep 08 2009

A325813 a(n) = gcd(A034448(n)-n, n-A048146(n)), where A034448 and A048146 are respectively the sum of unitary and non-unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 12, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 7, 3, 6, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 12, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 21, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, May 23 2019

nonn

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

Cf. A034460, A323166, A325812, A325814.

Cf. also A325385.

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));

a(n) = gcd(A034460(n), A325814(n)).

A325973 Arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}: a(n) = (1/2) * (A034448(n) + A048250(n)).

Original entry on oeis.org

1, 3, 4, 4, 6, 12, 8, 6, 7, 18, 12, 16, 14, 24, 24, 10, 18, 21, 20, 24, 32, 36, 24, 24, 16, 42, 16, 32, 30, 72, 32, 18, 48, 54, 48, 31, 38, 60, 56, 36, 42, 96, 44, 48, 42, 72, 48, 40, 29, 48, 72, 56, 54, 48, 72, 48, 80, 90, 60, 96, 62, 96, 56, 34, 84, 144, 68, 72, 96, 144, 72, 51, 74, 114, 64, 80, 96, 168, 80, 60, 43, 126

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

This is not multiplicative: a(4) = 4, a(9) = 7, but a(36) = 31, not 28. However, the function acts multiplicatively on certain subsequences of natural numbers, like for example when restricted to A048107, where this sequence coincides with A326043.

			For n = 36, its divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Of these, unitary divisors are 1, 4, 9 and 36, so A034448(36) = 1+4+9+36 = 50, while the squarefree divisors are 1, 2, 3 and 6, so A048250(36) = 1+2+3+6 = 12, thus a(36) = (50+12)/2 = 31.
For n = 495, its divisors are 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495. Of these, unitary are 1, 5, 9, 11, 45, 55, 99, 495, whose sum is A034448(495) = 720, while the squarefree divisors are 1, 3, 5, 11, 15, 33, 55, 165, and their sum is A048250(495) = 288. Thus a(495) = (720+288)/2 = 504. Also for 495, whose prime factorization is 3^2 * 5^1 * 11^1 this can be computed faster as the average of ((3^2)+1)*(5+1)*(11+1) and (3+1)*(5+1)*(11+1), thus (1/2)*(3+(3^2)+2)*(5+1)*(11+1) = 504.

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

Cf. A000203, A034448, A048107, A048250, A052548, A289521, A306633, A325974, A325975, A325977, A325978, A325981, A326043.

Array[(1/2) If[# == 1, 2, Times @@ (1 + Power @@@ #) + Times @@ (1 + #[[;; , 1]]) &@ FactorInteger[#]] &, 90] (* Michael De Vlieger, Jun 06 2019, after Giovanni Resta at A034448 and Amiram Eldar at A048250. *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325973(n) = ((A034448(n)+A048250(n))/2);

A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));

a(n) = (1/2) * (A034448(n) + A048250(n)).
a(n) = A000203(n) - A325974(n).
a(n) = n + A325977(n).
a(A048107(n)) = A326043(A048107(n)).
For n >= 1, a(2^n) = A052548(n-1) = 2^(n-1) + 2.
For n >= 1, a(3^n) = A289521(n) = (3^n + 5)/2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) + 1)/4 = 0.5921081944... . - Amiram Eldar, Feb 22 2024

A064611 Partial sum of usigma is divisible by n, where usigma(n) = A034448(n) and summatory-usigma(n) = A064609(n).

Original entry on oeis.org

1, 2, 8, 11, 12, 174, 212, 524, 1567, 14096, 19795, 38466, 42114, 55575, 338809, 498001, 1175281, 2424880, 3994532, 7908519, 48453784, 696840720, 5497869355, 7479239685

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056550, A064605-A064607, A064610, A064612, A048290, A062982, A045345.

			udivisor sums[=usigma(j) values] from 1 to 8 are added: 1+3+4+5+6+12+8+9=48; it is divisible by 8, thus 8 is here.

Amiram Eldar, Table of n, a(n), A064609(a(n))/a(n) for n=1..24.

Cf. A034448, A064609, A050226, A056550, A064605, A064606, A064607, A064610, A064612, A048290, A062982, A045345.

s = Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 10^6}]; Module[{a = First@ s, b = {First@ s}}, Do[a += s[[i]]; If[Divisible[a, i], AppendTo[b, i]], {i, 2, Length@ s}]; b] (* Michael De Vlieger, Mar 18 2017 *)

A064609(n) mod n = 0.

a(17)-a(22) from Donovan Johnson, Jul 20 2012

a(23)-a(24) from Amiram Eldar, Mar 17 2019

A064609 Partial sums of A034448: sum of unitary divisors from 1 to n.

Original entry on oeis.org

1, 4, 8, 13, 19, 31, 39, 48, 58, 76, 88, 108, 122, 146, 170, 187, 205, 235, 255, 285, 317, 353, 377, 413, 439, 481, 509, 549, 579, 651, 683, 716, 764, 818, 866, 916, 954, 1014, 1070, 1124, 1166, 1262, 1306, 1366, 1426, 1498, 1546, 1614, 1664, 1742, 1814, 1884

Offset: 1

Labos Elemer, Sep 24 2001

nonn

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

Cf. A034448, A064611.

Accumulate@ Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 52}] (* Michael De Vlieger, Mar 18 2017 *)

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) }
{ a=0; for (n=1, 1000, a+=usigma(n); write("b064609.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 20 2009

from sympy.ntheory.factor_ import udivisor_sigma
def a(n): return sum(udivisor_sigma(j,1) for j in range(1,n + 1))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 18 2017

a(n) = a(n-1) + A034448(n) = Sum_{j=1..n} usigma(j) where usigma(j) = A034448(j).

a(n) ~ Pi^2 * n^2 / (12*Zeta(3)). - Vaclav Kotesovec, Jan 11 2019

A323166 Greatest common divisor of Product (1+(p_i^e_i)) and n, when n = Product (p_i^e_i); a(n) = gcd(A034448(n), n).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 6, 1, 10, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 4, 15, 2, 1, 4, 1, 2, 3, 2, 1, 6, 1, 8, 1, 2, 1, 60, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 18, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 90, 7, 4, 1, 2, 5, 12, 1, 2, 3, 10, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A034448, A323160, A323163.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323166(n) = gcd(n, A034448(n));

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

A285614 Unitary highly abundant numbers: numbers n such that usigma(n) > usigma(m) for all m < n, where usigma(n) = sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 12, 14, 18, 21, 22, 26, 30, 42, 60, 66, 78, 90, 102, 114, 130, 138, 150, 170, 174, 186, 210, 294, 318, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1050, 1110, 1218, 1230, 1290, 1410, 1470, 1554, 1590, 1722, 1770, 1830, 1974

Offset: 1

Amiram Eldar, Apr 22 2017

nonn

Corresponds to A002093 (Highly abundant numbers), with usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1, A034448) instead of sigma(n) (sum of divisors, A000203).

Contains many terms of A280013 (sum of squarefree divisors instead of unitary divisors), but not all of them - the first terms of A280013 that are not in this sequence are 16530, 26070, 8734110, 8757210,...

			The first 9 values of usigma(n) for n=1..9 are: 1, 3, 4, 5, 6, 12, 8, 9, 10. usigma(10)=18 is higher than these 9 values, thus 10 is in the sequence.

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

Cf. A034448, A002093, A034683, A129499, A280013.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; a = {}; k = 0; Do[s = usigma[n]; If[s > k, AppendTo[a, n]; k = s], {n, 1000}]; a

A327158 Unitary multiply-perfect numbers: n divides usigma(n), where usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 6, 60, 90, 87360

Offset: 1

Antti Karttunen, Aug 28 2019

10^13 < a(6) <= 146361946186458562560000. - Giovanni Resta, Aug 29 2019

Peter Hagis, Jr., Lower bounds for unitary multiperfect numbers, Fibonacci Quarterly, Vol. 22, No. 2 (1984), pp. 140-143.

Fixed points of A323166, positions of zeros in A327164.
Cf. A002827 (a subsequence), A034448, A327163.
Cf. also A007691.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
isA327158(n) = (gcd(n,A034448(n))==n);

A063974 Number of terms in inverse set of usigma = sum of unitary divisors = A034448.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 2, 0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 0, 3, 0, 1, 0, 1, 0, 3, 0, 2, 1, 0, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 6, 0, 1, 0, 0, 0, 1, 0, 3, 0, 1, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 1, 0, 1, 0, 2, 0, 2, 0

Offset: 1

Labos Elemer, Sep 05 2001

nonn

			usigma(x) = 288, invusigma(288) = {138,154,165,168,213,235,248,253}, so a(288) = 8, the number of all terms in the inverse set.

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

Cf. A034444, A034448, A051444, A054973, A057637, A308041.

Size of set {x; usigma(x) = n}.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A308041. - Amiram Eldar, Dec 23 2024

cp's OEIS Frontend

A034460 a(n) = usigma(n) - n, where usigma(n) = sum of unitary divisors of n (A034448).

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A064125 Numbers k such that k and k+1 have the same sum of unitary divisors (A034448).

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A325813 a(n) = gcd(A034448(n)-n, n-A048146(n)), where A034448 and A048146 are respectively the sum of unitary and non-unitary divisors of n.

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A325973 Arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}: a(n) = (1/2) * (A034448(n) + A048250(n)).

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A064611 Partial sum of usigma is divisible by n, where usigma(n) = A034448(n) and summatory-usigma(n) = A064609(n).

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A064609 Partial sums of A034448: sum of unitary divisors from 1 to n.

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A323166 Greatest common divisor of Product (1+(p_i^e_i)) and n, when n = Product (p_i^e_i); a(n) = gcd(A034448(n), n).

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A285614 Unitary highly abundant numbers: numbers n such that usigma(n) > usigma(m) for all m < n, where usigma(n) = sum of unitary divisors of n (A034448).

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