A139256 -id:A139256 - cp's OEIS frontend

A341039 Numbers k such that k divides A341038(k).

1, 12, 18, 56, 117, 810, 992, 2091, 16256, 796797, 2488734, 22122192, 57636005, 67100672, 114243682

Offset: 1

J. M. Bergot and Robert Israel, Feb 03 2021

Contains no primes.

			a(3) = 18 is a term because A341038(18) = 198 is divisible by 18.

Includes A139256.

Cf. A341038.

f:= proc(n) local D,S,i;
  D:= sort(convert(numtheory:-divisors(n),list));
  S:= ListTools:-PartialSums(D);
  add(S[-i]*D[i],i=1..nops(D))
end proc:
select(t -> f(t) mod t = 0, [$1..20000]);

A348412 Numbers whose even divisors have an integer harmonic mean.

Original entry on oeis.org

2, 6, 12, 30, 56, 84, 168, 270, 280, 540, 616, 840, 992, 1092, 1344, 2856, 2976, 3276, 3780, 4590, 5320, 5940, 7560, 12400, 12420, 14880, 16256, 16380, 18848, 24360, 26784, 36036, 37200, 37240, 41664, 48768, 49140, 55692, 60480, 65520, 86304, 86800, 111720, 128520

Offset: 1

Amiram Eldar, Oct 17 2021

nonn

The corresponding harmonic means are 2, 3, 4, 5, 6, 7, 9, 9, 10, 12, 11, 15, 10, 13, 16, 17, 15, ...

Equivalently, even numbers k such that the harmonic mean of the divisors of k/2 is either an integer (A001599) or a half-integer (A348411).

			6 is a term since its even divisors are 2 and 6, and their harmonic mean, 1/((1/2 + 1/6)/2) = 3, is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..569

Cf. A001599, A074400, A348411.

A139256 is a subsequence.

Select[Range[2, 10^5, 2], IntegerQ[HarmonicMean[Select[Divisors[#], EvenQ]]] &]

isok(m) = if (! (m%2), my(d=select(x->!(x%2), divisors(m))); denominator(#d/sum(k=1, #d, 1/d[k])) == 1); \\ Michel Marcus, Oct 31 2021

from sympy import gcd, divisor_sigma
A348412_list = [2*n for n in range(1,10**3) if (lambda x, y: 2*gcd(x,y*n)>=x)(divisor_sigma(n),divisor_sigma(n,0))] # Chai Wah Wu, Oct 20 2021

A377001 Integers k equal to the sum over A000203(t) mod t, for some steps, starting with t = k and then using the result to feed the next calculation.

Original entry on oeis.org

4, 8, 32, 72, 94, 118, 128, 144, 147, 204, 284, 1017, 1102, 1210, 1462, 1968, 2294, 2342, 2457, 2486, 2670, 2924, 5564, 6128, 6368, 7008, 8192, 10856, 12216, 12914, 14066, 14595, 16694, 18416, 18825, 19668, 21870, 22401, 22713, 23388, 26234, 26966, 29038, 31806

Offset: 1

Paolo P. Lava, Oct 12 2024

Up to 10^7, the longest process takes place with 2813292 which needs 23 steps.
Numbers of the form 2^A000043(n) or 1+A000668(n) are a subsequence.
If we multiply instead of adding A000203(t) mod t, we get the twice even perfect numbers (A139256).
E.g. k = 12 -> sigma(12) mod 12 = 4; sigma(4) mod 4 = 3 and 4 * 3 = 12.

			k = 72 (2 steps):
sigma(72) mod 72 = 51;
sigma(51) mod 51 = 21 and 51 + 21 = 72.
k = 147  (6 steps):
sigma(147) mod 147 = 81;
sigma(81) mod 81 = 40;
sigma(40) mod 40 = 10;
sigma(10) mod 10 = 8;
sigma(8) mod 8 = 7;
sigma(7) mod 7 = 1 and 81 + 40 + 10 + 8 + 7 + 1 = 147.

Cf. A000043, A000203, A000668, A139256, A377002.

with(numtheory): P:=proc(q) local a,b,n,v; v:=[];
for n from 1 to q do a:=0; b:=n; while a

A066190 Numbers k such that the sum of the even aliquot parts of k divides k.

Original entry on oeis.org

4, 6, 10, 12, 14, 22, 26, 34, 38, 46, 56, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502

Offset: 1

Robert G. Wilson v, Dec 15 2001

nonn

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

Union of A100484 and A139256.

Cf. A053813.

Do[ d = Drop[ Divisors[ n ], -1 ]; l = Length[ d ]; ev = 0; k = 2; While[ k <= l, If[ EvenQ[ d[ [ k ] ] ], ev = ev + d[ [ k ] ] ]; k++ ]; If[ IntegerQ[ n/ev ], Print[ n ] ], {n, 2, 10^3, 2} ]
Select[Range[550],Divisible[#,Total[Select[Most[Divisors[#]],EvenQ]]]&]//Quiet  (* Harvey P. Dale, Mar 11 2011 *)

{ n=0; for (m=1, 10^9, d=divisors(m); s=0; for (i=2, numdiv(m) - 1, if (d[i]%2 == 0, s += d[i])); if (s > 0 && m%s == 0, write("b066190.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 05 2010

a(n) = 2*A053813(n). - Amiram Eldar, Jul 07 2022

a(56) from Harry J. Smith, Feb 05 2010

A347155 Sum of divisors of nontriangular numbers.

Original entry on oeis.org

3, 7, 6, 8, 15, 13, 12, 28, 14, 24, 31, 18, 39, 20, 42, 36, 24, 60, 31, 42, 40, 30, 72, 32, 63, 48, 54, 48, 38, 60, 56, 90, 42, 96, 44, 84, 72, 48, 124, 57, 93, 72, 98, 54, 120, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 68, 126, 96, 144, 72, 195, 74, 114, 124, 140

Offset: 1

Omar E. Pol, Aug 20 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys.
So knowing this characteristic shape we can know if a number is a nontriangular number (or not) just by looking at the diagram, even ignoring the concept of nontriangular number.
Therefore we can see a geometric pattern of the distribution of the nontriangular numbers in the stepped pyramid described in A245092.
If both Dyck paths have peaks on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317303.
If both Dyck paths have valleys on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317304.

			a(6) = 13 because the sum of divisors of the 6th nontriangular (i.e., 9) is 1 + 3 + 9 = 13.
On the other we can see that in the main diagonal of the diagrams both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys as shown below.
Illustration of initial terms:
m(n) = A014132(n).
.
   n   m(n) a(n)   Diagram
.                    _   _ _   _ _ _   _ _ _ _   _ _ _ _ _   _ _ _ _ _ _
                   _| | | | | | | | | | | | | | | | | | | | | | | | | | |
   1    2    3    |_ _|_| | | | | | | | | | | | | | | | | | | | | | | | |
                   _ _|  _|_| | | | | | | | | | | | | | | | | | | | | | |
   2    4    7    |_ _ _|    _|_| | | | | | | | | | | | | | | | | | | | |
   3    5    6    |_ _ _|  _|  _ _|_| | | | | | | | | | | | | | | | | | |
                   _ _ _ _|  _| |  _ _|_| | | | | | | | | | | | | | | | |
   4    7    8    |_ _ _ _| |_ _|_|    _ _|_| | | | | | | | | | | | | | |
   5    8   15    |_ _ _ _ _|  _|     |  _ _ _|_| | | | | | | | | | | | |
   6    9   13    |_ _ _ _ _| |      _|_| |  _ _ _|_| | | | | | | | | | |
                   _ _ _ _ _ _|  _ _|    _| |    _ _ _|_| | | | | | | | |
   7   11   12    |_ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|_| | | | | | |
   8   12   28    |_ _ _ _ _ _ _| |_ _|  _|  _ _| | |  _ _ _ _|_| | | | |
   9   13   14    |_ _ _ _ _ _ _| |  _ _|  _|    _| | |    _ _ _ _|_| | |
  10   14   24    |_ _ _ _ _ _ _ _| |     |     |  _|_|   |  _ _ _ _ _|_|
                   _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       | | |
  11   16   31    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|_| |
  12   17   18    |_ _ _ _ _ _ _ _ _| | |_ _ _|      _| |  _ _|
  13   18   39    |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|  _|_|
  14   19   20    |_ _ _ _ _ _ _ _ _ _| | |       |_ _|
  15   20   42    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|
                   _ _ _ _ _ _ _ _ _ _ _| | |  _ _| |
  16   22   36    |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _|
  17   23   24    |_ _ _ _ _ _ _ _ _ _ _ _| | |
  18   24   60    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  19   25   31    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  20   26   42    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
  21   27   40    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column m gives the nontriangular numbers.
Also the diagrams have on the main diagonal the following property: diagram [1] has peaks, diagrams [2, 3] have valleys, diagrams [4, 5, 6] have peaks, diagrams [7, 8, 9, 10] have valleys, and so on.
a(n) is also the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 3 + 3 = 6, so a(3) = 6.
For more information see A237593.

Cf. A000203, A000217, A014132, A237591, A237593, A245092, A262626, A317303, A317304, A346873.

Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors).

Array[DivisorSigma[1,#+Round@Sqrt[2#]]&,100] (* Giorgos Kalogeropoulos, Aug 20 2021 *)

a(n) = sigma(n + round(sqrt(2*n))); \\ Michel Marcus, Aug 21 2021

a(n) = A000203(A014132(n)).

A068495 Composite n such that n reduced mod(phi(n)) = sigma(n) reduced mod(n).

Original entry on oeis.org

6, 56, 260, 780, 992, 1976, 2156, 3696, 5520, 13800, 16256, 19872, 22560, 23688, 25232, 41072, 87000, 89964, 133984, 145888, 366720, 785808, 851760, 1100864, 1235052, 1270208, 1439552, 1470720, 2129400, 2237888, 4729664, 8690112

Offset: 1

Benoit Cloitre, Mar 11 2002

Are all terms even? - Robert Israel, Jan 13 2017

Contains A139256(n) for n > 1.

select(n -> not isprime(n) and n mod numtheory:-phi(n) = numtheory:-sigma(n) mod n, [$2..10^7]); # Robert Israel, Jan 13 2017

for(n=3,500000, if(n%eulerphi(n)==(1-isprime(n))*sigma(n)%n,print1(n,",")))

a(31) and a(32) from Robert Israel, Jan 13 2017

A084699 Composite integers j such that binomial(2*j,j) == 2^j (mod j).

Original entry on oeis.org

12, 30, 56, 424, 992, 16256, 58288, 119984, 356992, 1194649, 9973504, 12327121, 13141696, 22891184, 67100672, 233850649

Offset: 1

Benoit Cloitre, Oct 15 2003

If p is prime, binomial(2*p,p) == 2^p (mod p).
a(17) > 10^9.
From Gabriel Guedes and Ricardo Machado, Nov 16 2023: (Start)
Theorem. Let j = (2^k)*p, where p is an odd prime and k is in N; then binomial(2*j,j) == 2^j (mod j) if and only if p satisfies the following conditions:
a) p divides binomial(2^(k+1),2^k) - 2^(2^k);
b) p has at least k 1's in its binary expansion.
Theorem. If m is an even perfect number then j = 2m satisfies the congruence binomial(2*j,j) == 2^j (mod j). See A000396.
Theorem. Let j = p^2 with p a prime number. Then p is a Wieferich prime if and only if binomial(2*j,j) == 2^j (mod j). See A001220. (End)
Contains 17179738112 and 274877382656 (from Guedes-Machado paper). - Michael De Vlieger, Nov 22 2023
Contains 3386741824, 750984028672, 33029195197184, 1145067923695616, 422612863956511744. - Ricardo Machado, Nov 23 2023
Contains 84385517065596416, 62648180117928433664, 273984397779878971648, 36506097537257040703232. - Max Alekseyev, Dec 07 2023

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems, sect. III: Binomial coefficients modulo integers, binomod.gp (V. 1.4, 11/2015).
Gabriel Guedes and Ricardo Machado, Perfect numbers, Wieferich primes and the solutions of binomial(2n, n) congruent to 2^n mod n, Notes Num. Theor. Disc. Math. (NNTDM 2023) Vol. 29, No. 4, 705-712.

Contains A139256 as a subsequence.

Cf. A015910, A059288, A080469, A082180, A109760, A109769, A228562, A328497.

lista(nn) = {forcomposite(n=1, nn, if (binomod(2*n, n, n) == Mod(2, n)^n, print1(n, ", ")));} \\ Michel Marcus, Dec 06 2013 and Dec 03 2023

More terms from David Wasserman, Jan 03 2005

a(11)-a(16) from Max Alekseyev, Aug 05 2011

A348584 Numbers k such that k | A328258(k).

Original entry on oeis.org

1, 12, 56, 180, 992, 16256, 127400, 441000, 2646000, 67100672, 325458000, 2758909440, 17179738112, 274877382656

Offset: 1

Amiram Eldar, Oct 24 2021

The corresponding ratios A113184(k)/k are 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, -1, -1, ...
If p is a Mersenne exponent (A000043), then 2^p*(2^p-1) (twice an even perfect number) is a term with ratio A328258(k)/k = -1.
If there exists an odd term k, then it is a unitary multiply-perfect number (A327158), since A328258(k) = A034448(k) for an odd k.

			12 is a term since A328258(12) = -12 is divisible by 12.

The unitary version of A348583.
A139256 is a subsequence.
Cf. A000043, A034448, A327158, A328258.

f[p_, e_] := 1 - (-1)^p*(p^e); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[3*10^6], Divisible[s[#], #] &]

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A341039 Numbers k such that k divides A341038(k).

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A348412 Numbers whose even divisors have an integer harmonic mean.

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A377001 Integers k equal to the sum over A000203(t) mod t, for some steps, starting with t = k and then using the result to feed the next calculation.

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A066190 Numbers k such that the sum of the even aliquot parts of k divides k.

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A347155 Sum of divisors of nontriangular numbers.

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A068495 Composite n such that n reduced mod(phi(n)) = sigma(n) reduced mod(n).

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A084699 Composite integers j such that binomial(2*j,j) == 2^j (mod j).

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A348584 Numbers k such that k | A328258(k).

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