A002093 -id:A002093 - cp's OEIS frontend

A093890 Number of primes arising as the sum of one or more divisors of n.

0, 2, 1, 4, 1, 5, 1, 6, 2, 7, 1, 9, 1, 5, 4, 11, 1, 12, 1, 13, 5, 5, 1, 17, 2, 5, 4, 16, 1, 20, 1, 18, 4, 6, 6, 24, 1, 5, 5, 24, 1, 24, 1, 18, 11, 5, 1, 30, 1, 15, 3, 18, 1, 30, 6, 30, 5, 7, 1, 39, 1, 3, 18, 31, 6, 34, 1, 16, 3, 34, 1, 44, 1, 4, 13, 16, 4, 39, 1, 42, 5, 5, 1, 48, 5, 5, 2, 41, 1, 51, 2

Offset: 1

Amarnath Murthy, Apr 23 2004

nonn

a(2^n) = pi(2^(n+1)-1).

Except for n=3 and n=42, it appears that the records occur at the highly abundant numbers A002093. The record values appear to be pi(sigma(n)) for n in A002093, which means that these n are members of A093891. [T. D. Noe, Mar 19 2010]

			a(4) = 4, the divisors of 4 are 1, 2 and 4.
Primes arising are 2, 3 = 1 + 2, 5 = 1 + 4 and 7 = 1 + 2 + 4.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A093891, A093892.

Cf. A161510 (primes counted with repetition). [T. D. Noe, Mar 19 2010]

Do[l = Subsets[Divisors[n]]; l = Union[Map[Plus @@ #&, l]]; Print[Length[Select[l, PrimeQ]]], {n, 100}] (* Ryan Propper, Jun 04 2006 *)
CountPrimes[n_] := Module[{d=Divisors[n],t,lim,x}, t=CoefficientList[Product[1+x^i, {i,d}], x]; lim=PrimePi[Length[t]-1]; Count[t[[1+Prime[Range[lim]]]], ?(#>0 &)]]; Table[CountPrimes[n], {n,100}] (* _T. D. Noe, Mar 19 2010 *)

Corrected and extended by Ryan Propper, Jun 04 2006

A307866 K-champion numbers: numbers m such that K(m) > K(j) for all j < m, where K(m) is the Kalmár function (A074206).

Original entry on oeis.org

0, 1, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 192, 240, 288, 360, 432, 480, 576, 720, 864, 960, 1152, 1440, 1728, 1920, 2160, 2304, 2880, 3456, 4320, 5760, 6912, 8640, 11520, 17280, 23040, 25920, 30240, 34560, 46080, 51840, 60480, 69120, 86400, 103680, 120960

Offset: 1

Amiram Eldar, May 02 2019

nonn

The corresponding record values are 0, 1, 2, 3, 4, 8, 20, 26, 48, 76, 112, 132, 208, ... (see the link for more values).
Deléglise et al. (2008) proved that the number of powerful (A001694) terms in this sequence is finite. They ask whether a(391) = 485432135516160000 (the 112th powerful term) is the largest. - Amiram Eldar, Aug 20 2019
Is abs(omega(a(n)) - omega(a(n+1))) <= 1? (Cf. A001221.) - David A. Corneth, Apr 16 2020

David A. Corneth, Table of n, a(n) for n = 1..884 (terms 1..762 from Amiram Eldar, calculated by Deléglise et al.)
David A. Corneth, Table of n, a(n) for n = 1..1544 if abs(omega(a(n)) - omega(a(n + 1))) <= 1
M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Grandes valeurs et nombres champions de la fonction arithmétique de Kalmár, Journal of Number Theory, Vol. 128, No. 6 (2008), pp. 1676-1716.
M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Tables des 761 premiers champions de la fonction de Kalmar, alternative link.
Amiram Eldar, Table of n, a(n), A074206(a(n)), calculated by Deléglise et al.
T. M. A. Fink, Number of ordered factorizations and recursive divisors, arXiv:2307.16691 [math.NT], 2023.

Cf. A001221, A001694, A002093, A033833, A074206, A163272, A330686 (after primorial deflation).

a[0] = 0; a[1] = 1; a[n_] := a[n] = Total[a /@ Most[Divisors[n]]]; s = {}; am=-1; Do[a1 = a[n]; If[a1>am, am=a1; AppendTo[s, n]], {n, 0, 10000}]; s

For n >= 1, a(1+n) = A108951(A330686(n)). - Antti Karttunen, Dec 31 2019

A327634 Infinitary highly abundant numbers: numbers m such that isigma(m) > isigma(k) for all k < m, where isigma(k) is the sum of infinitary divisors of n (A049417).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 21, 22, 24, 30, 40, 42, 54, 66, 72, 78, 88, 96, 102, 114, 120, 168, 210, 216, 264, 312, 330, 360, 378, 384, 408, 456, 480, 510, 546, 552, 600, 672, 690, 696, 744, 840, 1080, 1320, 1512, 1560, 1848, 1920, 2040, 2184, 2280, 2688

Offset: 1

Amiram Eldar, Sep 20 2019

The infinitary version of A002093.

			The first 10 values of isigma(k) for k = 1 to 10 are: 1, 3, 4, 5, 6, 12, 8, 15, 10, 18. Record values are reached for all these values of k except for 7 and 9, therefore the sequence begins with 1, 2, 3, 4, 5, 6, 8, 10, ...

Amiram Eldar, Table of n, a(n) for n = 1..930

Cf. A002093, A037992, A049417, A285614, A292983.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); seq = {};sm = 0; Do[s = isigma[n]; If[s > sm, sm = s; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A007626 Sum of divisors of superabundant numbers (A004394).

Original entry on oeis.org

1, 3, 7, 12, 28, 60, 91, 124, 168, 360, 546, 744, 1170, 2418, 2880, 4368, 5952, 9360, 19344, 39312, 59520, 99944, 112320, 232128, 471744, 714240, 1199328, 1451520, 2437344, 2926080, 3249792, 6604416, 9999360

Offset: 1

Walter Nissen

Local maxima of sigma(n), the sum of divisors function A000203.

Same as A063072 for the first 19 terms. - T. D. Noe, Jul 01 2008

Amiram Eldar, Table of n, a(n) for n = 1..6863 (terms 1..500 from T. D. Noe)

Cf. A002192, A000203, A051444, A004394.

See A034885 and A002093 for another version.

Reap[ For[ n=1; a=0, n <= 3*10^6, n++, s = DivisorSigma[1, n]; b = s/n; If[ b>a, a=b; Print[s]; Sow[s]]]][[2, 1]] (* Jean-François Alcover, Apr 02 2013 *)
Join[{1},DeleteDuplicates[Select[{#[[1]],#[[2]],#[[2]]/#[[1]]}&/@Table[ {n,DivisorSigma[1,n]}, {n,10^6}],#[[3]]>1&],GreaterEqual[#1[[3]],#2[[3]]]&][[All,2]]] (* The program generates the first 31 terms of the sequence. *) (* Harvey P. Dale, Oct 04 2022 *)

a(n) = A000203(A004394(n)). - Amiram Eldar, Sep 25 2021

A087315 a(n) = Product_{k=1..n} prime(k)^prime(n-k+1).

Original entry on oeis.org

1, 4, 72, 21600, 190512000, 580909190400000, 428616352408083840000000, 859278392084450410309036800000000000, 2097197194438629126172451944256706311040000000000000

Offset: 0

Amarnath Murthy, Sep 03 2003

nonn

			a(3) = 2^5*3^3*5^2 = 21600.

G. C. Greubel, Table of n, a(n) for n = 0..23

Cf. A006939, A002110, A025487, A004394, A002093, A002182, A055932, A025487, A089247, A071364, A097320, A046523, A056166, A114129, A053810.

[1] cat [(&*[NthPrime(k)^(NthPrime(n-k+1)): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

seq(product(ithprime(k)^ithprime(n-k+1), k=1..n), n=0..10);

Table[Product[Prime[k]^Prime[n - k + 1], {k, 1, n}], {n, 0, 10}] (* G. C. Greubel, Oct 14 2018 *)

for(n=0, 10, print1(prod(k=1,n, prime(k)^prime(n-k+1)), ", ")) \\ G. C. Greubel, Oct 14 2018

[prod(nth_prime(i)^nth_prime(k-i+1) for i in (1..k)) for k in (0..10)] # Giuseppe Coppoletta, Nov 03 2014

More terms from Jorge Coveiro, Dec 22 2004

Corrected by David Wasserman, May 02 2005

A328134 Exponential highly abundant numbers: numbers m such that esigma(m) > esigma(k) for all k < m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 12, 16, 18, 20, 28, 36, 52, 60, 68, 72, 84, 92, 100, 124, 132, 140, 144, 180, 244, 252, 300, 324, 360, 396, 468, 588, 612, 684, 828, 900, 1116, 1260, 1332, 1476, 1548, 1692, 1764, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988

Offset: 1

Amiram Eldar, Oct 04 2019

nonn

The exponential version of A002093.

			The first 10 values of esigma(k) for k = 1 to 10 are 1, 2, 3, 6, 5, 6, 7, 10, 12, 10. The record values are reached for 1, 2, 3, 4, 7, 8, 9.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A002093, A051377, A285614 (unitary), A292983 (bi-unitary), A327634 (infinitary).

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; s = {}; em = 0; Do[e = esigma[n]; If[e > em, em = e; AppendTo[s, n]], {n, 1, 3000}]; s

A095849 Numbers j where sigma_k(j) increases to a record for all real values of k.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 840, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 720720, 1441440, 2162160, 3603600, 7207200, 10810800, 36756720, 61261200, 122522400, 183783600, 698377680

Offset: 1

Matthew Vandermast, Jun 09 2004

nonn

For any value of k, sigma_k(j) > sigma_k(m) for all m < j, where the function sigma_k(j) is the sum of the k-th powers of all divisors of j.

Conjecture: a number is in this sequence if and only if it is in both A002182 and A095848. - J. Lowell, Jun 21 2008

T. D. Noe, Table of n, a(n) for n = 1..74 (complete) [I assume this is only conjectured to be complete. - _N. J. A. Sloane_, Jan 02 2019]

Cf. A002093 (highly abundant numbers), A002182 (highly composite numbers) and A004394 (superabundant numbers), consisting of numbers that establish records for sigma_k(j) where k equals 1, 0 and -1 respectively. See also A095848.

Cf. also A166981 (numbers that establish records for both k=0 and k=-1).

Extended by T. D. Noe, Apr 22 2010
Corrected by T. D. Noe and Matthew Vandermast, Oct 04 2010
Removed keyword "fini", since it appears that as yet there is no proof. - N. J. A. Sloane, Sep 17 2022

A119616 Second elementary symmetric function of divisors of n.

Original entry on oeis.org

0, 2, 3, 14, 5, 47, 7, 70, 39, 97, 11, 287, 13, 163, 158, 310, 17, 533, 19, 609, 262, 343, 23, 1375, 155, 457, 390, 1043, 29, 1942, 31, 1302, 542, 733, 502, 3185, 37, 895, 718, 2945, 41, 3358, 43, 2247, 1859, 1267, 47, 5983, 399, 2697, 1142, 3017, 53, 5150, 1006

Offset: 1

N. J. A. Sloane, based on email from Neven Juric (neven.juric(AT)apis-it.hr), Jun 07 2006

a(p)=p if p is prime and records are A002093 (highly abundant numbers). - Robert G. Wilson v, Jun 07 2006

			|-------+------------------------------------------+---------------------|
|...n...|................divisors(n)...............|..s2(divisors.(n))...|
|-------+------------------------------------------+---------------------|
|...1...|....................1.....................|..........0..........|
|...2...|...................1,2....................|..........2..........|
|...3...|...................1,3....................|..........3..........|
|...4...|..................1,2,4...................|.........14..........|
|...5...|...................1,5....................|..........5..........|
|...6...|.................1,2,3,6..................|.........47..........|

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A002093, A000203, A001157, A002117, A067692.

Column k=2 of A224381.

a:= n-> (l-> add(add(l[i]*l[j], i=1..j-1), j=2..nops(l)))
        (sort([numtheory[divisors](n)[]])):
seq(a(n), n=1..80);  # Alois P. Heinz, Jun 25 2014

f[n_] := Block[{d = Divisors@n}, Sum[ d[[u]]*d[[v]], {v, 2, Length@d}, {u, v - 1}]]; Array[f, 55] (* Robert G. Wilson v *)

a(n)=my(d=divisors(n));sum(i=1,#d-1,sum(j=i+1,#d,d[i]*d[j])) \\ Charles R Greathouse IV, Mar 05 2013

a(n)=(sigma(n)^2-sigma(n,2))/2 \\ Charles R Greathouse IV, Mar 05 2013

a(n) = Sum_{u|n, v|n, u

a(n) = (sigma_1(n)^2-sigma_2(n))/2, cf. A000203, A001157. - Vladeta Jovovic, Jun 07 2006

Sum_{k=1..n} a(k) = zeta(3) * n^3 / 4 + O(n^2*log(n)^2). - Amiram Eldar, Dec 15 2023

A193988 Numbers n such that sigma_2(n) > sigma_2(k) for all k < n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 28, 30, 32, 34, 36, 40, 42, 44, 46, 48, 52, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 90, 96, 100, 102, 108, 114, 120, 126, 132, 138, 140, 144, 150, 156, 162, 168, 180, 192, 198, 204, 210, 216, 228

Offset: 1

T. D. Noe, Aug 17 2011

nonn

Where record values of sigma_2(n) occur. Records transform of A001157.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A001157, A002093 (highly abundant numbers), A193989.

mx = 0; t = {}; Do[u = DivisorSigma[2, n]; If[u > mx, mx = u; AppendTo[t, n]], {n, 1000}]; t
DeleteDuplicates[Table[{n,DivisorSigma[2,n]},{n,1000}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, May 14 2022 *)

A280013 Numbers k such that sum of squarefree divisors of k > sum of squarefree divisors of m for all m < k.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 14, 21, 22, 26, 30, 42, 66, 78, 102, 114, 130, 138, 170, 174, 186, 210, 318, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1110, 1218, 1230, 1290, 1410, 1554, 1590, 1722, 1770, 1830, 1974, 2010, 2130, 2190, 2310, 2730, 3390, 3570, 3990, 4290, 4830, 5610

Offset: 1

Ilya Gutkovskiy, Apr 14 2017

nonn

Numbers k such that psi(rad(k)) > psi(rad(m)) for all m < k, where psi() is the Dedekind psi function (A001615) and rad() is the squarefree kernel (A007947).
Numbers k such that Sum_{d|k} mu(d)^2*d > Sum_{d|m} mu(d)^2*d for all m < k, where mu() is the Moebius function (A008683).
All terms are squarefree. - Robert Israel, Apr 19 2017

Robert Israel, Table of n, a(n) for n = 1..500
Index entries for sequences related to sums of divisors

Cf. A001615, A002093, A002182, A007947, A008683, A034090, A048250, A174572.

ssd:= n -> convert(select(numtheory:-issqrfree,numtheory:-divisors(n)),`+`):
M:= 0: A:= NULL:
for n from 1 to 10^5 do
    r:= ssd(n);
    if r > M then M:= r; A:= A, n fi
od:
A; # Robert Israel, Apr 19 2017

mx = 0; t = {}; Do[u = DivisorSum[n, # &, SquareFreeQ[#] &]; If[u > mx, mx = u; AppendTo[t, n]], {n, 6000}]; t

from sympy.ntheory.factor_ import core
from sympy import divisors
def s(n): return sum(list(filter(lambda i: core(i) == i, divisors(n))))
def ok(n):
    m=1
    while ms(m): return False
        m+=1
    return True # Indranil Ghosh, Apr 16 2017

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cp's OEIS Frontend

A093890 Number of primes arising as the sum of one or more divisors of n.

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A307866 K-champion numbers: numbers m such that K(m) > K(j) for all j < m, where K(m) is the Kalmár function (A074206).

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A327634 Infinitary highly abundant numbers: numbers m such that isigma(m) > isigma(k) for all k < m, where isigma(k) is the sum of infinitary divisors of n (A049417).

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A007626 Sum of divisors of superabundant numbers (A004394).

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A087315 a(n) = Product_{k=1..n} prime(k)^prime(n-k+1).

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A328134 Exponential highly abundant numbers: numbers m such that esigma(m) > esigma(k) for all k < m, where esigma(m) is the sum of exponential divisors of m (A051377).

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A095849 Numbers j where sigma_k(j) increases to a record for all real values of k.

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A119616 Second elementary symmetric function of divisors of n.

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