A162296 -id:A162296 - cp's OEIS frontend

A325314 a(n) = n - A162296(n), where A162296(n) is the sum of divisors of n that have a square factor.

1, 2, 3, 0, 5, 6, 7, -4, 0, 10, 11, -4, 13, 14, 15, -12, 17, -9, 19, -4, 21, 22, 23, -24, 0, 26, -9, -4, 29, 30, 31, -28, 33, 34, 35, -43, 37, 38, 39, -32, 41, 42, 43, -4, -9, 46, 47, -64, 0, -25, 51, -4, 53, -54, 55, -40, 57, 58, 59, -36, 61, 62, -9, -60, 65, 66, 67, -4, 69, 70, 71, -111, 73, 74, -25, -4, 77, 78, 79, -88, -36, 82, 83

Offset: 1

Antti Karttunen, Apr 21 2019

sign

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

Cf. A013661, A033879, A162296, A228058, A325313, A325315, A325320.

Array[# - DivisorSigma[1, #] + Times @@ (1 + FactorInteger[#][[;; , 1]]) - Boole[# == 1] &, 83] (* Michael De Vlieger, Jun 06 2019, after Amiram Eldar at A162296 *)

A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));

a(n) = n - A162296(n).
a(n) = A033879(n) + A325313(n).
a(A228058(n)) = -A325320(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.1775329665... . - Amiram Eldar, Feb 22 2024

A325385 a(n) = gcd(n-A048250(n), n-A162296(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 5, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 19, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 41, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 3, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, Apr 24 2019

nonn

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

Cf. A048250, A126795, A162296, A228058, A325313, A325314, A325375.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325385(n) = gcd(n-A048250(n),n-A162296(n));

a(n) = gcd(n-A048250(n), n-A162296(n)).
a(n) = gcd(A325313(n), A325314(n)).
a(A228058(n)) = A325375(n).

A325974 Arithmetic mean of {sum of non-unitary divisors} and {sum of nonsquarefree divisors}: a(n) = (1/2)*(A048146(n) + A162296(n)).

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 0, 9, 6, 0, 0, 12, 0, 0, 0, 21, 0, 18, 0, 18, 0, 0, 0, 36, 15, 0, 24, 24, 0, 0, 0, 45, 0, 0, 0, 60, 0, 0, 0, 54, 0, 0, 0, 36, 36, 0, 0, 84, 28, 45, 0, 42, 0, 72, 0, 72, 0, 0, 0, 72, 0, 0, 48, 93, 0, 0, 0, 54, 0, 0, 0, 144, 0, 0, 60, 60, 0, 0, 0, 126, 78, 0, 0, 96, 0, 0, 0, 108, 0, 108, 0, 72, 0, 0, 0, 180, 0, 84, 72

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

			For n = 36, its divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Of these, non-unitary divisors are 2, 3, 6, 12 and 18 so A048146(36) = 2+3+6+12+18 = 41, while the nonsquarefree divisors are 4, 9, 12, 18 and 36, so A162296(36) = 4+9+12+18+36 = 79, thus a(36) = (41+79)/2 = 60.

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

Cf. A000203, A005117, A034448, A048146, A162296, A325973, A325975, A325978.

Cf. A002117, A013661.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048146(n) = (sigma(n)-A034448(n));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325974(n) = ((A048146(n)+A162296(n))/2);

a(n) = (1/2)*(A048146(n) + A162296(n)).
a(n) = A000203(n) - A325973(n).
a(n) = n - A325978(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)*(1/2 - 1/(4*zeta(3))) - 1/4 = 0.2303588390... . - Amiram Eldar, Feb 22 2024

A322609 Numbers k such that s(k) = 2*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

24, 54, 112, 150, 294, 726, 1014, 1734, 1984, 2166, 3174, 5046, 5766, 8214, 10086, 11094, 13254, 16854, 19900, 20886, 22326, 26934, 30246, 31974, 32512, 37446, 41334, 47526, 56454, 61206, 63654, 68694, 71286, 76614, 96774, 102966, 112614, 115926, 133206

Offset: 1

Amiram Eldar, Dec 20 2018

nonn

This sequence is infinite since 6*p^2 is included for all primes p. Terms that are not of the form 6*p^2: 112, 1984, 19900, 32512, 134201344, ...
Includes 4*k if k is an even perfect number: see A000396. - Robert Israel, Jan 06 2019
From Amiram Eldar, Oct 01 2022: (Start)
24 = 6*prime(1)^2 = 4*A000396(1) is the only term that is common to the two forms that are mentioned above.
19900 is the only term below 10^11 which is not of any of these two forms. Are there any other such terms?
All the known nonunitary perfect numbers (A064591) are also of the form 4*k, where k is an even perfect number.
Equivalently, numbers k such that A325314(k) = -k. (End)

			24 is a term since A162296(24) = 48 = 2*24.

Amiram Eldar, Table of n, a(n) for n = 1..12091 (terms below 10^11; terms 1..300 from Robert Israel)

Cf. A162296, A325314.
Subsequence of A005101 and A013929.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), this sequence (m=2), A357493 (m=3), A357494 (m=4).
Similar sequences: A000396, A002827, A007357, A054979, A064591, A322486, A323757, A324707, A327633.

filter:= proc(n) convert(remove(numtheory:-issqrfree,numtheory:-divisors(n)),`+`)=2*n end proc:
select(filter, [$1..200000]); # Robert Israel, Jan 06 2019

s[1]=0; s[n_] := DivisorSigma[1,n] - Times@@(1+FactorInteger[n][[;;,1]]); Select[Range[10000], s[#] == 2# &]

s(n) = sumdiv(n, d, d*(1-moebius(d)^2)); \\ A162296
isok(n) = s(n) == 2*n; \\ Michel Marcus, Dec 20 2018

from sympy import divisors, factorint
A322609_list = [k for k in range(1,10**3) if sum(d for d in divisors(k,generator=True) if max(factorint(d).values(),default=1) >= 2) == 2*k] # Chai Wah Wu, Sep 19 2021

A325378 a(n) = A162296(A228058(n)) - A048250(A228058(n)).

Original entry on oeis.org

30, 70, 90, 246, 150, 266, 190, 210, 678, 342, 270, 310, 654, 574, 370, 570, 450, 738, 930, 490, 510, 722, 550, 570, 798, 1582, 690, 1026, 750, 2034, 790, 1230, 1626, 1178, 870, 1526, 910, 970, 990, 2046, 1558, 1406, 1722, 1962, 1150, 1170, 1210, 4062, 1710, 1290, 3390, 1350, 1862, 1390, 1938, 1410, 2214, 1470, 3030

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..25000

Cf. A048250, A162296, A228058, A325319, A325320, A325379.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
k=0; n=0; while(k<100,n++; if(isA228058(n), k++; print1(A162296(n) - A048250(n),", ")));

a(n) = A162296(A228058(n)) - A048250(A228058(n)).

a(n) = A325319(n) + A325320(n).

A357493 Numbers k such that s(k) = 3*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

480, 2688, 56304, 89400, 195216, 2095104, 9724032, 69441408, 1839272960, 5905219584

Offset: 1

Amiram Eldar, Oct 01 2022

Analogous to 3-perfect numbers (A005820) with nonsquarefree divisors.
Equivalently, numbers k such that A325314(k) = -2*k.
a(11) > 10^11, if it exists.
If k is one of the 6 known 3-perfect numbers, then 4*k is a term.

			480 is a term since A162296(480) = 1440 = 3*480.

Cf. A162296, A325314.
Subsequence of A013929 and A068403.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), A322609 (m=2), this sequence (m=3), A357494 (m=4).
Similar sequence: A005820.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) == 3*n]; Select[Range[2, 10^7], q]

A357494 Numbers k such that s(k) = 4*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

902880, 1534680, 361674720, 767685600, 4530770640, 4941414720, 5405788800, 5517818880, 16993944000, 20429240832, 94820077440

Offset: 1

Amiram Eldar, Oct 01 2022

Analogous to 4-perfect numbers (A027687) with nonsquarefree divisors.
Equivalently, numbers k such that A325314(k) = -3*k.
There are no numbers k below 10^11 such that A162296(k) = m*k for integers m > 4.

			902880 is a term since A162296(902880) = 3611520 = 4*902880.

Cf. A162296, A325314.
Subsequence of A013929 and A023198.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), A322609 (m=2), A357493 (m=3), this sequence (m=4).
Similar sequence: A027687.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) == 4*n]; Select[Range[2, 2*10^6], q]

A357605 Numbers k such that A162296(k) > 2*k.

Original entry on oeis.org

36, 48, 72, 80, 96, 108, 120, 144, 160, 162, 168, 180, 192, 200, 216, 224, 240, 252, 264, 270, 280, 288, 300, 312, 320, 324, 336, 352, 360, 378, 384, 392, 396, 400, 408, 416, 432, 448, 450, 456, 468, 480, 486, 500, 504, 528, 540, 552, 560, 576, 588, 594, 600, 612

Offset: 1

Amiram Eldar, Oct 06 2022

nonn

The least odd term is a(470) = A357607(1) = 4725.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 5, 92, 1011, 10160, 102125, 1022881, 10231151, 102249758, 1022781199, 10229781638, ... . Apparently, the asymptotic density of this sequence exists and equals 0.102... .
An analog of abundant numbers, in which the divisor sum is restricted to nonsquarefree divisors. - Peter Munn, Oct 26 2022

			36 is a term since A162296(36) = 79 > 2*36.

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

Cf. A162296.
Subsequence of A005101 and A013929.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; Select[Range[2, 1000], q]

A362401 Numbers in the range of A162296, where A162296(n) is the sum of divisors of n that have a square factor larger than 1.

Original entry on oeis.org

0, 4, 9, 12, 16, 24, 25, 27, 28, 32, 36, 48, 49, 54, 56, 60, 72, 75, 79, 80, 96, 100, 108, 112, 117, 120, 121, 124, 126, 128, 144, 147, 150, 152, 162, 168, 169, 176, 180, 183, 192, 196, 199, 200, 216, 224, 240, 248, 252, 268, 270, 272, 288, 289, 294, 296, 300

Offset: 1

Amiram Eldar, Apr 18 2023

nonn

Possible values of A162296 in increasing order.

			0 is a term since A162296(k) = 0 if k is squarefree (A005117).

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

Cf. A005117, A162296.

Similar sequences: A078923, A002191, A002202, A002174, A274790.

s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; m = 300; Select[Union[Array[s, m]], # <= m &]

s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = select(x -> (x < kmax), Set(vector(kmax, k, s(k))))

A325315 Bitwise-XOR of absolute values of (n - A048250(n)) and (n - A162296(n)).

Original entry on oeis.org

1, 3, 2, 1, 4, 0, 6, 1, 5, 2, 10, 4, 12, 4, 6, 1, 16, 15, 18, 6, 30, 24, 22, 20, 19, 10, 30, 0, 28, 52, 30, 1, 46, 54, 46, 51, 36, 48, 54, 54, 40, 28, 42, 12, 28, 52, 46, 100, 41, 57, 38, 14, 52, 28, 38, 8, 46, 26, 58, 40, 60, 28, 22, 1, 82, 12, 66, 10, 94, 12, 70, 83, 72, 98, 42, 20, 94, 20, 78, 102, 105, 126, 82, 32, 66, 120, 118, 12

Offset: 1

Antti Karttunen, Apr 21 2019

nonn

Cf. A000396, A003987, A028982 (positions of odd terms), A048250, A162296, A228058, A325310, A325313, A325314.

Array[BitXor @@ Abs[#1 - Map[Total, {#3, Complement[#2, #3]}]] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 88] (* Michael De Vlieger, Apr 21 2019 *)

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A325315(n) = bitxor(abs(A325313(n)),abs(A325314(n)));

a(n) = A003987(abs(A325313(n)), abs(A325314(n))).

cp's OEIS Frontend

A325314 a(n) = n - A162296(n), where A162296(n) is the sum of divisors of n that have a square factor.

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A325385 a(n) = gcd(n-A048250(n), n-A162296(n)).

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A325974 Arithmetic mean of {sum of non-unitary divisors} and {sum of nonsquarefree divisors}: a(n) = (1/2)*(A048146(n) + A162296(n)).

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A322609 Numbers k such that s(k) = 2*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

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A325378 a(n) = A162296(A228058(n)) - A048250(A228058(n)).

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A357493 Numbers k such that s(k) = 3*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

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A357494 Numbers k such that s(k) = 4*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

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A357605 Numbers k such that A162296(k) > 2*k.

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