Asher Auel - cp's OEIS frontend

A057529 Numbers n with property that n is divisible by the number of divisors of n and by the sum of the digits of n.

1, 2, 8, 9, 12, 18, 24, 36, 40, 60, 72, 80, 84, 108, 132, 152, 156, 180, 204, 225, 228, 240, 252, 288, 360, 372, 396, 441, 444, 448, 450, 468, 480, 504, 516, 600, 612, 640, 684, 720, 732, 792, 804, 828, 864, 880, 882, 936, 972, 1016, 1040, 1044, 1056, 1116

Offset: 1

Asher Auel, Sep 03 2000

Cf. A005349, A033950, A057530.

A057530 n is odd and divisible by number of divisors of n and sum of digits of n.

Original entry on oeis.org

1, 9, 225, 441, 1521, 2025, 2601, 12321, 40401, 62001, 99225, 103041, 251001, 321489, 585225, 893025, 1022121, 1108809, 1212201, 1320201, 1946025, 2368521, 2480625, 2772225, 3101121, 3744225, 4473225, 4862025, 5517801, 6125625

Offset: 1

Asher Auel, Sep 03 2000

For most values (except 9,2025 and 99225) number of divisors of n = sum of digits of n, see A057531.
The above comment is wrong: for 16 out of the first 34 terms of the sequence, the number of divisors of n does not equal the sum of the digits of n. - Harvey P. Dale, Dec 31 2015
Since A000005(n) is odd, n must be a square. - Robert Israel, Oct 31 2019

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

Cf. A000005, A005349, A033950, A057529.

[k:k in [1..6000001 by 2]| IsIntegral(k/NumberOfDivisors(k)) and IsIntegral(k/&+Intseq(k))]; // Marius A. Burtea, Oct 31 2019

filter:= proc(m) local n;
  n:= m^2;
  n mod numtheory:-tau(n) = 0 and n mod convert(convert(n,base,10),`+`) = 0
end proc:
map(`^`, select(filter, [seq(i,i=1..10000,2)]),2); # Robert Israel, Oct 31 2019

Select[Range[1,5*10^6,2],Divisible[#,DivisorSigma[0,#]] && Divisible[ #,Total[ IntegerDigits[#]]]&] (* Harvey P. Dale, Dec 31 2015 *)

More terms from Harvey P. Dale, Dec 31 2015

A057531 Numbers whose sum of digits and number of divisors are equal.

Original entry on oeis.org

1, 2, 11, 22, 36, 84, 101, 152, 156, 170, 202, 208, 225, 228, 288, 301, 372, 396, 441, 444, 468, 516, 525, 530, 602, 684, 710, 732, 804, 828, 882, 952, 972, 1003, 1016, 1034, 1070, 1072, 1106, 1111, 1164, 1236, 1304, 1308, 1425, 1472, 1476, 1521, 1524

Offset: 1

Asher Auel, Sep 03 2000

[A007953(n)/A000005(n) = c] AND [A000005(n)/A007953(n) = c], c an integer. - Ctibor O. Zizka, Jun 26 2009

			36 is a term as the sum of the digits of 36 is 3+6 = 9 and the number of divisors is 9 too.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 210 terms from Daniel Arribas)
Code Golf StackExchange, Find the nth number where the digit sum equals the number of factors, coding challenge started Nov 28 2022.

Cf. A000005, A007953, A057532, A050689, A070274, A070275, A063737, A067077.

Select[ Range[ 1000 ], DivisorSigma[ 0, # ]==Plus@@IntegerDigits[ # ]& ] (* Harvey P. Dale, Feb 19 2004 *)

A057532 n is odd and sum of digits of n equals the numbers of divisors of n.

Original entry on oeis.org

1, 11, 101, 225, 301, 441, 525, 1003, 1111, 1425, 1521, 1575, 1911, 2015, 2101, 2325, 2475, 2541, 2601, 2925, 3225, 3311, 3825, 4275, 4301, 4851, 5025, 5175, 5445, 5733, 5775, 6075, 6321, 6525, 7315, 7605, 7623, 8325, 8925, 9225, 9555, 10003, 10021

Offset: 1

Asher Auel, Sep 03 2000

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000005, A007953, A033950, A057531.

Select[Range[1, 10^4, 2], Plus @@ IntegerDigits[#] == DivisorSigma[0, #] &] (* Giovanni Resta, Apr 24 2017 *)

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

A059026 Table B(n,m) read by rows: B(n,m) = LCM(n,m)/n + LCM(n,m)/m - 1 for all 1<=m<=n.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 2, 6, 1, 5, 6, 7, 8, 1, 6, 3, 2, 4, 10, 1, 7, 8, 9, 10, 11, 12, 1, 8, 4, 10, 2, 12, 6, 14, 1, 9, 10, 3, 12, 13, 4, 15, 16, 1, 10, 5, 12, 6, 2, 7, 16, 8, 18, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 1, 12, 6, 4, 3, 16, 2, 18, 4, 6, 10, 22, 1, 13, 14, 15, 16

Offset: 1

Asher Auel, Dec 15 2000

In an n X m box, a ball makes B(n,m) "bounces" starting at one corner until it reaches another corner, only allowed to travel on diagonal grid lines. B(n+2,n) = A022998(n+1) for all n >= 1. B(2n-1,n) = A016777(n) = 3n + 1 for all n >= 1 (central vertical).

Cf. A022998, A059028, A059029, A059030, A059031.

B := (n,m) -> lcm(n,m)/n + lcm(n,m)/m - 1: seq(seq(B(n,m), n=1..m),m=1..15);

A059030 Fourth main diagonal of A059026: a(n) = B(n+3,n) = lcm(n+3,n)/(n+3) + lcm(n+3,n)/n - 1 for all n >= 1.

Original entry on oeis.org

4, 6, 2, 10, 12, 4, 16, 18, 6, 22, 24, 8, 28, 30, 10, 34, 36, 12, 40, 42, 14, 46, 48, 16, 52, 54, 18, 58, 60, 20, 64, 66, 22, 70, 72, 24, 76, 78, 26, 82, 84, 28, 88, 90, 30, 94, 96, 32, 100, 102, 34, 106, 108, 36, 112, 114, 38, 118, 120, 40, 124, 126, 42, 130

Offset: 1

Asher Auel, Dec 15 2000

Cf. A059026, A059029, A059031.

B := (n,m) -> lcm(n,m)/n + lcm(n,m)/m - 1: seq(B(m+3,m),m=1..90);

G.f.: x(2x^3+2x^2+6x+4)/(1-x^3)^2.

A061806 Numbers n such that the iterative cycle: n -> sum of digits of n^2 has only three distinct elements.

Original entry on oeis.org

4, 5, 17, 26, 28, 32, 37, 40, 43, 44, 49, 50, 53, 62, 63, 64, 67, 73, 74, 76, 77, 82, 83, 86, 87, 88, 89, 91, 92, 93, 94, 97, 98, 107, 109, 113, 114, 116, 117, 118, 122, 124, 125, 126, 127, 128, 133, 137, 141, 143, 149, 154, 157, 158, 161, 164, 166, 167, 169, 170

Offset: 1

Asher Auel, May 17 2001

			4 -> 1+6 = 7 -> 4+9 = 13 -> 1+6+9 = 16 -> 2+5+6 = 13, thus {7,13,16} are the only distinct elements of the iterative cycle of 4.

Cf. A007953, A004159, A061903 - A061910.

A061899 Fibonacci numbers that are not squarefree.

Original entry on oeis.org

8, 144, 2584, 46368, 75025, 832040, 14930352, 267914296, 4807526976, 12586269025, 86267571272, 225851433717, 1548008755920, 27777890035288, 498454011879264, 2111485077978050, 8944394323791464, 160500643816367088, 2880067194370816120, 4660046610375530309, 51680708854858323072

Offset: 1

Asher Auel, May 20 2001

nonn

a(n) <= Fibonacci(6n) since 4 | Fibonacci(6n). Using other residue classes it can be shown that a(n) << 1.134^n. How far can this method be taken? - Charles R Greathouse IV, Dec 13 2011

			144 and 2584 are Fibonacci numbers (A000045) and are not squarefree: 144 = 2^4*3^2, 2584 = 2^3*17*19.

Amiram Eldar, Table of n, a(n) for n = 1..328 (terms 1..87 from Harry J. Smith)

Cf. A000045, A061305.

Select[Fibonacci[Range[100]],!SquareFreeQ[#]&] (* Harvey P. Dale, May 01 2018 *)

{ n=0; h=g=1; for (i=0, 375, f=g + h; h=g; g=f; if (!issquarefree(f), write("b061899.txt", n++, " ", f)) ) } \\ Harry J. Smith, Jul 28 2009

a(n) = A000045(A037917(n)). - R. J. Mathar, Jan 13 2014

A061900 Triangular numbers that are not squarefree.

Original entry on oeis.org

28, 36, 45, 120, 136, 153, 171, 276, 300, 325, 351, 378, 496, 528, 630, 666, 780, 820, 990, 1035, 1128, 1176, 1225, 1275, 1431, 1485, 1540, 1596, 1953, 2016, 2080, 2556, 2628, 2775, 2850, 3160, 3240, 3321, 3828, 3916, 4005, 4095, 4560, 4656, 4753, 4851

Offset: 1

Asher Auel, May 20 2001

nonn

			36 and 45 are triangular numbers (A000217) and are not squarefree: 36 = (2^2)(3^2), 45 = (3^2)(5).

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

Cf. A000217, A061304.

{ n=t=0; for (i=1, 1926, t+=i; if (!issquarefree(t), write("b061900.txt", n++, " ", t)) ) } \\ Harry J. Smith, Jul 28 2009

cp's OEIS Frontend

User: Asher Auel

A057529 Numbers n with property that n is divisible by the number of divisors of n and by the sum of the digits of n.

Views

Author

Keywords

Crossrefs

A057530 n is odd and divisible by number of divisors of n and sum of digits of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Extensions

A057531 Numbers whose sum of digits and number of divisors are equal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A057532 n is odd and sum of digits of n equals the numbers of divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A059026 Table B(n,m) read by rows: B(n,m) = LCM(n,m)/n + LCM(n,m)/m - 1 for all 1<=m<=n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

A059030 Fourth main diagonal of A059026: a(n) = B(n+3,n) = lcm(n+3,n)/(n+3) + lcm(n+3,n)/n - 1 for all n >= 1.

Views

Author

Keywords

Crossrefs

Programs

Maple

Formula

A061806 Numbers n such that the iterative cycle: n -> sum of digits of n^2 has only three distinct elements.

Views

Author

Keywords

Examples

Crossrefs

A061899 Fibonacci numbers that are not squarefree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula