A299174 -id:A299174 - cp's OEIS frontend

A119434 Odd n such that 2*phi(n) < n.

105, 165, 195, 315, 495, 525, 585, 735, 825, 945, 975, 1155, 1365, 1485, 1575, 1755, 1785, 1815, 1995, 2145, 2205, 2415, 2475, 2535, 2625, 2805, 2835, 2925, 3003, 3045, 3135, 3255, 3315, 3465, 3675, 3705, 3795, 3885, 3927, 4095, 4125, 4305, 4389, 4455

Offset: 1

Franklin T. Adams-Watters, May 19 2006

nonn

Obviously 2*phi(n) = n is impossible for odd n. Odd elements of A054741 and A119432. This is not the same as A036798. 684411 = 3*7*13*23*109 is in this sequence but not in A036798. (This is may not be the smallest such value.) The primitive elements of this sequence are A119433, excluding the initial 2
If n is in the sequence, then so is every odd multiple of n. - Robert Israel, Jan 06 2017
The asymptotic density of this sequence is in the interval (0.01120, 0.01176) (Kobayashi, 2016). It is 1/2 less than the asymptotic density of A119432. The number of terms below 10^k for k = 3, 4, ... are 11, 109, 1152, 11076, 111927, 1124091, 11224403, 112074112, ... - Amiram Eldar, Oct 15 2020

Robert Israel, Table of n, a(n) for n = 1..10000
Mitsuo Kobayashi, A generalization of a series for the density of abundant numbers, International Journal of Number Theory, Vol. 12, No. 3 (2016), pp. 671-677.

Cf. A000010, A036798, A054741, A118700, A119432, A119433, A299174.

select(t -> numtheory:-phi(t) < t/2, [seq(t,t=1..10000,2)]);

Select[Range[1, 10^4, 2], 2 EulerPhi[#] < #&] (* Jean-François Alcover, Apr 12 2019 *)

lista(nn) = forstep (n=1, nn, 2, if (n > 2*eulerphi(n), print1(n, ", "))) \\ Michel Marcus, Jul 04 2015

A036798 UNION A118700. - R. J. Mathar, Aug 08 2007

A119432 \ A299174. - Amiram Eldar, Oct 15 2020

A350803 Numbers k with at least one partition into two parts (s,t), s<=t such that t | s*k.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116

Offset: 1

Wesley Ivan Hurt, Jan 16 2022

nonn

From Bernard Schott, Jan 22 2022: (Start)
A299174 is a subsequence because, if k = 2*u, we have s=t=u, s<=t, and u | u*k.
A082663 is another subsequence because, if k = p*q with p < q < 2p, then with s = k-p^2 = p*(q-p) and t = p^2, we have s <= t and p^2 | p*(q-p) * (pq).
It seems that A090196 is the subsequence of odd terms. (End)
gcd(s, t) > 1 where s and t and k > 2 are as in name. - David A. Corneth, Jan 22 2022
Numbers k such that k^2 has at least one divisor d with k/2 <= d < k. - Robert Israel, Jan 08 2025

			15 is in the sequence since 15 = 6+9 where 9 | 6*15 = 90.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A338021, A350804 (exactly one).
Subsequences: A082663, A299174.
Cf. A090196.

filter:= proc(n) nops(select(t -> t >= n/2 and t < n, numtheory:-divisors(n^2)))>=1 end proc:
select(filter, [$1..300]); # Robert Israel, Jan 08 2025

f(n) = sum(s=1, n\2, !((s*n)%(n-s))); \\ A338021
isok(k) = f(k) >= 1; \\ Michel Marcus, Jan 17 2022

A320537 Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.

Original entry on oeis.org

2, 4, 10, 6, 14, 50, 8, 22, 70, 230, 12, 26, 98, 250, 1150, 16, 34, 110, 290, 1250, 5050, 18, 38, 130, 310, 1450, 5150, 22310, 20, 44, 154, 370, 1550, 5290, 23230, 106030, 24, 46, 170, 406, 1850, 5350, 23690, 106490, 510050, 28, 52, 182, 410, 2030, 5450, 24610, 107410, 513130, 2065450

Offset: 1

Omar E. Pol, Oct 15 2018

This is a permutation of the positive even numbers (A299174).
The union of all odd-indexed columns gives A319796, the even numbers in A071562.
The union of all even-indexed columns gives A319802, the even numbers in A071561.

			From _Hartmut F. W. Hoft_, Oct 06 2021: (Start)
The 10x10 section of table T(n,k):
(Table with first 20 terms from _Omar E. Pol_)
------------------------------------------------------------------
n\k | 1   2   3    4    5     6     7      8       9       10  ...
------------------------------------------------------------------
  1 | 2   10  50   230  1150  5050  22310  106030  510050  2065450
  2 | 4   14  70   250  1250  5150  23230  106490  513130  2115950
  3 | 6   22  98   290  1450  5290  23690  107410  520150  2126050
  4 | 8   26  110  310  1550  5350  24610  110170  530150  2157850
  5 | 12  34  130  370  1850  5450  25070  112010  530450  2164070
  6 | 16  38  154  406  2030  5650  25250  112930  532450  2168150
  7 | 18  44  170  410  2050  5750  25750  114770  534290  2176550
  8 | 20  46  182  430  2150  6250  25990  115690  537050  2186650
  9 | 24  52  190  434  2170  6350  26450  116150  540350  2216950
  10| 28  58  238  470  2350  6550  26750  117070  544870  2219650
   ... (End)

Row 1 is A320521.
Column 1 gives A174973 = A238443, without the 1.
Column 2 gives A244894.
Cf. A000203, A071561, A071562, A236104, A237270, A237271, A237593, A238443, A239663, A239665, A239929, A240062, A245092, A262626, A299174, A319796, A319802, A341969, A341970, A341971, A346969, A348171.

(* function a341969 is defined in A341969 *)
sArray[b_, pMax_] := Module[{list=Table[{}, pMax], i, p}, For[i=2, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax&&Length[list[[p]]]Hartmut F. W. Hoft, Oct 06 2021 *)

Terms a(21) and beyond from Hartmut F. W. Hoft, Oct 06 2021

A317108 Numbers missing from A317106.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128

Offset: 1

Daniël Karssen, Jul 24 2018

nonn

Equal to A299174 for n<=2930; a(2931)=5861, A299174(2931)=5862.
A317106 is finite, so this sequence is infinite.
See A317106 for further information.

Daniël Karssen, Table of n, a(n) for n = 1..67087

Cf. A299174, A317106.

A317440 Numbers missing from A317438.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128

Offset: 1

Daniël Karssen, Jul 28 2018

nonn

Equal to A299174 for n<=11361; a(11362)=22723, A299174(11362)=22724.
A317438 is finite, so this sequence is infinite.
See A317438 for further information.

Daniël Karssen, Table of n, a(n) for n = 1..245060

Cf. A317438, A299174.

A349767 Numbers m such that 2^m - m is divisible by 5.

Original entry on oeis.org

3, 14, 16, 17, 23, 34, 36, 37, 43, 54, 56, 57, 63, 74, 76, 77, 83, 94, 96, 97, 103, 114, 116, 117, 123, 134, 136, 137, 143, 154, 156, 157, 163, 174, 176, 177, 183, 194, 196, 197, 203, 214, 216, 217, 223, 234, 236, 237, 243, 254, 256, 257, 263, 274, 276, 277, 283, 294, 296, 297, 303

Offset: 1

Bernard Schott, Dec 10 2021

nonn

For every prime p, there are infinitely many numbers m such that 2^m - m (A000325) is divisible by p, here are numbers m corresponding to p = 5.

Equivalently, numbers that are congruent to {3, 14, 16, 17, 23, 34, 36, 37, 43, 54, 56, 57} mod 60, <==> numbers that are congruent to {+-3, +-14, +-16, +-17, +-23, +-34} mod 60.

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 4, 1983, page 158, 1993.

The IMO Compendium, Problem 4, 15th Canadian Mathematical Olympiad 1983.
Index to sequences related to Olympiads.

Cf. A000325, A070402, A010874.

Similar with: A299174 (p = 2), A047257 (p = 3), this sequence (p = 5).

filter:= n -> 2^n-n mod 5 = 0 : select(filter, [$1..400]);

Select[Range[300], PowerMod[2, #, 5] == Mod[#, 5] &] (* Amiram Eldar, Dec 10 2021 *)

isok(m) = Mod(2, 5)^m == Mod(m, 5); \\ Michel Marcus, Dec 10 2021

def ok(n): return pow(2, n, 5) == n%5
print([k for k in range(357) if ok(k)]) # Michael S. Branicky, Dec 10 2021

A321220 a(n) = n+2 if n is even, otherwise a(n) = 2*n+1 if n is odd.

Original entry on oeis.org

2, 3, 4, 7, 6, 11, 8, 15, 10, 19, 12, 23, 14, 27, 16, 31, 18, 35, 20, 39, 22, 43, 24, 47, 26, 51, 28, 55, 30, 59, 32, 63, 34, 67, 36, 71, 38, 75, 40, 79, 42, 83, 44, 87, 46, 91, 48, 95, 50, 99, 52, 103, 54, 107, 56, 111, 58, 115, 60, 119, 62, 123, 64, 127, 66

Offset: 0

Michel Marcus, Oct 31 2018

For n >= 3, a(n) is the Harborth Constant for the Dihedral groups D2n. See Balachandra link, Theorem 1 p. 2.

Colin Barker, Table of n, a(n) for n = 0..1000
Niranjan Balachandran, Eshita Mazumdar, Kevin Zhao, The Harborth Constant for Dihedral Groups, arXiv:1803.08286 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

A299174 and A004767 interleaved.

Cf. A043547, A114752.

[IsOdd(n) select (2*n+1) else n+2: n in [0..80]]; // Vincenzo Librandi, Nov 01 2018

a:=n->`if`(modp(n,2)=0,n+2,2*n+1): seq(a(n),n=0..70); # Muniru A Asiru, Oct 31 2018

CoefficientList[Series[(2 + 3 x + x^3)/(1 - x^2)^2, {x, 0, 64}], x] (* Michael De Vlieger, Oct 31 2018 *)
Table[If[OddQ[n], (2 n + 1), n + 2], {n, 0, 80}] (* Vincenzo Librandi, Nov 01 2018 *)

PARI
```
a(n) = if (n%2, 2*n+1, n+2);
```

Vec((2 + 3*x + x^3) / ((1 - x)^2*(1 + x)^2) + O(x^80)) \\ Colin Barker, Oct 31 2018

a(n) = A043547(n+1) + 1.
From Colin Barker, Oct 31 2018: (Start)
G.f.: (2 + 3*x + x^3) / (1-x^2)^2.
a(n) = 2*a(n-2) - a(n-4) for n > 3.
(End)

A369395 AGM transform of the even positive numbers.

Original entry on oeis.org

0, 4, 432, 61696, 12300000, 3339123264, 1195810789376, 549031054934016, 315439869711260160, 222215334010000000000, 188664842745174745939968, 190234762349632291168321536, 224946256003775354246877765632, 308520390288000443379128425267200, 486093585063330330624000000000000000

Offset: 1

Hugo Pfoertner, Jan 24 2024

See A368366 for the definition of the AGM transform.

Paolo Xausa, Table of n, a(n) for n = 1..210

Cf. A299174, A368366, A368367-A368371, A369394.

A369395[n_] := n^n*((n+1)^n - (2*n)!!);
Array[A369395, 15] (* Paolo Xausa, Jan 29 2024 *)

a369395(n) = {my(v=vector(n,i,i+i)); vecsum(v)^n - n^n*vecprod(v)};

from math import factorial
def A369395(n): return n**n*((n+1)**n-(factorial(n)<Chai Wah Wu, Jan 25 2024

A376108 Non-Leonardo numbers.

Original entry on oeis.org

2, 4, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Chai Wah Wu, Sep 10 2024

A299174 UNION A005408(A001690 - 1).

Cf. A001595, A299174, A001690.

def A376108(n):
    def f(x):
        a, b, c = 1, 1, n
        while True:
            if b > x: return c
            a, b = b, a+b+1
            c +=1
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m

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A119434 Odd n such that 2*phi(n) < n.

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A350803 Numbers k with at least one partition into two parts (s,t), s<=t such that t | s*k.

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A320537 Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.

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A317108 Numbers missing from A317106.

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A317440 Numbers missing from A317438.

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A349767 Numbers m such that 2^m - m is divisible by 5.

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A321220 a(n) = n+2 if n is even, otherwise a(n) = 2*n+1 if n is odd.

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A369395 AGM transform of the even positive numbers.

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