A157502 -id:A157502 - cp's OEIS frontend

A157512 Partial sums of A157502.

2, 8, 16, 26, 38, 52, 70, 90, 112, 136, 162, 190, 220, 252, 286, 324, 364, 406, 450, 496, 544, 594, 646, 700, 756, 814, 874, 936, 1002, 1070, 1140, 1212, 1286, 1362, 1440, 1520, 1602, 1686, 1772, 1860, 1950, 2042, 2136, 2232, 2330, 2432, 2536

Offset: 1

Gerald Hillier, Mar 02 2009

nonn

			a(4) = 2 + 6 + 8 + 10 = 26.

Cf. A157502.

lista(nn) = {s = 0; forstep (i = 2, nn, 2, if (! issquare(i), s+=i; print1(s, ", ");););} \\ Michel Marcus, Aug 26 2013

Set R(n) = floor(A157502(n)/2) and S(n) = floor(sqrt(A157502(n))/2); then a(n) = R(n)*(R(n) + 1) - (4*S(n)^3 + 6*S(n)^2 + 2*S(n))/3.

More terms from Michel Marcus, Aug 26 2013

A182664 a(n) = A088828(n) + A157502(n).

Original entry on oeis.org

5, 11, 15, 21, 25, 29, 35, 39, 43, 47, 53, 57, 61, 65, 69, 75, 79, 83, 87, 91, 95, 101, 105, 109, 113, 117, 121, 125, 131, 135, 139, 143, 147, 151, 155, 159, 165, 169, 173, 177, 181, 185, 189, 193, 197, 203, 207, 211, 215, 219, 223, 227, 231, 235, 239, 245

Offset: 1

Ed Smiley, Dec 23 2012

nonn

Cf. A088828 and A157502. Also alternate generation formula related to A000217, A010054.

Module[{nn=60,tb},tb=2*Accumulate[Table[If[IntegerQ[(Sqrt[8n+1]-1)/2],1,0],{n,nn}]];Join[{5},Table[1+4i+tb[[i-1]],{i,2,nn}]]] (* Harvey P. Dale, Jul 22 2013 *)

Let T(n) be 1 if positive n is a triangular number, else 0; we also define T(0) as 0.  Then we can also write this sequence as 1 + 4*n + 2*[T(1) + T(2) ... T(n-1)]. (Other than the special definition for T(0), T(n) is essentially A010054.)
The sequence can also be seen visually as
+ 4
+ 4 + 6
+ 4 + 6 + 4
+ 4 + 6 + 4 + 4
+ 4 + 6 + 4 + 4 + 6
+ 4 + 6 + 4 + 4 + 6 + 4
+ 4 + 6 + 4 + 4 + 6 + 4 + 4
+ 4 + 6 + 4 + 4 + 6 + 4 + 4 + 4
+ 4 + 6 + 4 + 4 + 6 + 4 + 4 + 4 + 6...

A088828 Nonsquare positive odd numbers.

Original entry on oeis.org

3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139

Offset: 1

Labos Elemer, Oct 28 2003

Odd numbers with even abundance: primes and some composites too.
Odd numbers with odd abundance are in A016754. Even numbers with odd abundance are in A088827. Even numbers with even abundance are in A088829.
Or, odd numbers without the squares. - Gerald Hillier, Apr 12 2009

			n = p prime, abundance = 1 - p = even and negative;
n = 21, sigma = 1 + 3 + 7 + 21 = 32, abundance = 32 - 42 = -20.

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

Cf. A005408, A014105, A016754, A088827, A088829, A157502.

[ n: n in [1..140 by 2] | IsEven(SumOfDivisors(n)-2*n) ]; // Klaus Brockhaus, Apr 15 2009

Do[s=DivisorSigma[1, n]-2*n; If[ !OddQ[s]&&OddQ[n], Print[{n, s}]], {n, 1, 1000}]
Select[Range[1, 500, 2], EvenQ[DivisorSigma[1, #] - 2 #] &] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)

isok(n) = (n>0) && (n % 2) && ! issquare(n); \\ Michel Marcus, Aug 28 2013

from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A088828_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not is_square(n),count(max(startvalue+(startvalue&1^1),1),2))
A088828_list = list(islice(A088828_gen(),30)) # Chai Wah Wu, Jul 06 2023

from math import isqrt
def A088828(n): return (s:=(m:=isqrt(k:=(n<<1)-1))+(k-m*(m+1)>=1))+k+(s&1) # Chai Wah Wu, Jun 19 2024

a(n) = 2*n + s - ((s+1) mod 2) where s = round(sqrt(2*n-1)). - Gerald Hillier, Apr 15 2009
A005408 SETMINUS A016754. - R. J. Mathar, Jun 16 2018
a(n) = 2*(n+h) + 1 where h = floor((1/4)*(sqrt(8*n) - 1)) is the largest value such that A014105(h) < n. - John Tyler Rascoe, Jul 05 2022

Entry revised by N. J. A. Sloane, Jan 31 2014 at the suggestion of Omar E. Pol

A128201 Union of positive squares and the odd numbers.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123

Offset: 1

Reinhard Zumkeller, Mar 04 2007

Range of A128200.

Positive numbers n such that n^((1 + n)/2) is an integer. - Gionata Neri, May 07 2016

R. Zumkeller, Table of n, a(n) for n = 1..10000

Partial sums given by A157130. - Gerald Hillier, Feb 25 2009
See A176693 for the union of even numbers and the squares. - M. F. Hasler, Apr 19 2015
Cf. A157502, A157512.

f[n_] := Block[{s = Range[n]^2, t}, Union[s, Range[1, Last@ s, 2]] // Sort]; f@ 12 (* Michael De Vlieger, Apr 16 2015 *)

A128201(n)=!(bittest(n=2*n-round(sqrt(2*n)),0)||issquare(n))+n \\ Based on Hiliers's formula. - M. F. Hasler, Apr 19 2015

is_A128201(n)=bittest(n,0)||issquare(n) \\ M. F. Hasler, Apr 19 2015

from math import isqrt
def A128201(n):
    def f(x): return n+(x>>1)-(isqrt(x)>>1)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 02 2024

a(n) = f(n,1,1,2), where f(n,i,m,x) = if i=n then m; else if m+1=x^2 then f(n,i+1,m+1,x); else if m+1>x^2 then f(n,i+1,m+1,x+2); else f(n,i+1,m+2,x).

Set R(n) = 2*n - round(sqrt(2*n)); then a(n) = R(n) + sign(frac(sqrt(R(n)))) * (not(R(n) mod 2)). - Gerald Hillier, Apr 16 2015

A346316 Composite numbers with primitive root 6.

Original entry on oeis.org

121, 169, 289, 1331, 1681, 2197, 3481, 3721, 4913, 6241, 6889, 7921, 10609, 11449, 11881, 12769, 14641, 16129, 17161, 18769, 22801, 24649, 28561, 32041, 39601, 49729, 51529, 52441, 54289, 63001, 66049, 68921, 73441, 76729, 83521, 120409, 134689, 139129, 157609

Offset: 1

Robert Hutchins, Jul 13 2021

nonn

An alternative description: Numbers k such that 1/k in base 6 generates a repeating fraction with period phi(n) and n/2 < phi(n) < n-1.

For example, in base 6, 1/121 has repeat length 110 = phi(121) which is > 121/2 but less than 121-1.

Robert Hutchins, PrimRoot.c

Subsequence of A244623.
Subsequence of A167794.
Cf. A108989 (for base 2),  A158248 (for base 10).
Cf. A157502.

isA033948 := proc(n)
    if n in {1,2,4} then
        true;
    elif type(n,'odd') and nops(numtheory[factorset](n)) = 1 then
        true;
    elif type(n,'even') and type(n/2,'odd') and nops(numtheory[factorset](n/2)) = 1 then
        true;
    else
        false;
    end if;
end proc:
isA167794 := proc(n)
    if not isA033948(n) or n = 1 then
        false;
    elif numtheory[order](6,n) = numtheory[phi](n) then
        true;
    else
        false;
    end if;
end proc:
A346316 := proc(n)
    option remember;
    local a;
    if n = 1 then
        121;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) and isA167794(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A346316(n),n=1..20) ; # R. J. Mathar, Sep 15 2021

Select[Range[160000], CompositeQ[#] && PrimitiveRoot[#, 6] == 6 &] (* Amiram Eldar, Jul 13 2021 *)

isok(m) = (m>1) && !isprime(m) && (gcd(m, 6)==1) && (znorder(Mod(6, m))==eulerphi(m)); \\ Michel Marcus, Aug 12 2021

A167794 INTERSECT A002808.

A377539 The number of iterations of the map x -> x + A000005(x), starting from n, until reaching an even number, and always at least one iteration taken.

Original entry on oeis.org

1, 1, 4, 3, 3, 1, 2, 1, 1, 1, 6, 1, 5, 1, 4, 3, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1, 16, 1, 16, 1, 15, 1, 14, 1, 13, 11, 13, 1, 12, 1, 12, 1, 11, 1, 10, 1, 2, 1, 1, 1, 9, 1, 9, 1, 8, 1, 7, 1, 7, 1, 6, 1, 5, 5, 5, 1, 5, 1, 4, 1, 4, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 38, 1, 37, 1, 36, 1, 36, 1, 35, 1, 35

Offset: 1

Ctibor O. Zizka, Oct 31 2024

The iteration step is x -> A062249(x).
a(n) = 1 if and only if n is an odd square (A016754) or an even nonsquare (A157502). - Robert Israel, Oct 31 2024
Therefore, a(n) = 1 <=> A323158(n) = 0. - Antti Karttunen, Jan 15 2025

			For n = 2, there is a(2) = 1 iteration to an even number: 2 -> 4 (with at least one iteration so 2 itself is not the even number target).
For n = 3 there are a(3) = 4 iterations to reach an even number: 3 -> 5 -> 7 -> 9 -> 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A062249 (step), A064491 (trajectory of 1), A016754, A157502, A323158.

f:= proc(n) local x,i;
  x:= n;
  for i from 1 do x:= x + numtheory:-tau(x); if x::even then return i fi od
end proc:
map(f, [$1..200]); # Robert Israel, Oct 31 2024

a[n_] := -1 + Length@ NestWhileList[# + DivisorSigma[0, #] &, n, OddQ, {2, 1}]; Array[a, 100] (* Amiram Eldar, Oct 31 2024 *)

A377539(n) = for(i=1,oo,if(!((n=(n+numdiv(n)))%2),return(i))); \\ Antti Karttunen, Jan 15 2025

cp's OEIS Frontend

A157512 Partial sums of A157502.

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A182664 a(n) = A088828(n) + A157502(n).

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A088828 Nonsquare positive odd numbers.

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A128201 Union of positive squares and the odd numbers.

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A346316 Composite numbers with primitive root 6.

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A377539 The number of iterations of the map x -> x + A000005(x), starting from n, until reaching an even number, and always at least one iteration taken.

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Maple

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