A162296 -id:A162296 - cp's OEIS frontend

A325310 a(n) = A001511(A325315(n)), except when A325315(n) = 0, then a(n) = 0.

1, 1, 2, 1, 3, 0, 2, 1, 1, 2, 2, 3, 3, 3, 2, 1, 5, 1, 2, 2, 2, 4, 2, 3, 1, 2, 2, 0, 3, 3, 2, 1, 2, 2, 2, 1, 3, 5, 2, 2, 4, 3, 2, 3, 3, 3, 2, 3, 1, 1, 2, 2, 3, 3, 2, 4, 2, 2, 2, 4, 3, 3, 2, 1, 2, 3, 2, 2, 2, 3, 2, 1, 4, 2, 2, 3, 2, 3, 2, 2, 1, 2, 2, 6, 2, 4, 2, 3, 4, 2, 2, 5, 2, 3, 2, 3, 6, 1, 2, 1, 3, 3, 2, 2, 2

Offset: 1

Antti Karttunen, Apr 21 2019

nonn

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

Cf. A000396, A001511, A028982 (gives the positions of 1's), A048250, A162296, A228058, A325313, A325314, A325315, A325378, A325379.

Array[If[# == 0, 0, IntegerExponent[2 #, 2]] &[BitXor @@ Abs[#1 - Map[Total, {#3, Complement[#2, #3]}]]] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 105] (* Michael De Vlieger, Apr 21 2019 *)

A001511ext(n) = if(!n,n,sign(n)*(1+valuation(n,2))); \\ Like A001511 but gives 0 for 0 and -A001511(-n) for negative numbers.
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)));
A325310(n) = A001511ext(A325315(n));

If A325315(n) = 0, then a(n) = 0, otherwise a(n) = A001511(A325315(n)).

a(A228058(n)) = A001511(abs(A325379(n))), assuming there are no odd perfect numbers, in which case a(A228058(n)) >= 3 for all n.

A326070 Numbers k such that A325977(k) has the same sign as A325978(k).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 100

Offset: 1

Antti Karttunen, Jun 07 2019

nonn

Here A325977(k) = A325973(k) - k and A325978(k) = k - A325974(k), where A325973(k) is the average of {sum of unitary divisors} and {sum of squarefree divisors} = (1/2) * (A034448(k) + A048250(k)) while A325974(k) is the average of {sum of non-unitary divisors} and {sum of nonsquarefree divisors} = (1/2)*(A048146(k) + A162296(k)). Only if signs of A325977(k) and A325978(k) are equal can their difference A325978(k) - A325977(k) = (k - A325974(k)) - (A325973(k) - k) = 2k - (A325973(k) + A325974(k)) = 2k - A000203(k) = A033879(k) be zero, which happens when k is a perfect number (in A000396).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A033879, A325963, A325973, A325974, A325975, A325977, A325978, A325981.

Cf. A326071 (complement), A326072, A000396 (a subsequence).

isA326070(n) = (sign(A325977(n))==sign(A325978(n))); \\ For other needed code, see under A325975.

A357608 Numbers k such that k and k+1 are both in A357605.

Original entry on oeis.org

76544, 104895, 126224, 165375, 170624, 174824, 201824, 245024, 257984, 271215, 273104, 316575, 338624, 387855, 447615, 469664, 477224, 540224, 618975, 633555, 641024, 659295, 705375, 752895, 770175, 842624, 843975, 862784, 870975, 893024, 913275, 957824, 1047375

Offset: 1

Amiram Eldar, Oct 06 2022

nonn

Numbers k such that A162296(k) > 2*k and A162296(k+1) > 2*(k+1).

			76544 is a term since 76544 and 76545 are both in A357605: A162296(76544) = 170688 > 2*76544 and A162296(76545) = 157248 > 2*76545.

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

Cf. A162296.
Subsequence of A013929, A096399 and A357605.
Similar sequences: A096399, A283418, A318167, A327635, A327942, A331412, A333951.

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, 10^6], q[#] && q[#+1] &]

A335989 Terms of A301517 that are not exponentially odd numbers (A268335).

Original entry on oeis.org

12500, 18252, 21600, 37500, 50000, 67228, 84500, 87500, 91260, 127764, 137500, 146016, 150000, 151200, 162500, 200000, 200772, 201684, 212500, 231868, 237500, 237600, 253500, 262500, 268912, 274400, 280800, 287500, 310284, 336140, 337500, 346788, 350000, 362500

Offset: 1

Amiram Eldar, Jul 03 2020

nonn

If k = Product p^e, then A162296(k) / A048250(k) = -1 + Product (p^(e+1) - 1)/(p^2 - 1). If k is exponentially odd, then e = 2*m - 1 is odd for all the prime factors p of k and p^(e+1) - 1 = (p^2)^m - 1 is divisible by p^2 - 1. Therefore, A162296(k) / A048250(k) is an integer for all exponentially odd numbers, and it is a positive integer for all the nonsquarefree (A013929) exponentially odd numbers.
It seems that most of the terms of A301517 are exponentially odd numbers. For example, the first 10^4 terms of A301517 include only 9 terms that are not exponentially odd numbers. Up to 10^8 there are 9660732 terms of A301517, and only 9107 of them are not exponentially odd numbers.
The number of terms of this sequence that do not exceed 10^k, for k = 5, 6, ... are 9, 92, 916, 9107, 91172, 911187, .... Apparently, this sequence has an asymptotic density c = 0.000091... If this is true, then the asymptotic density of A301517 is c + A065463 - A059956 = 0.096606... (A065463 is the density of the exponentially odd numbers, and A059956 is the density of the squarefree numbers which are a subset of the exponentially odd numbers).

			12500 = 2^2 * 5^5 is a term since the exponent of its prime factor 2 is 2 which even, and therefore it is not an exponentially odd number, and the sum of its squarefree divisors, A048250(12500) = 18 divides the sum of its nonsquarefree divisors, A162296(12500) = 27324 = 18 * 1518.

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

Cf. A048250, A059956, A065463, A162296, A268335, A301517.

f[p_, e_] := (p^(e + 1) - 1)/(p^2 - 1); Select[Range[2, 4*10^5], Max[Last /@ (fct = FactorInteger[#])] > 1 && ! AllTrue[Last /@ fct, OddQ] && (r =  Times @@ (f @@@ fct)) > 1 && IntegerQ[r] &]

A362404 Numbers k such that k and k+1 are both in A362401.

Original entry on oeis.org

24, 27, 48, 79, 120, 168, 199, 288, 350, 360, 378, 391, 447, 507, 528, 775, 840, 895, 960, 1088, 1136, 1368, 1638, 1639, 1680, 1848, 1849, 2095, 2127, 2208, 2322, 2749, 2808, 3720, 3726, 3798, 3799, 3919, 4050, 4087, 4488, 4550, 4872, 5040, 5328, 5448, 5631, 6240

Offset: 1

Amiram Eldar, Apr 18 2023

nonn

			24 is a term since 24 and 25 are both in the range of A162296: A162296(20) = 24 and A162296(25) = 25.

Amiram Eldar, Table of n, a(n) for n = 1..3557 (terms below 10^8)

Subsequence of A362401.
A362405 is a subsequence.
Cf. A162296.

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; seq[max_] := Module[{v = Select[Union[Array[s, max]], 0 < # <= max &], i}, i = Position[Differences[v], 1] // Flatten; v[[i]]]; seq[10^4]

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) = {my(v = select(x -> (x < kmax), Set(vector(kmax, k, s(k))))); for(k=1, #v-1, if(v[k+1] - v[k] == 1, print1(v[k], ", ")));}

A362405 Numbers k such that k, k+1 and k+2 are all in A362401.

Original entry on oeis.org

1638, 1848, 3798, 11448, 16854, 26910, 35574, 37248, 57120, 69678, 69822, 85848, 94248, 110526, 208848, 272214, 305046, 310248, 335478, 335479, 368448, 573048, 580680, 687240, 1017126, 1154270, 1230606, 1289358, 1423248, 1467414, 1697808, 1718880, 1776750, 1777248

Offset: 1

Amiram Eldar, Apr 18 2023

nonn

Up to 10^8, k = 335478 is the only number k such that k, k+1, k+2 and k+3 are all in A362401. Are there any other such terms?

			1638 is a term since 1638, 1639 and 1640 are all in the range of A162296: A162296(1053) = 1638, A162296(576) = 1639 and A162296(1636) = 1640.

Amiram Eldar, Table of n, a(n) for n = 1..132 (terms below 10^8)

Subsequence of A362401 and A362404.

Cf. A162296.

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; seq[max_] := Module[{v = Select[Union[Array[s, max]], 0 < # <= max &], w, i, j}, i = Position[Differences[v], 1] // Flatten; w = v[[i]]; j = Position[Differences[w], 1] // Flatten; w[[j]]]; seq[10^6]

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) = {my(v = select(x -> (x < kmax), Set(vector(kmax, k, s(k))))); for(k=1, #v-2, if(v[k+1] - v[k] == 1 && v[k+2] - v[k+1] == 1, print1(v[k], ", ")));}

A300984 Numbers whose sum of squarefree divisors and sum of nonsquarefree divisors are both squarefree numbers.

Original entry on oeis.org

676, 1352, 2704, 5408, 5476, 8788, 10816, 10952, 14884, 21316, 21632, 21904, 29768, 35152, 42632, 43264, 43808, 59536, 70304, 85264, 86528, 95048, 114244, 119072, 140608, 148996, 170528, 173056, 175232, 190096, 202612, 209764, 228488, 238144, 262088, 281216

Offset: 1

Michel Lagneau, Mar 17 2018

nonn

Conjecture: a(n) is of the form a(n) = 2^i*p^j with i, j integers and p prime. This has been verified for n up to 10^7.
Observation: For n < = 10^7, p belongs to the set E = {13, 37, 61, 73, 109, 157, 181, 193, 229, 277, 313, 373, 397, 409, 421, 433, 457, 541, 601, 613, 661, 673, 709, 733, 757, 769, 829, 853, 877, 997, 1009, 1021, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1297, 1381, 1429, 1453, 1489}. We observe that E minus {181, 433, 601, 769, 853, 1021, 1429} belongs to A082539.
Generalization: For n <= 10^m with m > 7, it is conjectured that a majority of primes p where a(n) = 2^i*p^j are in A082539. For example, with m = 7, 84% of the primes p are in A082539.

			676 is in the sequence because A048250(676) = 42 = 2*3*7 and A162296(676) = 1239 = 3*7*59 are both squarefree numbers.

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

Cf. A013929, A048250, A082539, A162296.

lst={};Do[If[SquareFreeQ[Total[Select[Divisors[n],SquareFreeQ]]]&& SquareFreeQ[DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]]],AppendTo[lst,n]],{n,300000}];lst

isok(n) = my(sd = sumdiv(n,d,d*issquarefree(d))); issquarefree(sd) && issquarefree(sigma(n) - sd); \\ Michel Marcus, Mar 17 2018

A318807 Numbers whose sum of squarefree divisors and sum of nonsquarefree divisors are both perfect squares.

Original entry on oeis.org

1, 3, 9, 22, 27, 66, 70, 88, 94, 115, 119, 170, 198, 210, 214, 217, 264, 265, 280, 282, 310, 322, 345, 357, 376, 382, 385, 497, 510, 517, 527, 594, 630, 642, 651, 679, 680, 710, 729, 742, 745, 782, 795, 840, 846, 856, 862, 889, 930, 935, 966, 970, 1035, 1066

Offset: 1

Michel Lagneau, Sep 04 2018

nonn

Let s be the sum of the squarefree divisors of a number m. The sequence lists the numbers m such that s and sigma(m) - s are both a perfect square.
Or numbers m such that A048250(m) and A162296(m) are perfect squares.
The corresponding pairs of squares (s, sigma(m) - s) are (1, 0), (4, 0), (4, 9), (36, 0), (4, 36), (144, 0), (144, 0), (36, 144), (144, 0), (144, 0), (144, 0), (324, 0), (144, 324), ...
The subsequence b(n) where s and sigma(m) - s are strictly positive begins with 9, 27, 88, 198, 264, 280, 376, 594, 630, ... b(n) is not squarefree (subsequence of A013929).
The subsequence c(n) where the ratio r = (sigma(a(n)) - s)/s is an integer begins with 27, 88, 264, 280, 376, 594, 680, 840, 856, 1128, 1240, ... and the corresponding r are 3^2, 2^2, 2^2, 2^2, 2^2, 3^2, 2^2, 2^2, 2^2, 2^2, 2^2, 2^2, 2^2, 5^2, 3^2, 2^2, 7^2, 3^2, 2^2, 11^2, ... It is conjectured that r belongs to A001248.

			27 is in the sequence because A048250(27) = 4 and A162296(27) = 36 are both a perfect square.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001248, A013929, A048250, A162296.

filter:= proc(n) local F, SF, NSF, t;
  F:= ifactors(n)[2];
  SF:= mul(1+t[1],t=F);
  if not issqr(SF) then return false fi;
  NSF:= mul((1-t[1]^(1+t[2]))/(1-t[1]), t=F) - SF;
  issqr(NSF);
end proc:
select(filter, [$1..2000]); # Robert Israel, Sep 05 2018

lst={};Do[If[IntegerQ[Sqrt[Total[Select[Divisors[n],SquareFreeQ]]]]&&IntegerQ[Sqrt[DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]]]],AppendTo[lst,n]],{n,1100}];lst
sdsndQ[n_]:=Module[{d=Divisors[n],sf,nsf},sf=Select[d,SquareFreeQ];nsf= Complement[ d,sf];AllTrue[ {Sqrt[ Total[sf]],Sqrt[ Total[nsf]]},IntegerQ]]; Select[Range[1500],sdsndQ] (* Harvey P. Dale, Sep 13 2024 *)

isok(n) = {my(sd=sumdiv(n, d, issquarefree(d)*d)); issquare(sd) && issquare(sigma(n) - sd);} \\ Michel Marcus, Sep 04 2018

Definition modified by Harvey P. Dale, Sep 13 2024

A357497 Nonsquarefree numbers whose harmonic mean of nonsquarefree divisors in an integer.

Original entry on oeis.org

4, 9, 12, 18, 24, 25, 28, 45, 49, 54, 60, 90, 112, 121, 126, 132, 150, 153, 168, 169, 198, 270, 289, 294, 336, 361, 364, 414, 529, 560, 594, 630, 637, 684, 726, 841, 918, 961, 1014, 1140, 1232, 1305, 1350, 1369, 1512, 1521, 1638, 1680, 1681, 1710, 1734, 1849, 1984

Offset: 1

Amiram Eldar, Oct 01 2022

nonn

Analogous to harmonic numbers (A001599) with nonsquarefree divisors.
The squares of primes (A001248) are terms since they have a single nonsquarefree divisor.
If p is a prime then 6*p^2 is a term.

			12 is a term since its nonsquarefree divisors are 4 and 12 and their harmonic mean is 6 which is an integer.

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

Cf. A162296, A322609, A357493, A357494, A357495, A357496.
Subsequence of A013929.
Subsequence: A001248.
Similar sequences: A001599 (harmonic numbers), A006086 (unitary), A063947 (infinitary), A286325 (bi-unitary), A319745 (nonunitary), A335387 (tri-unitary).

q[n_] := Length[d = Select[Divisors[n], ! SquareFreeQ[#] &]] > 0 && IntegerQ[HarmonicMean[d]]; Select[Range[2000], q]

A357606 Primitive terms of A357605: numbers in A357605 with no proper divisor in A357605.

Original entry on oeis.org

36, 48, 80, 120, 162, 168, 200, 224, 264, 270, 280, 300, 312, 352, 378, 392, 408, 416, 450, 456, 500, 552, 588, 594, 630, 696, 700, 702, 744, 750, 882, 888, 918, 968, 980, 984, 1026, 1032, 1050, 1088, 1100, 1128, 1216, 1232, 1242, 1272, 1300, 1372, 1416, 1452

Offset: 1

Amiram Eldar, Oct 06 2022

nonn

Numbers k such that A162296(k) > 2*k but for all the aliquot divisors d of k (i.e., d | k, d < k), A162296(d) <= 2*d.
If k is a term then all the positive multiples of k are terms of A357605.
The least odd term is a(144) = 4725.

			36 is a term since A162296(36) = 79 > 2*36, but for all the divisors d of 36, 1, 2, 3, 4, 6, 9, 12 and 18, A162296(d) <= 2*d. E.g., A162296(18) = 28 < 2*18.

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

Cf. A162296.
Subsequence of A005101, A013929 and A357605.
Similar sequences: A091191, A302574.

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]; q[1] = False; primQ[n_] := q[n] && AllTrue[Most @ Divisors[n], ! q[#] &]; Select[Range[1500], primQ]

cp's OEIS Frontend

A325310 a(n) = A001511(A325315(n)), except when A325315(n) = 0, then a(n) = 0.

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A326070 Numbers k such that A325977(k) has the same sign as A325978(k).

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A357608 Numbers k such that k and k+1 are both in A357605.

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A335989 Terms of A301517 that are not exponentially odd numbers (A268335).

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A362404 Numbers k such that k and k+1 are both in A362401.

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A362405 Numbers k such that k, k+1 and k+2 are all in A362401.

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A300984 Numbers whose sum of squarefree divisors and sum of nonsquarefree divisors are both squarefree numbers.

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A318807 Numbers whose sum of squarefree divisors and sum of nonsquarefree divisors are both perfect squares.

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