A048050 -id:A048050 - cp's OEIS frontend

A054017 Chowla's function of n modulo n is a square (excluding 0's).

14, 20, 39, 40, 46, 55, 80, 94, 100, 104, 117, 130, 155, 158, 183, 190, 200, 203, 291, 292, 295, 299, 320, 323, 334, 416, 430, 446, 464, 475, 488, 530, 539, 549, 567, 579, 583, 638, 650, 695, 718, 799, 873, 878, 890, 943, 955, 959, 964, 979, 1030, 1118

Offset: 1

Asher Auel, Jan 19 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

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

Cf. A048050, A054013, A054018.

aQ[n_] := (c = DivisorSigma[1, n] - n - 1) > 0 && IntegerQ @ Sqrt @ Mod[c, n]; Select[Range[1000], aQ] (* Amiram Eldar, Aug 28 2019 *)

A054018 Squares mentioned in A054017.

Original entry on oeis.org

9, 1, 16, 9, 25, 16, 25, 49, 16, 1, 64, 121, 36, 81, 64, 169, 64, 36, 100, 225, 64, 36, 121, 36, 169, 49, 361, 225, 1, 144, 441, 441, 144, 256, 400, 196, 64, 441, 1, 144, 361, 64, 400, 441, 729, 64, 196, 144, 729, 100, 841, 729, 25, 400, 256, 1225, 100, 729, 1225

Offset: 1

Asher Auel, Jan 19 2000

nonn

If Chowla's function of n read (modulo n) is a nonzero square, print this square.

Chowla's function (A048050) = sum of divisors of n except 1 and n.

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

Cf. A048050, A054013, A054017, A054019.

chowla[n_] := DivisorSigma[1, n] - n - 1; aQ[n_] := (c = chowla[n]) > 0 && IntegerQ @ Sqrt @ Mod[c, n]; Mod[chowla[#], #] & /@ Select[Range[1000], aQ] (* Amiram Eldar, Aug 28 2019 *)

A054022 Chowla function of n is divisible by the number of divisors of n.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 27, 29, 31, 32, 35, 36, 37, 39, 41, 43, 47, 50, 51, 53, 55, 59, 61, 67, 71, 73, 75, 79, 83, 87, 89, 91, 95, 97, 98, 101, 103, 107, 109, 111, 113, 115, 119, 123, 127, 131, 135, 137, 139, 143, 149, 151, 155, 157, 159, 162, 163

Offset: 1

Asher Auel, Jan 19 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Complement is A054023 Cf. A000005, A048050, A054014.

with(numtheory):
[seq(`if`((sigma(i)-i-1) mod tau(i)=0,i,print( )),i=1..1000)];

Select[Range[200],Divisible[DivisorSigma[1,#]-1-#,DivisorSigma[0,#]]&] (* Harvey P. Dale, Mar 11 2012 *)

A057533 Values of n for which iteration of Chowla's function loops.

Original entry on oeis.org

48, 75, 92, 140, 146, 176, 195, 215, 255, 267, 287, 312, 332, 369, 386, 407, 411, 519, 527, 551, 627, 734, 744, 818, 972, 973, 984, 1027, 1050, 1078, 1096, 1149, 1175, 1185, 1387, 1408, 1472, 1474, 1535, 1575, 1648, 1651, 1784, 1792, 1880, 1888, 1891

Offset: 1

Asher Auel, Sep 03 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A048050.

Cf. A005276 (subsequence). - Reinhard Zumkeller, Feb 09 2013

a057533 n = a057533_list !! (n-1)
a057533_list = filter (\z -> p z [z]) [1..] where
   p x ts = y > 0 && (y `elem` ts || p y (y:ts)) where y = a048050 x
-- Reinhard Zumkeller, Feb 09 2013

A070160 Nonprime numbers k such that phi(k)/(sigma(k) - k - 1) is an integer.

Original entry on oeis.org

4, 9, 15, 25, 35, 49, 95, 119, 121, 143, 169, 209, 287, 289, 319, 323, 361, 377, 527, 529, 559, 779, 841, 899, 903, 923, 961, 989, 1007, 1189, 1199, 1343, 1349, 1369, 1681, 1763, 1849, 1919, 2159, 2209, 2507, 2759, 2809, 2911, 3239, 3481, 3599, 3721

Offset: 1

Labos Elemer, Apr 26 2002

nonn

Euler phi value divided by Chowla function gives integer.

			In A062972, n=15: q = 8/8 = 1; n=101: q = 100/1 = 100. While integer quotient chowla(n)/phi(n) gives only 5 nonprime solutions below 20000000 (see A070037), here, the integer reciprocals, q = phi(n)/chowla(n) obtained with squared primes and with other composites. If n=p^2, q = p(p-1)/p = p-1. So for squared primes, the quotients give A006093.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A000010, A001065, A000203, A020492, A068418, A062972, A055940, A070159, A070037.

Union of A000040 (primes) and A070161.

Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n-1); If[IntegerQ[s], Print[n]], {n, 2, 100000}]

{k : A000010(k)/A048050(k) is an integer}.

A070161 Nonprime numbers n such that q=phi(n)/(sigma(n)-n-1) is an integer and n is not a prime square.

Original entry on oeis.org

15, 35, 95, 119, 143, 209, 287, 319, 323, 377, 527, 559, 779, 899, 903, 923, 989, 1007, 1189, 1199, 1343, 1349, 1763, 1919, 2159, 2507, 2759, 2911, 3239, 3599, 3827, 4031, 4607, 5183, 5207, 5249, 5459, 5543, 6439, 6887, 7067, 7279, 7739, 8159, 8639, 9179

Offset: 1

Labos Elemer, Apr 26 2002

nonn

			n=35: phi(35)=24, sigma(35)=1+5+7+35=48, chowla(35)=12, quotient=2

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A000010, A001065, A000203, A048050, A055940, A070159, A070160, A070037 A020492, A068418, A062972.

Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n-1); If[ !PrimeQ[n]&&!PrimeQ[Sqrt[n]]&&IntegerQ[s], Print[n]], {n, 2, 100000}]

q=A000010(n)/A048050(n) and n is not in A001248.

A085842 Numbers k whose divisors (apart from 1 and k) sum to a prime.

Original entry on oeis.org

4, 6, 9, 10, 22, 25, 30, 34, 42, 49, 58, 60, 70, 78, 82, 84, 102, 118, 120, 121, 138, 142, 168, 169, 186, 198, 202, 214, 216, 220, 222, 228, 234, 238, 240, 246, 258, 270, 274, 280, 282, 289, 294, 298, 348, 358, 360, 361, 364, 370, 372, 382, 390, 394, 406, 414, 438, 442, 444

Offset: 1

Chuck Seggelin, Jul 05 2003

nonn

			102 is a member since the divisors of 102 are {1, 2, 3, 6, 17, 34, 51, 102} and 2 + 3 + ... + 51 = 113, a prime.

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

Cf. A037020, A048050.

with(numtheory); b := []; for n from 3 to 2000 do t1 := divisors(n); t2 := convert(t1,list); t3 := add(t2[i],i=1..nops(t2)); if isprime(t3-1-n) then b := [op(b),n]; fi; od: b;

f[n_]:=Plus@@Divisors[n]-n-1; lst={};Do[a=f[n];If[PrimeQ[a],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 21 2009 *)
Select[Range[500], PrimeQ[DivisorSigma[1, #] - # - 1] &] (* Amiram Eldar, Dec 04 2020 *)

isok(k) = isprime(sigma(k)-1-k); \\ Michel Marcus, Dec 04 2020

A137510 Irregular triangle read by rows in which row n lists the divisors of n in the range 1 < d < n; or 0 if there are no such divisors.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 3, 0, 2, 4, 3, 2, 5, 0, 2, 3, 4, 6, 0, 2, 7, 3, 5, 2, 4, 8, 0, 2, 3, 6, 9, 0, 2, 4, 5, 10, 3, 7, 2, 11, 0, 2, 3, 4, 6, 8, 12, 5, 2, 13, 3, 9, 2, 4, 7, 14, 0, 2, 3, 5, 6, 10, 15, 0, 2, 4, 8, 16, 3, 11, 2, 17, 5, 7, 2, 3, 4, 6, 9, 12, 18, 0, 2, 19, 3, 13, 2, 4, 5, 8, 10, 20, 0, 2, 3, 6, 7

Offset: 1

N. J. A. Sloane, May 08 2008

The length of row n is A264440(n). - Wolfdieter Lang, Jan 16 2016

Row n lists the nontrivial divisors of n, or 0 if there are no such divisors. - Omar E. Pol, Nov 22 2010

			From _Omar E. Pol_, Nov 22 2010: (Start)
The irregular triangle begins:
0;
0;
0;
2;
0;
2, 3;
0;
2, 4;
3;
2, 5;
0,
2, 3, 4, 6;
(End)

Cf. A070824, A027750, A027751, A264440 (row length). Row sums give A048050.

for n from 1 to 80 do if isprime(n) or n = 1 then printf("0,") ; else dvs := sort(convert(numtheory[divisors](n) minus {1,n},list) ) ; for d in dvs do printf("%d,",d) ; od: fi ; od: # R. J. Mathar, May 23 2008
with(numtheory): A:=proc (n) local div: div:=divisors(n): `minus`(div, {div[tau(n)], div[1]}) end proc: for n to 35 do A(n) end do: a:=proc (n) if A(n)={} then 0 else seq(A(n)[j],j=1..tau(n)-2) end if end proc: for n to 35 do a(n) end do; # yields sequence in triangular form - Emeric Deutsch, May 25 2008

Array[Complement[Divisors@ #, {1, #}] &, {42}] /. {} -> {0} // Flatten (* Michael De Vlieger, Jan 16 2016 *)

More terms from R. J. Mathar and Emeric Deutsch, May 23 2008

A174514 Partial sums of A048995.

Original entry on oeis.org

1, 5, 56, 143, 238, 357, 480, 625, 786, 973, 1178, 1387, 1602, 1839, 2084, 2331, 2592, 2859, 3134, 3421, 3710, 4001, 4304, 4609, 4930, 5253, 5578, 5913, 6254, 6625, 7030, 7437, 7862, 8291, 8738, 9209, 9682, 10179, 10694, 11211, 11730, 12259, 12798, 13349, 13904, 14465, 15040, 15623, 16234, 16857, 17482, 18109, 18766

Offset: 1

Jonathan Vos Post, Nov 28 2010

nonn

The subsequence of values which are themselves in A048995 begins: 1, 625. The subsequence of primes begins 5, 4001, 8291, 9209.

			a(8) = 1 + 4 + 51 + 87 + 95 + 119 + 123 + 145 = 625 = 5^4.

Cf. A048050, A048995.

a(n) = SUM[i=1..n] A048995(i) = SUM[i=1..n] (numbers that are not the sum of the nontrivial factors, excluding 1 and i, of some natural number).

A187086 Numbers n with property that sum of divisors of n except 1 and n is a positive square.

Original entry on oeis.org

14, 39, 40, 46, 55, 94, 117, 130, 155, 158, 183, 190, 203, 208, 291, 292, 295, 299, 320, 323, 334, 430, 446, 475, 488, 530, 539, 549, 567, 579, 583, 638, 695, 718, 799, 873, 878, 890, 928, 943, 955, 959, 964, 979, 1030, 1118, 1191, 1255, 1384, 1411, 1454

Offset: 1

Zak Seidov, Mar 04 2011

nonn

Or, Chowla's function of n is a positive square.

			divisors(14)={1,2,7,14}, 2+7=9=3^2 (= A187087(1));
divisors(40)={1,2,4,5,8,10,20,40}, 2+4+5+8+10+20=49=7^2 (= A187087(3)).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A048050, A187087.

IsA187086:=func< n | not IsPrime(n) and IsSquare(&+[ a: a in Divisors(n) | a ne 1 and a ne n ]) >; [ n: n in [2..2000] | IsA187086(n) ]; // Klaus Brockhaus, Mar 04 2011

select(t -> not isprime(t) and issqr(numtheory:-sigma(t)-1-t), [$2..2000]); # Robert Israel, Oct 25 2017

Select[Range@ 1500, And[! PrimeQ@ #, IntegerQ@ Sqrt[DivisorSigma[1, #] - # - 1]] &] (* Michael De Vlieger, Oct 25 2017 *)

{for(n=1,2000,spf=sumdiv(n,x,x)-1-n;if(spf>0&&issquare(spf),print1(n", ")))}

cp's OEIS Frontend

A054017 Chowla's function of n modulo n is a square (excluding 0's).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A054018 Squares mentioned in A054017.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A054022 Chowla function of n is divisible by the number of divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A057533 Values of n for which iteration of Chowla's function loops.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A070160 Nonprime numbers k such that phi(k)/(sigma(k) - k - 1) is an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A070161 Nonprime numbers n such that q=phi(n)/(sigma(n)-n-1) is an integer and n is not a prime square.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A085842 Numbers k whose divisors (apart from 1 and k) sum to a prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A137510 Irregular triangle read by rows in which row n lists the divisors of n in the range 1 < d < n; or 0 if there are no such divisors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica