cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A341529 a(n) = sigma(n) * A003961(n), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of the divisors of n.

Original entry on oeis.org

1, 9, 20, 63, 42, 180, 88, 405, 325, 378, 156, 1260, 238, 792, 840, 2511, 342, 2925, 460, 2646, 1760, 1404, 696, 8100, 1519, 2142, 5000, 5544, 930, 7560, 1184, 15309, 3120, 3078, 3696, 20475, 1558, 4140, 4760, 17010, 1806, 15840, 2068, 9828, 13650, 6264, 2544, 50220, 6897, 13671, 6840, 14994, 3186, 45000, 6552, 35640
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2021

Keywords

Comments

Question: Does the maximum value of ratio A341529(n)/A341528(n) stay below 2?
From Amiram Eldar and Antti Karttunen, Jan 28 2023: (Start)
Answer to the above question is yes: Sup_{n>=1} A341529(n)/A341528(n) = 2.
Proof:
f(n) = A341529(n)/A341528(n) is a multiplicative function with f(p^e) = (1 + 1/p + ... + 1/p^e)/(1 + 1/q + ... + 1/q^e), where q = nextprime(p).
First we prove a lemma which states that f(p^(1+e)) / f(p^e) > 1, for any prime p, and exponent e.
We note that (sigma(p^(1+e))/(p^(1+e))) / (sigma(p^e)/(p^e)) = (sigma(p^(1+e))/(p*sigma(p^e))) = sigma(p^(1+e)) / (sigma(p^(1+e)) - 1), so setting q = nextprime(p), we can write the ratio f(p^(1+e)) / f(p^e) as (sigma(p^(1+e))/(sigma(p^(1+e))-1)) / (sigma(q^(1+e))/(sigma(q^(1+e))-1)), and to prove this to be > 1, we just note that the denominator is less than the numerator, because sigma(p^e) is monotonically growing with respect to the increasing prime p.
Since q > p, we have f(p^e) > 1 for all p and all e>=1, and together with the above lemma this shows that f(n) <= f(n*m) for all m>=1.
Suppose n = Product_i p_i^e_i, and let pmax = max(p_i), emax = max(e_i), so n is a divisor of m = (pmax#)^emax, and f(n) < f(m), where p# = 2 * 3 * ... * p is the primorial of p, A034386(p).
Then f(m) = f(2^emax) * f(3^emax) * ... * f(pmax^emax) = (1 + 1/2 + ... + 1/2^emax)/(1 + 1/3 + ... + 1/3^emax)) * (1 + 1/3 + ... + 1/3^emax)/(1 + 1/5 + ... + 1/5^emax)) * ... * (1 + 1/p + ... + 1/p^emax)/(1 + 1/q + ... + 1/q^emax))[telescoping product] = (1 + 1/2 + ... + 1/2^emax)/(1 + 1/qmax + ... + 1/qmax^emax) <= (1 + 1/2 + ... + 1/2^emax) < 2, where qmax = nextprime(pmax).
So we have f(n) < 2 for all n.
To prove that 2 is the supremum, we have lim_{e,k -> oo) f(prime(k)#^e) = 2.
(End)

Links

Crossrefs

Cf. A000203, A003961, A003973, A028982 (positions of odd terms), A072861, A151800, A286385, A341512, A341526, A341527, A341528, A341530, A342661, A342667, A342671, A342672.
Cf. also arrays A341605/A341606.

Programs

  • Mathematica
    Array[DivisorSigma[1, #]*Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &, 56] (* Michael De Vlieger, Feb 22 2021 *)
  • PARI
    A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341529(n) = (sigma(n)*A003961(n));

Formula

Multiplicative with a(p^e) = q^e * (p^(e+1)-1)/(p-1), where q = nextPrime(p).
a(n) = A000203(n) * A003961(n).
For all n > 1, a(n) > A341528(n).
For all n >= 1, A072861(n) <= a(n) <= A003961(n)^2. [See A286385].
a(n) = A341528(n) + A341512(n) = A342671(n) * A342672(n) = A342661(A003961(n)). - Antti Karttunen, Mar 22 2021
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} p^4*(p-1)/((p^3-nextprime(p))*(p^2-nextprime(p))) = 3.0664809..., where nextprime is A151800. - Amiram Eldar, Dec 08 2022

A080257 Numbers having at least two distinct or a total of at least three prime factors.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 10 2003

Keywords

Comments

Complement of A000430; A080256(a(n)) > 3.
A084114(a(n)) > 0, see also A084110.
Also numbers greater than the square of their smallest prime-factor: a(n)>A020639(a(n))^2=A088377(a(n));
a(n)>A000430(k) for n<=13, a(n) < A000430(k) for n>13.
Numbers with at least 4 divisors. - Franklin T. Adams-Watters, Jul 28 2006
Union of A024619 and A033942; A211110(a(n)) > 2. - Reinhard Zumkeller, Apr 02 2012
Also numbers > 1 that are neither prime nor a square of a prime. Also numbers whose omega-sequence (A323023) has sum > 3. Numbers with omega-sequence summing to m are: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7). - Gus Wiseman, Jul 03 2019
Numbers n such that sigma_2(n)*tau(n) = A001157(n)*A000005(n) >= 4*n^2. Note that sigma_2(n)*tau(n) >= sigma(n)^2 = A072861 for all n. - Joshua Zelinsky, Jan 23 2025

Examples

			8=2*2*2 and 10=2*5 are terms; 4=2*2 is not a term.
From _Gus Wiseman_, Jul 03 2019: (Start)
The sequence of terms together with their prime indices begins:
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
(End)

Links

Crossrefs

Cf. A001248, A001221, A001222, A006881, A030078, A088381, A088383, A000005.
Cf. A060687, A118914, A323014, A323023, A325249.

Programs

  • Haskell
    a080257 n = a080257_list !! (n-1)
a080257_list = m a024619_list a033942_list where
   m xs'@(x:xs) ys'@(y:ys) | x < y  = x : m xs ys'
                           | x == y = x : m xs ys
                           | x > y  = y : m xs' ys
-- Reinhard Zumkeller, Apr 02 2012
  • Mathematica
    Select[Range[100],PrimeNu[#]>1||PrimeOmega[#]>2&] (* Harvey P. Dale, Jul 23 2013 *)
  • PARI
    is(n)=omega(n)>1 || isprimepower(n)>2
  • PARI
    is(n)=my(k=isprimepower(n)); if(k, k>2, !isprime(n)) \\ Charles R Greathouse IV, Jan 23 2025

Formula

a(n) = n + O(n/log n). - Charles R Greathouse IV, Sep 14 2015

Extensions

Definition clarified by Harvey P. Dale, Jul 23 2013

A156302 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^2*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 5, 10, 30, 57, 152, 289, 676, 1304, 2809, 5335, 10961, 20487, 40329, 74476, 141914, 258094, 479638, 860025, 1563716, 2767982, 4940567, 8636563, 15173805, 26217392, 45416811, 77629455, 132800937, 224695510, 380079521, 637006921
Offset: 0

Views

Author

Paul D. Hanna, Feb 08 2009

Keywords

Comments

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 57*x^5 + 152*x^6 +...
log(A(x)) = x + 3^2*x^2/2 + 4^2*x^3/3 + 7^2*x^4/4 + 6^2*x^5/5 + 12^2*x^6/6 +...
Also log(A(x)) = (x + 3*x^2 + 4*x^3 + 7*x^4 +...+ sigma(k)*x^k +...)/1 +
(3*x^2 + 7*x^4 + 12*x^6 + 15*x^8 + 18*x^10 +...+ sigma(2*k)*x^(2*k) +...)/2 +
(4*x^3 + 12*x^6 + 13*x^9 + 28*x^12 + 24*x^15 +...+ sigma(3*k)*x^(3*k) +...)/3 +
(7*x^4 + 15*x^8 + 28*x^12 + 31*x^16 + 42*x^20 +...+ sigma(4*k)*x^(4*k) +...)/4 +
(6*x^5 + 18*x^10 + 24*x^15 + 42*x^20 + 31*x^25 +...+ sigma(5*k)*x^(5*k) +...)/5 +...

Links

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A072861, A178933, A205797, A382125.

Programs

  • Mathematica
    nmax = 40; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1, k]^2*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2024 *)
  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n,sigma(k)^2*x^k/k)+x*O(x^n)),n)}
  • PARI
    {a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k)^2*a(n-k)))}
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,sum(k=1,n\m,sigma(m*k)*x^(m*k)/m)+x*O(x^n))),n)}

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(k)^2*a(n-k) for n>0, with a(0) = 1.
Euler transform of A060648. [From Vladeta Jovovic, Feb 14 2009]
It appears that G.f.: A(x)=prod(n=1,infinity, E(x^n)^(-A001615(n))) where E(x) = prod(n=1,infinity,1-x^n). [From Joerg Arndt, Dec 30 2010]
G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k) * x^(n*k) / n ). [From Paul D. Hanna, Jan 23 2012]
log(a(n)) ~ 3*(5*zeta(3))^(1/3) * n^(2/3) / 2. - Vaclav Kotesovec, Oct 29 2024

A072379 Sum_{k<=n} (sigma(k)^2), where sigma(k) denotes the sum of the divisors of k A000203.

Original entry on oeis.org

1, 10, 26, 75, 111, 255, 319, 544, 713, 1037, 1181, 1965, 2161, 2737, 3313, 4274, 4598, 6119, 6519, 8283, 9307, 10603, 11179, 14779, 15740, 17504, 19104, 22240, 23140, 28324, 29348, 33317, 35621, 38537, 40841, 49122, 50566, 54166, 57302
Offset: 1

Views

Author

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 20 2002

Keywords

Links

Crossrefs

Cf. A024916, A072861.
Cf. A057434, A061502, A074789.

Programs

  • Maple
    A072379 := proc(n)
    add( numtheory[sigma](k)^2,k=0..n) ;
end proc:
seq(A072379(n),n=1..80) ; # R. J. Mathar, Jul 09 2024
  • Mathematica
    Accumulate[Table[DivisorSigma[1, k]^2, {k, 1, 50}]] (* Vaclav Kotesovec, Sep 10 2018 *)
  • PARI
    a(n) = sum(k=1, n, sigma(k)^2) \\ Michel Marcus, Jun 20 2013

Formula

Ramanujan's asymptotic formula: (5/6)*Zeta(3)*n^3+O(n^2*log(n)^2)

A065018 a(n) = Sum_{d|n} sigma(d)^2.

Original entry on oeis.org

1, 10, 17, 59, 37, 170, 65, 284, 186, 370, 145, 1003, 197, 650, 629, 1245, 325, 1860, 401, 2183, 1105, 1450, 577, 4828, 998, 1970, 1786, 3835, 901, 6290, 1025, 5214, 2465, 3250, 2405, 10974, 1445, 4010, 3349, 10508, 1765, 11050, 1937, 8555, 6882
Offset: 1

Views

Author

Vladeta Jovovic, Nov 19 2001

Keywords

Links

Crossrefs

Cf. A029939, A062367.

Programs

  • PARI
    { for (n=1, 1000, a=sumdiv(n, d, sigma(d)^2); write("b065018.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 03 2009
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^2*x^k/(1-x^k))) \\ Seiichi Manyama, May 08 2021

Formula

Dirichlet convolution of A072861 and A000012. Dirichlet g.f.: zeta^2(s)*zeta^2(s-1)*zeta(s-2)/zeta(2s-2). - R. J. Mathar, Feb 03 2011
Sum_{k=1..n} a(k) ~ 5 * Zeta(3)^2 * n^3 / 6. - Vaclav Kotesovec, Feb 01 2019
From Seiichi Manyama, May 08 2021: (Start)
G.f.: Sum_{k >= 1} sigma(k)^2 * x^k/(1 - x^k).
If p is prime, a(p) = 1 + (p+1)^2. (End)

A356533 a(n) = sigma_2(n)^2.

Original entry on oeis.org

1, 25, 100, 441, 676, 2500, 2500, 7225, 8281, 16900, 14884, 44100, 28900, 62500, 67600, 116281, 84100, 207025, 131044, 298116, 250000, 372100, 280900, 722500, 423801, 722500, 672400, 1102500, 708964, 1690000, 925444, 1863225, 1488400, 2102500, 1690000, 3651921
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 11 2022

Keywords

Links

Crossrefs

Cf. A001157, A127473, A035116, A072861, A356535 (partial sums).

Programs

  • Mathematica
    Table[DivisorSigma[2, n]^2, {n, 1, 40}]
  • PARI
    a(n) = sigma(n, 2)^2; \\ Michel Marcus, Aug 11 2022

Formula

Dirichlet g.f.: zeta(s) * zeta(s-2)^2 * zeta(s-4) / zeta(2*s-4).
Multiplicative with a(p^e) = ((p^(2*e+2)-1)/(p^2-1))^2. - Amiram Eldar, Aug 11 2022
a(n) = A001157(n)^2. - R. J. Mathar, Aug 18 2022

A356534 a(n) = sigma_3(n)^2.

Original entry on oeis.org

1, 81, 784, 5329, 15876, 63504, 118336, 342225, 573049, 1285956, 1774224, 4177936, 4831204, 9585216, 12446784, 21911761, 24147396, 46416969, 47059600, 84603204, 92775424, 143712144, 148060224, 268304400, 248094001, 391327524, 417793600, 630612544, 594872100
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 11 2022

Keywords

Links

Crossrefs

Cf. A001158, A127473, A035116, A072861, A356536 (partial sums).

Programs

  • Mathematica
    Table[DivisorSigma[3, n]^2, {n, 1, 40}]
  • PARI
    a(n) = sigma(n, 3)^2; \\ Michel Marcus, Aug 11 2022

Formula

Dirichlet g.f.: zeta(s) * zeta(s-3)^2 * zeta(s-6) / zeta(2*s-6).
Multiplicative with a(p^e) = ((p^(3*e+3)-1)/(p^3-1))^2. - Amiram Eldar, Aug 11 2022

A277521 Numbers k such that number of divisors of k and sum of divisors of k divides product of divisors of k and the average of the divisors of k is an integer.

Original entry on oeis.org

1, 6, 30, 66, 102, 210, 270, 318, 330, 420, 462, 510, 546, 570, 642, 672, 690, 714, 840, 870, 924, 930, 966, 1122, 1320, 1410, 1428, 1518, 1590, 1638, 1722, 1770, 1890, 1932, 2130, 2226, 2280, 2310, 2346, 2370, 2670, 2730, 2760, 2838, 2970, 2982, 3102, 3162, 3210, 3360, 3444, 3486, 3498, 3570, 3720, 3780, 3948, 3990
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 19 2016

Keywords

Comments

Intersection of A003601, A120736 and A145551.
Numbers k such that A000005(k)|A007955(k), A000203(k)|A007955(k) and A000005(k)| A000203(k).
Numbers k such that A000005(k)|A062981(k), A072861(k)|A062758(k) and A245656(k) = 1.

Examples

			a(2) = 6 because 6 has 4 divisors {1,2,3,6}, 1*2*3*6/4 = 9, 1*2*3*6/(1 + 2 + 3 + 6) = 3 and (1 + 2 + 3 + 6)/4 = 3 are integer.

Links

Crossrefs

Cf. A000005, A000203, A003601, A007955, A062758, A062981, A072861, A120736, A145551, A245656.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,k,n;for n from 1 to q do
a:=divisors(n); b:=mul(a[k],k=1..nops(a));
if type(sigma(n)/tau(n),integer) and type(b/sigma(n),integer) and
type(b/tau(n),integer) then print(n); fi;
od; end: P(10^5); # Paolo P. Lava, Oct 20 2016
  • Mathematica
    Select[Range[4000], Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[1, #1]] && Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[0, #1]] && Divisible[DivisorSigma[1, #1], DivisorSigma[0, #1]] & ]

A361179 a(n) = sigma(n)^4.

Original entry on oeis.org

1, 81, 256, 2401, 1296, 20736, 4096, 50625, 28561, 104976, 20736, 614656, 38416, 331776, 331776, 923521, 104976, 2313441, 160000, 3111696, 1048576, 1679616, 331776, 12960000, 923521, 3111696, 2560000, 9834496, 810000, 26873856, 1048576, 15752961, 5308416
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 03 2023

Keywords

Comments

In general, for k>=1, Sum_{m=1..n} sigma(m)^k ~ c(k) * z(k) * n^(k+1) / (k+1), where z(k) = Product_{j=2..k+1} zeta(j).
z(k) tends to A021002 = 2.29485659167331379418351583... if k tends to infinity.
Table of logarithms of the first twenty constants c(k):
log(c1) = 0
log(c2) = 0.4185904294034097177091498674425959208785022862606440306200960821...
log(c3) = 1.0423888168104400391462790418324165821902123159643681963298587386...
log(c4) = 1.7991790110714031081639242851527957388041981665455193670488985855...
log(c5) = 2.6531418047626712704435945717713008165192112256395129469527055461...
log(c6) = 3.5826667694785981489341382260447390026333883927530294731356708082...
log(c7) = 4.5733843557245275039380976990636718508529417039225677910093512418...
log(c8) = 5.6152065176325962438798772352645945078887296036246579568363264836...
log(c9) = 6.7007695219862872061684609152917692899880931107656334442026270254...
log(c10) = 7.8245175718301572361518558972457980392624870372412384620464547480...
log(c11) = 8.9821318589248960303876549202030018215854310738197659104984082438...
log(c12) = 10.170161510396427442300796140752106239603402200741405656518889304...
log(c13) = 11.385778844373902103940190311048453116470874526205115584130363228...
log(c14) = 12.626614423444098003503814842580453502016287945932183786430620101...
log(c15) = 13.890644760144907314506933347339629337810929043024214330654043796...
log(c16) = 15.176115136560648867246990011975416479066956527530401883224856531...
log(c17) = 16.481485806132270823150284520463000397265757050340939883069076823...
log(c18) = 17.805393674783928883671133007206209125657866860089528876021281793...
log(c19) = 19.146624201995507049618714377273936711664382470319966849198205155...
log(c20) = 20.504090088752226662590920186246482636058069128320785639131816842...
c1 = 1, c2 = 5/(2*zeta(2)) = 15/Pi^2.

Crossrefs

Cf. A000203, A072861, A361147.
Cf. A361132, A361148.

Programs

  • Mathematica
    Table[DivisorSigma[1, n]^4, {n, 1, 50}]
  • PARI
    a(n) = sigma(n)^4;
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 + p^2*X)*(1 + 3*p*X + 4*p^2*X + 3*p^3*X + p^4*X^2)/((1 - X)*(1 - p*X)*(1 - p^2*X)*(1 - p^3*X)*(1 - p^4*X)))[n], ", "))

Formula

Multiplicative with a(p^e) = ((p^(e+1)-1)/(p-1))^4.
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) * zeta(s-4) * Product_{primes p} (1 + 1/p^(3*s-6) + 3/p^(2*s-3) + 5/p^(2*s-4) + 3/p^(2*s-5) + 3/p^(s-1) + 5/p^(s-2) + 3/p^(s-3)).
Sum_{k=1..n} a(k) ~ c * Pi^6 * zeta(3) * zeta(5) * n^5 / 2700, where c = Product_{primes p} (1 + 3/p^2 + 5/p^3 + 3/p^4 + 3/p^5 + 5/p^6 + 3/p^7 + 1/p^9) = 6.0446828090651437986928739783339791032197283386377841627594461874871547391...
a(n) = A000583(A000203(n)).

A077101 a(n) = A051612(n)*A065387(n) = sigma(n)^2-phi(n)^2, where A051612(n) = sigma(n) - phi(n) and A065387(n) = sigma(n) + phi(n).

Original entry on oeis.org

0, 8, 12, 45, 20, 140, 28, 209, 133, 308, 44, 768, 52, 540, 512, 897, 68, 1485, 76, 1700, 880, 1196, 92, 3536, 561, 1620, 1276, 2992, 116, 5120, 124, 3713, 1904, 2660, 1728, 8137, 148, 3276, 2560, 7844, 164, 9072, 172, 6656, 5508, 4700, 188, 15120, 1485
Offset: 1

Views

Author

Labos Elemer, Nov 06 2002

Keywords

Comments

If n is prime, then a(n) = 4n.

Links

Crossrefs

Cf. A000010, A000203, A002117, A051612, A062354, A065387, A065464, A072861, A077099, A077100, A127473.

Programs

Formula

a(n) = A077099(n) * A077100(n). - Antti Karttunen, May 26 2017
From Amiram Eldar, Dec 04 2023: (Start)
a(n) = A072861(n) - A127473(n).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 5*zeta(3)/2 - Product_{p prime}(1 - (2*p-1)/p^3) = (5/2)*A002117 - A065464 = 2.576892... . (End)

Extensions

Edited by Dean Hickerson, Nov 07 2002
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