A070824 -id:A070824 - cp's OEIS frontend

A346510 a(n) is the number of nontrivial divisors of A346507(n) ending with 1.

1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 4, 2

Offset: 1

Stefano Spezia, Jul 21 2021

			a(42) = 4 since there are 4 nontrivial divisors of A346507(42) = 2541 ending with 1: 11, 21, 121 and 231.

Cf. A017281, A070824, A346388 (ending with 5), A346389 (ending with 6), A346392, A346507, A346508, A346509.

b={}; For[n=1, n<=500, n++, For[k=1, kMax[b], AppendTo[b, 10n+1]]]]; (* A346507 *) a={}; For[i =1, i<=Length[b], i++, AppendTo[a, Length[Drop[Select[Divisors[Part[b, i]], (Mod[#, 10]==1&)], -1]]-1]]; a

f(n) = sumdiv(n, d, (d>1) && (d(f(x)), [1..5000])) \\ Michel Marcus, Jul 28 2021

from sympy import divisors
def f(n): return sum(d%10 == 1 for d in divisors(n)[1:-1])
def A346507upto(lim): return sorted(set(a*b for a in range(11, lim//11+1, 10) for b in range(a, lim//a+1, 10)))
print(list(map(f, A346507upto(5000)))) # Michael S. Branicky, Jul 31 2021

a(n) = A346392(A346507(n)) - 1.

A382325 Numbers with a record ratio of proper factorizations to nontrivial divisors.

Original entry on oeis.org

4, 16, 32, 64, 128, 192, 256, 384, 512, 576, 768, 864, 1024, 1152, 1536, 1728, 2304, 3456, 4608, 5184, 5760, 6912, 8640, 9216, 10368, 11520, 13824, 17280, 20736, 23040, 25920, 27648, 34560, 41472, 51840, 62208, 69120, 82944, 103680, 138240, 165888, 172800

Offset: 1

Charles L. Hohn, Mar 21 2025

nonn

Numbers k that give a record value for A028422(k)/A070824(k).

a(n) = 0 (mod 4), and with prime factors of terms clustering around the smallest primes, it is observed that as n increases, the gcd of a(n)..a(oo) remains among the largest divisors of a(n).

			a(1)=4: |{{2, 2}}| / |{2}| = 1/1.
a(2)=16: |{{2, 2, 2, 2}, {2, 2, 4}, {2, 8}, {4, 4}}| / |{2, 4, 8}| = 4/3.
a(3)=32: |{{2, 2, 2, 2, 2}, {2, 2, 2, 4}, {2, 2, 8}, {2, 4, 4}, {2, 16}, {4, 8}}| / |{2, 4, 8, 16}| = 6/4.

Cf. A028422, A070824, A033833, A002182, A382326, A382327.

Subsequence of A025487.

f_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, return(if(#f, 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && dmx, mx=m; print1(x, ", ")))

A382326 Numbers with a record ratio of nontrivial divisors to prime factors (counted with multiplicity).

Original entry on oeis.org

4, 6, 12, 24, 30, 60, 120, 180, 210, 360, 420, 840, 1260, 2310, 2520, 4620, 7560, 9240, 13860, 27720, 55440, 60060, 83160, 110880, 120120, 138600, 166320, 180180, 277200, 360360, 720720, 1081080, 1441440, 1801800, 2162160, 3063060, 3603600, 5405400, 6126120

Offset: 1

Charles L. Hohn, Mar 21 2025

nonn

Numbers k that give a record value for A070824(k)/A001222(k).

			a(1) = 4: |{2}| / |{2, 2}| = 1/2.
a(2) = 6: |{2, 3}| / |{2, 3}| = 2/2.
a(3) = 12: |{2, 3, 4, 6}| / |{2, 2, 3}| = 4/3.

Cf. A070824, A001222, A002182, A382325, A382327.

Subsequence of A025487.

my(mx=0); for(x=2, 10^5, my(f=factor(x), m=(numdiv(f)-2)/bigomega(f)); if(m>mx, mx=m; print1(x, ", ")))

A095808 Number of ways to write n in the form m + (m+1) + ... + (m+k-1) + (m+k) + (m+k-1) + ... + (m+1) + m with integers m>= 1, k>=1. Or, number of divisors d of 4n-1 with 0 < (d-1)^2 < 4n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 1, 0, 0, 3, 0, 1, 2, 0, 1, 1, 0, 0, 2, 2, 0, 1, 1, 0, 3, 0, 1, 2, 0, 1, 1, 0, 0, 3, 1, 0, 2, 1, 0, 3, 1, 0, 1, 0, 2, 2, 0, 1, 1, 1, 1, 1, 0, 0, 5, 1, 1, 1, 0, 1, 1, 1, 0, 3, 1, 0, 2, 0, 1, 3, 0, 0, 2, 1, 1, 3

Offset: 1

Alfred Heiligenbrunner, Jun 15 2004

nonn

n = m + (m+1) + ... + (m+k-1) + (m+k) + (m+k-1) + ... + (m+1) + m means n = k^2 + m*(2k+1) or 4n-1 = (2k+1)*(4m+2k-1). So if 4n-1 disparts into two odd factors a*b, then k = (a-1)/2, m=(n-k^2)/(2k+1) give the solution of the origin equation. We only count solutions with k^2 < n, such that m>0. This means we are taking into account only factors a < 2n+1.

Note that a(n) = 0 if 4n-1 is prime. - Alfred Heiligenbrunner, Mar 01 2016

			a(16) = 2 because 16 = 5+6+5 and 16 = 1+2+3+4+3+2+1.
The trivial case 16=16 (k=0, m=n) is not counted. The cases m=0, e.g. 16 = 0+1+2+3+4+3+2+1+0 are not counted. The cases m<0 e.g. 16 = -4+-3+-2+-1+0+1+2+3+4+5+6+5+4+3+2+1+0+-1+-2+-3+-4 are not counted.

Alfred Heiligenbrunner, Tower-Sums of adjacent numbers (in German).

Cf. A001227, A001620, A069283, A070824, A078703.

seq((numtheory[tau](4*n-1)-2)/2, n=1..100); # Ridouane Oudra, Jan 18 2025

h1 = Table[count = 0; For[k = 1, k^2 < n, k++, If[Mod[n - k^2, 2k + 1] == 0, count++ ]]; count, {n, 100}] - or - h2 = Table[Length[Select[Divisors[4n - 1], ((# - 1)^2 < 4n) &]] - 1, {n, 100}]
a[n_] := (DivisorSigma[0, 4*n-1] - 2)/2; Array[a, 100] (* Amiram Eldar, Jan 28 2025 *)

a(n) = (numdiv(4*n-1) - 2)/2; \\ Amiram Eldar, Jan 28 2025

From Ridouane Oudra, Jan 18 2025: (Start)
a(n) = (tau(4*n-1) - 2)/2.
a(n) = A070824(4*n-1)/2.
a(n) = A078703(n) - 1. (End)
Sum_{k=1..n} a(k) = (log(n) + 2*gamma - 5 + 4*log(2))*n/4 + O(n^(1/3)*log(n)), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 27 2025

A161116 a(n) is the number of nontrivial positive divisors of 2n+3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 2, 0, 2, 0, 0, 4, 0, 1, 2, 0, 2, 2, 0, 0, 4, 2, 0, 2, 0, 0, 4, 2, 0, 3, 0, 2, 2, 0, 2, 2, 2, 0, 4, 0, 0, 6, 0, 0, 2, 0, 2, 4, 2, 1, 2, 2, 0, 2, 0, 2, 6, 0, 0, 2, 2, 2, 4, 0, 0, 4, 2, 0, 2, 2, 0, 6, 0, 1, 4, 0, 4, 2, 0, 0, 2

Offset: 0

Vladimir Shevelev, Jun 02 2009

a(n)=0 iff n is in A067076, i.e., 2n+3 is prime; a(n) is the number of positive integers of the form (n-3k)/(2k+1), 1<=k<=n/3.

			Since for n=3 we have 2n+3=9 and only nontrivial divisor of 9 is 3, then a(3)=1.

Cf. A067076, A034953, A054269, A082749, A006254.

Cf. A160973. - Vladimir Shevelev, Jun 07 2009

[NumberOfDivisors(2*n+3)-2: n in [0..100]]; // Vincenzo Librandi, Feb 08 2016

a(n) = numdiv(2*n+3) - 2; \\ Michel Marcus, Feb 08 2016

For n>=1, a(n)=A160973(n)+A079978(n). [Vladimir Shevelev, Jun 07 2009]

a(n) = A070824(2n+3).

Edited by Charles R Greathouse IV, Oct 12 2009

More terms from Michel Marcus, Feb 08 2016

A163838 a(n) = (n-th composite) * (number of nontrivial divisors of n-th composite).

Original entry on oeis.org

4, 12, 16, 9, 20, 48, 28, 30, 48, 72, 80, 42, 44, 144, 25, 52, 54, 112, 180, 128, 66, 68, 70, 252, 76, 78, 240, 252, 176, 180, 92, 384, 49, 200, 102, 208, 324, 110, 336, 114, 116, 600, 124, 252, 320, 130, 396, 272, 138, 420, 720, 148, 300, 304, 154, 468, 640, 243

Offset: 1

Juri-Stepan Gerasimov, Aug 05 2009, Aug 06 2009

nonn

The trivial divisors of the n-th composite are 1 and the n-th composite.

			a(1) =  4 (= 4*1);
a(2) = 12 (= 6*2);
a(3) = 16 (= 8*2);
a(4) =  9 (= 9*1);
a(5) = 20 (= 10*2);
a(6) = 48 (= 12*4).

Cf. A002808, A070824, A144925.

A002808 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then return a; end if; end do: end if; end proc: A070824 := proc(n) numtheory[tau](n)-2 ; end: A144925 := proc(n) A070824(A002808(n)) ; end: A163838 := proc(n) A144925(n)*A002808(n) ; end: seq(A163838(n),n=1..80) ; # R. J. Mathar, Oct 10 2009

# (DivisorSigma[0,#]-2)&/@Select[Range[100],CompositeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 23 2021 *)

a(n) = A002808(n)*A144925(n).

56 replaced with 112, 310 replaced with 320, and 468 inserted by R. J. Mathar, Oct 10 2009

A259977 Number of proper divisors of A005381(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 1, 2, 2, 4, 2, 2, 2, 7, 2, 6, 4, 2, 1, 4, 2, 4, 2, 6, 2, 2, 4, 5, 2, 6, 2, 6, 4, 4, 2, 6, 3, 2, 2, 2, 2, 6, 2, 4, 2, 2, 2, 10, 4, 7, 6, 2, 2, 8, 2, 4, 4, 2, 2, 14, 1, 2, 2, 4, 2, 10, 2, 6, 2, 2, 6, 6, 2, 2, 2, 13, 2, 2, 4, 4, 4, 6, 2, 10, 2, 10, 2, 8, 6, 2, 1, 6, 4, 4, 4, 8, 2, 2, 2, 6

Offset: 1

N. J. A. Sloane, Jul 12 2015, following a suggestion from R. P. Boas, May 19 1974

nonn

R. P. Boas & N. J. A. Sloane, Correspondence, 1974

Cf. A005381, A070824.

A259977 := proc(n)
    A070824(A005381(n)) ;
end proc:
seq(A259977(n),n=1..100) ; # R. J. Mathar, Jul 14 2015

A291108 Expansion of Sum_{k>=2} k^2x^(2k)/(1 - x^k).

Original entry on oeis.org

0, 0, 0, 4, 0, 13, 0, 20, 9, 29, 0, 65, 0, 53, 34, 84, 0, 130, 0, 145, 58, 125, 0, 273, 25, 173, 90, 265, 0, 399, 0, 340, 130, 293, 74, 614, 0, 365, 178, 609, 0, 735, 0, 625, 340, 533, 0, 1105, 49, 754, 298, 865, 0, 1183, 146, 1113, 370, 845, 0, 1859, 0, 965, 580, 1364, 194, 1743, 0, 1465, 538, 1599, 0, 2550, 0, 1373, 884

Offset: 1

Ilya Gutkovskiy, Aug 17 2017

nonn

Sum of squares of divisors of n except 1 and n^2 (sum of squares of nontrivial divisors of n).

			a(6) = 13 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are nontrivial {2, 3} and 2^2 + 3^2 = 13.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors

Cf. A000290, A001157, A001248, A001358, A005063, A008578, A027751, A048050, A053807, A067558, A070824.

nmax = 75; Rest[CoefficientList[Series[Sum[k^2 x^(2 k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x]]
Join[{0}, Table[DivisorSigma[2, n] - n^2 - 1, {n, 2, 75}]]

A291108(n) = sumdiv(n,d,if((1==d)||(n==d),0,d^2)); \\ Antti Karttunen, Jan 22 2025

G.f.: Sum_{k>=2} k^2*x^(2*k)/(1 - x^k).
a(n) = A001157(n) -  A000290(n) - 1 for n > 1.
a(n) = A067558(n) - 1 for n > 1.
a(n) = A005063(n) if n is a semiprime (A001358).
a(n) = 0 if n is a prime or 1 (A008578).
a(n) = n if n is a square of prime (A001248).
a(p^k) = (p^(2*k) - p^2)/(p^2 - 1) for p is a prime and k > 0.

A328337 The number whose binary indices are the nontrivial divisors of n (greater than 1 and less than n).

Original entry on oeis.org

0, 0, 0, 2, 0, 6, 0, 10, 4, 18, 0, 46, 0, 66, 20, 138, 0, 294, 0, 538, 68, 1026, 0, 2222, 16, 4098, 260, 8266, 0, 16950, 0, 32906, 1028, 65538, 80, 133422, 0, 262146, 4100, 524954, 0, 1056870, 0, 2098186, 16660, 4194306, 0, 8423598, 64, 16777746, 65540

Offset: 1

Gus Wiseman, Oct 15 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The nontrivial divisors of 18 are {2, 3, 6, 9}, so a(18) = 2^1 + 2^2 + 2^5 + 2^8 = 294.

Removing zeros gives binary indices of rows of A163870.
The version for all divisors is A034729.
The version for proper divisors is A247146.
Cf. A000005, A000120, A027750, A029931, A033676, A048793, A060775, A070824, A070939, A275700, A326031.

Table[Total[(2^DeleteCases[Divisors[n],1|n])/2],{n,100}]

from sympy import divisors
def A328337(n): return sum(1<<(d-1) for d in divisors(n,generator=True) if 1Chai Wah Wu, Jul 15 2022

A000120(a(n)) = A070824(n).
A070939(a(n)) = A032742(n).
A001511(a(n)) = A107286(n).

A328459 Sorted positions of first appearances in A328458 (maximum run-length of nontrivial divisors) of each positive integer in the image.

Original entry on oeis.org

1, 2, 6, 12, 60, 420, 504, 840, 2520, 27720, 360360, 720720, 4084080

Offset: 0

Gus Wiseman, Oct 17 2019

			The sequence of terms > 1 together with their nontrivial divisors begins:
    2: {}
    6: {2,3}
   12: {2,3,4,6}
   60: {2,3,4,5,6,10,12,15,20,30}
  420: {2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210}
  504: {2,3,4,6,7,8,9,12,14,18,21,24,28,36,42,56,63,72,84,126,168,252}

Positions of first appearances in A328458.
The version for all divisors is A051451.
Cf. A000005, A033676, A060775, A070824, A119313, A129308, A181063, A199970, A163870, A328448, A328449.

dav=Table[Switch[n,1,1,_,Max@@Length/@Split[DeleteCases[Divisors[n],1|n],#2==#1+1&]],{n,1000}];
Table[Position[dav,i][[1,1]],{i,Union[dav]}]//Sort

a(12) from Robert Israel, Mar 31 2023

cp's OEIS Frontend

A346510 a(n) is the number of nontrivial divisors of A346507(n) ending with 1.

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A382325 Numbers with a record ratio of proper factorizations to nontrivial divisors.

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A382326 Numbers with a record ratio of nontrivial divisors to prime factors (counted with multiplicity).

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A095808 Number of ways to write n in the form m + (m+1) + ... + (m+k-1) + (m+k) + (m+k-1) + ... + (m+1) + m with integers m>= 1, k>=1. Or, number of divisors d of 4n-1 with 0 < (d-1)^2 < 4n.

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A161116 a(n) is the number of nontrivial positive divisors of 2n+3.

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A163838 a(n) = (n-th composite) * (number of nontrivial divisors of n-th composite).

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A259977 Number of proper divisors of A005381(n).

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A291108 Expansion of Sum_{k>=2} k^2*x^(2*k)/(1 - x^k).

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A291108 Expansion of Sum_{k>=2} k^2x^(2k)/(1 - x^k).