A088828 -id:A088828 - cp's OEIS frontend

A016754 Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.

1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569, 7921, 8281, 8649, 9025

Offset: 0

N. J. A. Sloane

The brown rat (rattus norwegicus) breeds very quickly. It can give birth to other rats 7 times a year, starting at the age of three months. The average number of pups is 8. The present sequence gives the total number of rats, when the intervals are 12/7 of a year and a young rat starts having offspring at 24/7 of a year. - Hans Isdahl, Jan 26 2008
Numbers n such that tau(n) is odd where tau(x) denotes the Ramanujan tau function (A000594). - Benoit Cloitre, May 01 2003
If Y is a fixed 2-subset of a (2n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007
Binomial transform of [1, 8, 8, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 8, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
All terms of this sequence are of the form 8k+1. For numbers 8k+1 which aren't squares see A138393. Numbers 8k+1 are squares iff k is a triangular number from A000217. And squares have form 4n(n+1)+1. - Artur Jasinski, Mar 27 2008
Sequence arises from reading the line from 1, in the direction 1, 25, ... and the line from 9, in the direction 9, 49, ..., in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008
Equals the triangular numbers convolved with [1, 6, 1, 0, 0, 0, ...]. - Gary W. Adamson & Alexander R. Povolotsky, May 29 2009
First differences: A008590(n) = a(n) - a(n-1) for n>0. - Reinhard Zumkeller, Nov 08 2009
Central terms of the triangle in A176271; cf. A000466, A053755. - Reinhard Zumkeller, Apr 13 2010
Odd numbers with odd abundance. Odd numbers with even abundance are in A088828. Even numbers with odd abundance are in A088827. Even numbers with even abundance are in A088829. - Jaroslav Krizek, May 07 2011
Appear as numerators in the non-simple continued fraction expansion of Pi-3: Pi-3 = K_{k>=1} (1-2*k)^2/6 = 1/(6+9/(6+25/(6+49/(6+...)))), see also the comment in A007509. - Alexander R. Povolotsky, Oct 12 2011
Ulam's spiral (SE spoke). - Robert G. Wilson v, Oct 31 2011
All terms end in 1, 5 or 9. Modulo 100, all terms are among { 1, 9, 21, 25, 29, 41, 49, 61, 69, 81, 89 }. - M. F. Hasler, Mar 19 2012
Right edge of both triangles A214604 and A214661: a(n) = A214604(n+1,n+1) = A214661(n+1,n+1). - Reinhard Zumkeller, Jul 25 2012
Also: Odd numbers which have an odd sum of divisors (= sigma = A000203). - M. F. Hasler, Feb 23 2013
Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c (A020882) and respective even leg b (A231100); sequence gives values c-b, sorted with duplicates removed. - K. G. Stier, Nov 04 2013
For n>1 a(n) is twice the area of the irregular quadrilateral created by the points ((n-2)*(n-1),(n-1)*n/2), ((n-1)*n/2,n*(n+1)/2), ((n+1)*(n+2)/2,n*(n+1)/2), and ((n+2)*(n+3)/2,(n+1)*(n+2)/2). - J. M. Bergot, May 27 2014
Number of pairs (x, y) of Z^2, such that max(abs(x), abs(y)) <= n. - Michel Marcus, Nov 28 2014
Except for a(1)=4, the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016
a(n) is the sum of 2n+1 consecutive numbers, the first of which is n+1. - Ivan N. Ianakiev, Dec 21 2016
a(n) is the number of 2 X 2 matrices with all elements in {0..n} with determinant = 2*permanent. - Indranil Ghosh, Dec 25 2016
Engel expansion of Pi*StruveL_0(1)/2 where StruveL_0(1) is A197037. - Benedict W. J. Irwin, Jun 21 2018
Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; the segments on the hypotenuse {p = a(n)/A001844(n), q = A060300(n)/A001844(n) = A001844(n) - p} and their ratio p/q = a(n)/A060300(n) are irreducible fractions in Q\Z. X values are A005408, Y values are A046092, Z values are A001844. - Ralf Steiner, Feb 25 2020
a(n) is the number of large or small squares that are used to tile primitive squares of type 2 (A344332). - Bernard Schott, Jun 03 2021
Also, positive odd integers with an odd number of odd divisors (for similar sequence with 'even', see A348005). - Bernard Schott, Nov 21 2021
a(n) is the least odd number k = x + y, with 0 < x < y, such that there are n distinct pairs (x,y) for which x*y/k is an integer; for example, a(2) = 25 and the two corresponding pairs are (5,20) and (10,15). The similar sequence with 'even' is A016742 (see Comment of Jan 26 2018). - Bernard Schott, Feb 24 2023
From Peter Bala, Jan 03 2024: (Start)
The sequence terms are the exponents of q in the series expansions of the following infinite products:
1) q*Product_{n >= 1} (1 - q^(16*n))*(1 + q^(8*n)) = q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + ....
2) q*Product_{n >= 1} (1 + q^(16*n))*(1 - q^(8*n)) = q - q^9 - q^25 + q^49 + q^81 - q^121 - q^169 + + - - ....
3) q*Product_{n >= 1} (1 - q^(8*n))^3 = q - 3*q^9 + 5*q^25 - 7*q^49 + 9*q^81 - 11*q^121 + 13*q^169 - + ....
4) q*Product_{n >= 1} ( (1 + q^(8*n))*(1 - q^(16*n))/(1 + q^(16*n)) )^3 = q + 3*q^9 - 5*q^25 - 7*q^49 + 9*q^81 + 11*q^121 - 13*q^169 - 15*q^225 + + - - .... (End)

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 586.

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe).
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
John Elias, Illustration: 8-fold Square Progression of Ulam's Spiral.
Milan Janjic, Two Enumerative Functions.
Scientific American, Cover of the March 1964 issue.
Amelia Carolina Sparavigna, Groupoids of OEIS A002378 and A016754 Numbers (oblong and odd square numbers), Politecnico di Torino (Italy, 2019).
Leo Tavares, Illustration: Diamond Triangles.
Leo Tavares, Illustration: Diamond Stars.
Eric Weisstein's World of Mathematics, Moore Neighborhood.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A000384, A001263, A001539, A001844, A003881, A005408, A006752, A014105, A016742, A016802, A016814, A016826, A016838, A033996, A046092, A060300, A138393, A167661, A167700.
Cf. A000447 (partial sums).
Cf. A005917, A344330, A344332.
Cf. A348005, A379481 [= a(A048673(n)-1)].
Partial sums of A022144.
Positions of odd terms in A341528.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A014634, A003154.

a016754 n = a016754_list !! n
a016754_list = scanl (+) 1 $ tail a008590_list
-- Reinhard Zumkeller, Apr 02 2012

[n^2: n in [1..100 by 2]]; // Vincenzo Librandi, Jan 03 2017

Range[1, 100, 2]^2 (* Paolo Xausa, Feb 13 2025 *)

A016754(n):=(n+n+1)^2$
makelist(A016754(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

A016754(n)=(n<<1+1)^2 \\ Charles R Greathouse IV, Jun 16 2011, corrected and edited by M. F. Hasler, Apr 11 2023

def A016754(n): return ((n<<1)|1)**2 # Chai Wah Wu, Jul 06 2023

a(n) = 1 + Sum_{i=1..n} 8*i = 1 + 8*A000217(n). - Xavier Acloque, Jan 21 2003; Zak Seidov, May 07 2006; Robert G. Wilson v, Dec 29 2010
O.g.f.: (1+6*x+x^2)/(1-x)^3. - R. J. Mathar, Jan 11 2008
a(n) = 4*n*(n + 1) + 1 = 4*n^2 + 4*n + 1. - Artur Jasinski, Mar 27 2008
a(n) = A061038(2+4n). - Paul Curtz, Oct 26 2008
Sum_{n>=0} 1/a(n) = Pi^2/8 = A111003. - Jaume Oliver Lafont, Mar 07 2009
a(n) = A000290(A005408(n)). - Reinhard Zumkeller, Nov 08 2009
a(n) = a(n-1) + 8*n with n>0, a(0)=1. - Vincenzo Librandi, Aug 01 2010
a(n) = A033951(n) + n. - Reinhard Zumkeller, May 17 2009
a(n) = A033996(n) + 1. - Omar E. Pol, Oct 03 2011
a(n) = (A005408(n))^2. - Zak Seidov, Nov 29 2011
From George F. Johnson, Sep 05 2012: (Start)
a(n+1) = a(n) + 4 + 4*sqrt(a(n)).
a(n-1) = a(n) + 4 - 4*sqrt(a(n)).
a(n+1) = 2*a(n) - a(n-1) + 8.
a(n+1) = 3*a(n) - 3*a(n-1) + a(n-2).
(a(n+1) - a(n-1))/8 = sqrt(a(n)).
a(n+1)*a(n-1) = (a(n)-4)^2.
a(n) = 2*A046092(n) + 1 = 2*A001844(n) - 1 = A046092(n) + A001844(n).
Limit_{n -> oo} a(n)/a(n-1) = 1. (End)
a(n) = binomial(2*n+2,2) + binomial(2*n+1,2). - John Molokach, Jul 12 2013
E.g.f.: (1 + 8*x + 4*x^2)*exp(x). - Ilya Gutkovskiy, May 23 2016
a(n) = A101321(8,n). - R. J. Mathar, Jul 28 2016
Product_{n>=1} A033996(n)/a(n) = Pi/4. - Daniel Suteu, Dec 25 2016
a(n) = A014105(n) + A000384(n+1). - Bruce J. Nicholson, Nov 11 2017
a(n) = A003215(n) + A002378(n). - Klaus Purath, Jun 09 2020
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=0} a(n)/n! = 13*e.
Sum_{n>=0} (-1)^(n+1)*a(n)/n! = 3/e. (End)
Sum_{n>=0} (-1)^n/a(n) = A006752. - Amiram Eldar, Oct 10 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = cosh(Pi/2).
Product_{n>=1} (1 - 1/a(n)) = Pi/4 (A003881). (End)
From Leo Tavares, Nov 24 2021: (Start)
a(n) = A014634(n) - A002943(n). See Diamond Triangles illustration.
a(n) = A003154(n+1) - A046092(n). See Diamond Stars illustration. (End)
From Peter Bala, Mar 11 2024: (Start)
Sum_{k = 1..n+1} 1/(k*a(k)*a(k-1)) = 1/(9 - 3/(17 - 60/(33 - 315/(57 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*2^2 ))))).
3/2 - 2*log(2) = Sum_{k >= 1} 1/(k*a(k)*a(k-1)) = 1/(9 - 3/(17 - 60/(33 - 315/(57 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*2^2 - ... ))))).
Row 2 of A142992. (End)
From Peter Bala, Mar 26 2024: (Start)
8*a(n) = (2*n + 1)*(a(n+1) - a(n-1)).
Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 1/2 - Pi/8 = 1/(9 + (1*3)/(8 + (3*5)/(8 + ... + (4*n^2 - 1)/(8 + ... )))). For the continued fraction use Lorentzen and Waadeland, p. 586, equation 4.7.9 with n = 1. Cf. A057813. (End)

Additional description from Terrel Trotter, Jr., Apr 06 2002

A324644 a(n) = gcd(sigma(n), A276086(n)).

Original entry on oeis.org

1, 3, 2, 1, 6, 1, 2, 15, 1, 9, 6, 1, 2, 3, 6, 1, 18, 1, 10, 3, 2, 9, 6, 5, 1, 3, 10, 1, 30, 1, 2, 21, 6, 9, 6, 7, 2, 15, 14, 45, 42, 1, 2, 21, 6, 9, 6, 1, 1, 3, 6, 7, 18, 5, 2, 15, 10, 45, 30, 7, 2, 3, 2, 1, 42, 1, 2, 21, 6, 9, 18, 5, 2, 3, 2, 35, 6, 7, 10, 3, 1, 63, 42, 7, 2, 3, 30, 45, 90, 1, 14, 21, 2, 9, 6, 7, 98, 3, 6, 7, 6, 1, 2, 105, 6

Offset: 1

Antti Karttunen, Mar 11 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30030

Cf. A000203, A009194, A276086, A324198, A324534, A324645, A324646, A364286.

Cf. A088828 (positions of even terms), A176693 (of odd terms).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324644(n) = gcd(sigma(n),A276086(n));

a(n) = gcd(A000203(n), A276086(n)).

A351458 Numbers k for which k * gcd(sigma(k), A276086(k)) is equal to sigma(k) * gcd(k, A276086(k)), where A276086 is the primorial base exp-function, and sigma gives the sum of divisors of its argument.

Original entry on oeis.org

1, 10, 56, 9196, 9504, 56160, 121176, 239096, 354892, 411264, 555520, 716040, 804384, 904704, 1063348, 1387386, 1444352, 1454112, 1884800, 2708640, 3317248, 3548920, 4009824, 4634784, 6179712, 6795360, 7285248, 14511744, 16328466, 28377216, 29855232, 31940280, 37444736, 42711552, 49762944, 52815744

Offset: 1

Antti Karttunen, Feb 13 2022

nonn

Numbers k such that k * A324644(k) = A000203(k) * A324198(k).
Numbers k such that gcd(A064987(k), A324580(k)) = gcd(A064987(k), A351252(k)).
Numbers k such that their abundancy index [sigma(k)/k] is equal to A324644(k)/A324198(k). See A364286.
A324644 gives odd values for even numbers and for the odd squares. A324198 is odd on all arguments, therefore on odd squares the above equation reduces to odd * odd = odd * odd, and on odd nonsquares as odd * even = even * odd. It is an open question whether there are any odd terms after the initial a(1)=1.
If k is even, but not a multiple of 3, then A276086(k) is a multiple of 3, but not even (i.e., is an odd multiple of 3). If for such k also sigma(k) = 3*k, then A007949(A324644(k)) = min(A007949(sigma(k)), A007949(A276086(k))) = 1, while A007949(A324198(k)) = min(A007949(k), A007949(A276086(k))) = 0, therefore all such k's do occur in this sequence, for example, the two known terms of A005820 (3-perfect numbers) that are not multiples of three: 459818240, 51001180160, but also any hypothetical term of A005820 of the form 4u+2, where 2u+1 is not multiple of 3, and which by necessity is then also an odd perfect number.
Similarly, of the 65 known 5-multiperfect numbers (A046060), those 20 that are not multiples of five are included in this sequence. Note that all 65 are multiples of six.
It is conjectured that the intersection of this sequence with the multiperfect numbers (A007691) gives A323653, see comments in the latter.
For all even terms k of this sequence, A007814(A000203(k)) = A007814(k), sigma preserves the 2-adic valuation, and A007949(A000203(k)) >= A007949(k), i.e., does not decrease the 3-adic valuation. The condition is equivalence (=) when k is a multiple of 6. With odd terms, any hypothetical odd perfect number x would yield a one greater 2-adic valuation for sigma(x) than for x, but would satisfy the main condition of this sequence. - Corrected Feb 17 2022
If k is a nonsquare positive odd number (in A088828), then it must be a term of A191218. - Antti Karttunen, Mar 10 2024

Cf. A000203, A005820, A007691, A007814, A007949, A046060, A088828, A191218, A276086, A323653, A324198, A324644, A327938, A364286, A351459 (intersection with a similar sequence, A349745).

Cf. also A351549.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA351458(n) = { my(s=sigma(n), z=A276086(n)); (n*gcd(s,z))==(s*gcd(n,z)); };

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]); \\ Works OK with rationals also!
isA351458(n) = { my(orgn=n, s=sigma(n), abi=s/n, p=2, q=A006530(abi), d, e1, e2); while((1!=abi)&&(p<=q), d = n%p; e1 = min(d, valuation(s, p)); e2 = min(d, valuation(orgn, p)); d = e1-e2; if(valuation(abi,p)!=d, return(0), abi /= (p^d)); n = n\p; p = nextprime(1+p)); (abi==1); }; \\ (This implementation does not require the construction of largish intermediate numbers, A276086, but might still be slower and return a few false positives on the long run, so please check the results with the above program). - Antti Karttunen, Feb 19 2022

A088827 Even numbers with odd abundance: even squares or two times squares.

Original entry on oeis.org

2, 4, 8, 16, 18, 32, 36, 50, 64, 72, 98, 100, 128, 144, 162, 196, 200, 242, 256, 288, 324, 338, 392, 400, 450, 484, 512, 576, 578, 648, 676, 722, 784, 800, 882, 900, 968, 1024, 1058, 1152, 1156, 1250, 1296, 1352, 1444, 1458, 1568, 1600, 1682, 1764, 1800, 1922

Offset: 1

Labos Elemer, Oct 28 2003

Sigma(k)-2k is odd means that sigma(k) is also odd.

Odd numbers with odd abundance are in A016754. Odd numbers with even abundance are in A088828. Even numbers with even abundance are in A088829.

			From _Michael De Vlieger_, May 14 2017: (Start)
4 is a term since it is even and the sum of its divisors {1,2,4} = 7 - 2(4) = -1 is odd. It is an even square.
18 is a term since it is even and the sum of its divisors {1,2,3,6,9,18} = 39 - 2(18) = 3 is odd. It is 2 times a square, i.e., 2(9). (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A016754, A088828, A088829, A111003, A191432.

Do[s=DivisorSigma[1, n]-2*n; If[OddQ[s]&&!OddQ[n], Print[{n, s}]], {n, 1, 1000}]
(* Second program: *)
Select[Range[2, 2000, 2], OddQ[DivisorSigma[1, #] - 2 #] &] (* Michael De Vlieger, May 14 2017 *)

from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A088827_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:is_square(n) or is_square(n>>1),count(max(startvalue+(startvalue&1),2),2))
A088827_list = list(islice(A088827_gen(),30)) # Chai Wah Wu, Jul 06 2023

Conjecture: a(n) = ((2*r) + 1)^2 * 2^(c+1) where r and c are the corresponding row and column of n in the table format of A191432, where the first row and column are 0. - John Tyler Rascoe, Jul 12 2022

Sum_{n>=1} 1/a(n) = Pi^2/8 (A111003). - Amiram Eldar, Jul 09 2023

A157502 Even numbers without the squares.

Original entry on oeis.org

2, 6, 8, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 32, 34, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130

Offset: 1

Gerald Hillier, Mar 01 2009

nonn

Complement of A128201.

			a(2) = 6 as the next even after 2, i.e., 4 (a perfect square), drops out and is replaced by following even.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A088828, A128201, A157512.

Cf. A000290 (squares), A005843 (even numbers).

Select[Range[2, 130, 2], ! IntegerQ@ Sqrt[#] &] (* Michael De Vlieger, Oct 01 2021 *)

isok(n) = ((n % 2) == 0) && ! issquare(n); \\ Michel Marcus, Aug 26 2013

from math import isqrt
def A157502(n): return (k:=n<<1)+(r:=(m:=isqrt(k))+int((k-m*(m+1)<<2)>=1))-(r&1) # Chai Wah Wu, Jul 30 2022

Set R(n) = round(sqrt(2*n)), then a(n) = 2*n + R(n) - (R(n) mod 2).

More terms from Michel Marcus, Aug 26 2013

A348741 Odd numbers k for which A161942(k) < k, where A161942 is the odd part of sigma.

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, 141, 143, 145, 147, 149, 151, 153, 155, 157

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Differs from A088828 for the first time at n=1080, where a(1080) = 2207, while A088828(1080) = 2205 = A348743(1), the value which is missing from this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..16291
Index entries for sequences related to sigma(n)

Setwise difference A088828 \ A348743.
Cf. A161942, A348742 (complement among the odd numbers).
Cf. also A348738, A348753, A348931.

odd[n_] := n/2^IntegerExponent[n, 2]; Select[Range[1, 150, 2], odd[DivisorSigma[1, #]] < # &] (* Amiram Eldar, Nov 02 2021 *)

A000265(n) = (n >> valuation(n, 2));
isA348741(n) = ((n%2)&&A000265(sigma(n))

A348743 Odd nonsquares k for which A161942(k) >= k, where A161942 is the odd part of sigma.

Original entry on oeis.org

2205, 19845, 108045, 143325, 178605, 187425, 236925, 266805, 319725, 353925, 372645, 407925, 452025, 462825, 584325, 637245, 646425, 658125, 672525, 789525, 796005, 804825, 845325, 920205, 972405, 981225, 1007325, 1055925, 1069425, 1102725, 1113525, 1116225, 1166445, 1201725, 1245825, 1289925, 1378125, 1380825, 1442925

Offset: 1

Antti Karttunen, Nov 02 2021

The first non-multiples of 5 are a(103) = 6243237 and a(125) = 8164233.
From Antti Karttunen, Nov 28 2024: (Start)
This is not a subsequence of A228058. At least k = A000040(28)*(A002110(27)/2)^2 = 15388519572341080054329140040512468358441210638435506649120749687401476705908239675 is a number of the form 4m+3 such that A161942(k) >= k.
Another such number is A000040(28)*81*(A002110(25)/6)^2 = 1279741205456530915782536871495922949062895982530933679752838870798129159675.
Question: What is the smallest term of this sequence that is of the form 4m+3, and thus not in A386427 (in A191218 and in A228058)?
(End)

Intersection of A088828 and A348742.
Cf. A386427 (a subsequence, which agrees for a very long time).
Cf. A033630, A065205, A083206, A161942, A191218, A228058, A336702, A348741.
Cf. also A065235, A162284.

A000265(n) = (n >> valuation(n, 2));
isA348743(n) = ((n%2)&&!issquare(n)&&A000265(sigma(n))>=n); \\ Edited Nov 28 2024

Definition changed (from > to >=) to formally include also any hypothetical odd perfect numbers - Antti Karttunen, Nov 28 2024

Comment removed, because it was more related to sequence A386427. - Antti Karttunen, Aug 21 2025

A088829 Even numbers with even abundance.

Original entry on oeis.org

6, 10, 12, 14, 20, 22, 24, 26, 28, 30, 34, 38, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 62, 66, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136, 138, 140, 142, 146, 148, 150, 152

Offset: 1

Labos Elemer, Oct 28 2003

nonn

Odd numbers with odd abundance are in A016754. Odd numbers with even abundance are in A088828. Even numbers with odd abundance are in A088827.

			Even perfect numbers are here.
n=10: sigma[10]=18, abundance=18-20=-2<0;
n=12: sigma[12]=28, abundance=28-24=+4>0.
sigma[n] is also even number; sign of abundance:<=>0.

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

Cf. A016754, A088827-A088828, A088830.

Do[s=DivisorSigma[1, n]-2*n; If[ !OddQ[s]&&!OddQ[n], Print[{n, s}]], {n, 1, 1000}]

A334748 Let p be the smallest odd prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller odd primes.

Original entry on oeis.org

3, 6, 5, 12, 15, 10, 21, 24, 27, 30, 33, 20, 39, 42, 7, 48, 51, 54, 57, 60, 35, 66, 69, 40, 75, 78, 45, 84, 87, 14, 93, 96, 55, 102, 105, 108, 111, 114, 65, 120, 123, 70, 129, 132, 135, 138, 141, 80, 147, 150, 85, 156, 159, 90, 165, 168, 95, 174, 177, 28, 183, 186, 189

Offset: 1

Peter Munn, May 09 2020

A permutation of A028983.
A007417 (which has asymptotic density 3/4) lists index n such that a(n) = 3n. The sequence maps the terms of A007417 1:1 onto A145204\{0}, defining a bijection between them.
Similarly, bijections are defined from the odd numbers (A005408) to the nonsquare odd numbers (A088828), from the positive even numbers (A299174) to A088829, from A003159 to the nonsquares in A003159, and from A325424 to the nonsquares in A036668. The latter two bijections are between sets where membership depends on whether a number's squarefree part divides by 2 and/or 3.

			84 = 21*4 has squarefree part 21 (and square part 4). The smallest odd prime absent from 21 = 3*7 is 5 and the product of all smaller odd primes is 3. So a(84) = 84*5/3 = 140.

Permutation of A028983.
Row 3, and therefore column 3, of A331590. Cf. A334747 (row 2).
A007913, A034386, A225546, A284723 are used in formulas defining the sequence.
The formula section details how the sequence maps the terms of A003961, A019565, A070826; and how f(a(n)) relates to f(n) for f = A008833, A048675, A267116; making use of A003986.
Subsequences: A016051, A145204\{0}, A329575.
Bijections are defined that relate to A003159, A005408, A007417, A036668, A088828, A088829, A299174, A325424.

a(n) = {my(c=core(n), m=n); forprime(p=3, , if(c % p, m*=p; break, m/=p)); m;} \\ Michel Marcus, May 22 2020

a(n) = n * p / (A034386(p-1)/2), where p = A284723(A007913(n)).
a(n) = A334747(A334747(n)).
a(n) = A331590(3, n) = A225546(4 * A225546(n)).
a(2*n) = 2 * a(n).
a(A019565(n)) = A019565(n+2).
a(k * m^2) = a(k) * m^2.
a(A003961(n)) = A003961(A334747(n)).
a(A070826(n)) = prime(n+1).
A048675(a(n)) = A048675(n) + 2.
A008833(a(n)) = A008833(n).
A267116(a(n)) = A267116(n) OR 1, where OR denotes the bitwise operation A003986.
a(A007417(n)) = A145204(n+1) = 3 * A007417(n).

cp's OEIS Frontend

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

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A016754 Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.

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A324644 a(n) = gcd(sigma(n), A276086(n)).

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A351458 Numbers k for which k * gcd(sigma(k), A276086(k)) is equal to sigma(k) * gcd(k, A276086(k)), where A276086 is the primorial base exp-function, and sigma gives the sum of divisors of its argument.

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A088827 Even numbers with odd abundance: even squares or two times squares.

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A157502 Even numbers without the squares.

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A348741 Odd numbers k for which A161942(k) < k, where A161942 is the odd part of sigma.

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