A180090 -id:A180090 - cp's OEIS frontend

A180114 a(n) = sigma(A001694(n)), sum of divisors of the powerful number A001694(n).

1, 7, 15, 13, 31, 31, 40, 63, 91, 57, 127, 195, 121, 217, 280, 133, 156, 255, 403, 183, 399, 465, 600, 403, 364, 511, 819, 307, 847, 400, 381, 855, 961, 1240, 741, 931, 1092, 1023, 553, 1651, 781, 1815, 1240, 1281, 1093, 1767, 1953, 871, 2520, 2821, 993, 1995

Offset: 1

Walter Kehowski, Aug 10 2010

			Sigma(2^2) = 7, sigma(2^3) = 15, sigma(3^2) = 13.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (7).

Cf. A001694, A000203, A090699, A180090, A180097, A180098, A362984.

emin := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); min(L) else 0 fi end: L:=[]: for w to 1 do for n from 1 to 144 do sn:=sigma(n); if emin(n)>1 then L:=[op(L),sn]; print(n,ifactor(n),sn,ifactor(sn)) fi; od; od;

pwfQ[n_] := n == 1 || Min[Last /@ FactorInteger[n]] > 1; DivisorSigma[1, Select[ Range@ 1000, pwfQ]] (* Giovanni Resta, Feb 06 2018 *)

lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 1, print1(sigma(k), ", "))); \\ Amiram Eldar, May 12 2023

from itertools import count, islice
from math import prod
from sympy import factorint
def A180114_gen(): # generator of terms
    for n in count(1):
        f = factorint(n)
        if all(e>1 for e in f.values()):
            yield prod((p**(e+1)-1)//(p-1) for p,e in f.items())
A180114_list = list(islice(A180114_gen(),20)) # Chai Wah Wu, May 21 2023

from math import isqrt
from sympy import mobius, integer_nthroot, divisor_sigma
def A180114(n):
    def squarefreepi(n):
        return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x, 3)[0])-l
        return c
    return divisor_sigma(bisection(f, n, n)) # Chai Wah Wu, Sep 10 2024

From Amiram Eldar, May 12 2023: (Start)
Sum_{A001694(k) < x} a(k) = c * x^(3/2) + O(x^(23/18 + eps)), where c = A362984 * A090699 / 3 = 1.5572721108... (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]
Sum_{k=1..n} a(k) ~ c * n^3, where c = A362984 / (3 * A090699^2) = 0.151716514097... . (End)

a(1)=1 prepended by and more terms from Giovanni Resta, Feb 06 2018

A337045 Indecomposable sigma-powerful numbers: powerful numbers k such that sigma(k) is also powerful, but restricted to terms that are not the product of 2 terms > 1 of A337044.

Original entry on oeis.org

81, 343, 400, 9261, 189728, 224939, 972000, 1705636, 2205472, 3087000, 3591200, 3648100, 7968032, 13645088, 15350724, 21161304, 24240600, 25992000, 26680500, 29184800, 32832900, 48586824, 51595489, 80802000, 103617387, 109215352, 110215125, 119604096, 122805792

Offset: 1

Hugo Pfoertner, Aug 15 2020

nonn

This is an implementation of the suggestion that Walter A. Kehowski made on his website (see link) with regard to so-called indecomposable sigma-powerful numbers. However, the results deviate from the table linked there. The table is considered to be deficient.

			From _David A. Corneth_, Aug 29 2020: (Start)
No two proper divisors of 400 are sigma-powerful and have the product of those divisors 400 so 400 is in the sequence.
27783 = 81 * 343 is sigma-powerful but 81 and 343 are sigma-powerful as well so 27783 can be decomposed into two sigma-powerful factors. So 27783 is not in the sequence. (End)

David A. Corneth, Table of n, a(n) for n = 1..6421 (terms < 10^18)
Walter A. Kehowski, Sigma-powerful numbers, Aug 09 2010.

Cf. A000203, A001694, A128607, A180090, A337044.

v=vector(50); n=0;
for(m=2, 150000000, my(is); if(ispowerful(m) && ispowerful(sigma(m)), v[n++]=m; is=1; for(j=1, n-1, if(v[n]%v[j], , if(vecsearch(v[1..n-1], v[n]/v[j]), is=0; break))); if(is, print1(v[n], ", "))))

A180097 Numbers n such that sigma(n) is powerful.

Original entry on oeis.org

1, 3, 7, 21, 22, 30, 31, 46, 51, 55, 66, 70, 71, 81, 85, 93, 94, 102, 107, 110, 115, 119, 127, 138, 142, 154, 156, 159, 165, 170, 187, 199, 210, 213, 214, 217, 230, 235, 238, 253, 255, 265, 282, 291, 310, 318, 321, 322, 330, 343, 345, 355, 357, 364, 371, 374

Offset: 1

Walter Kehowski, Aug 10 2010

			sigma(3)=2^2, sigma(7)=2^3, sigma(21)=2^5, sigma(66)=2^4*3^2.

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

Cf. A001694, A000203, A180090, A180098.

emin := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); min(L) else 0 fi end: L:=[]: for w to 1 do for n from 1 to 144 do sn:=numtheory[sigma](n); if emin(sn)>1 then L:=[op(L),n]; print(n,ifactor(n),sn,ifactor(sn)) fi; od; od; L;

sigmaPowerQ[1] = True; sigmaPowerQ[n_] := Min@FactorInteger[DivisorSigma[1, n]][[;; , 2]] > 1; Select[Range[1000], sigmaPowerQ] (* Amiram Eldar, Sep 08 2019 *)

isok(n) = ispowerful(sigma(n)); \\ Michel Marcus, Sep 08 2019

a(1) and more terms from Amiram Eldar, Sep 08 2019

A180098 Sigma(A180097(n)), sum of divisors of A180097(n), numbers n such that sigma(n) is powerful.

Original entry on oeis.org

1, 4, 8, 32, 36, 72, 32, 72, 72, 72, 144, 144, 72, 121, 108, 128, 144, 216, 108, 216, 144, 144, 128, 288, 216, 288, 392, 216, 288, 324, 216, 200, 576, 288, 324, 256, 432, 288, 432, 288, 432, 324, 576, 392, 576, 648, 432, 576, 864, 400, 576, 432, 576, 784, 432

Offset: 1

Walter Kehowski, Aug 10 2010

			Sigma(3)=2^2, sigma(7)=2^3, sigma(21)=2^5, sigma(66)=2^4*3^2.

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

Cf. A001694, A000203, A180090, A180097.

emin := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); min(L) else 0 fi end: L:=[]: for w to 1 do for n from 1 to 144 do sn:=numtheory[sigma](n); if emin(sn)>1 then L:=[op(L),n]; print(n,ifactor(n),sn,ifactor(sn)) fi; od; od; L; map(numtheory[sigma],L);

sigmaPowerQ[1] = True; sigmaPowerQ[n_] := Min@FactorInteger[DivisorSigma[1, n]][[;; , 2]] > 1; DivisorSigma[1, #] & /@ Select[Range[400], sigmaPowerQ] (* Amiram Eldar, Sep 08 2019 *)

lista(nn) = {for (n=1, nn, if (ispowerful(s=sigma(n)), print1(s, ", ")););} \\ Michel Marcus, Sep 08 2019

a(1) and more terms from Amiram Eldar, Sep 08 2019

A337044 Numbers k such that both k and sigma(k)=A000203(k) are powerful, i.e., are terms of A001694.

Original entry on oeis.org

1, 81, 343, 400, 9261, 27783, 32400, 137200, 189728, 224939, 972000, 1705636, 2205472, 3087000, 3591200, 3648100, 3704400, 7968032, 11113200, 13645088, 15350724, 15367968, 18220059, 21161304, 24240600, 25992000, 26680500, 29184800, 32832900, 48586824, 51595489

Offset: 1

Andrew Howroyd and Hugo Pfoertner, Aug 12 2020

nonn

From David A. Corneth, Aug 14 2020: (Start)
If coprime numbers k and m are in the sequence then k*m is in the sequence.
Up to 10^15, the largest prime divisor of a term is 178987 for which the product of the primes with multiplicity 1 of sigma(178987^2) is 16653 = 3 * 7 * 13 * 61. The second largest prime divisor is 25073 (for which sigma(25073^2) has a product of primes with multiplicity 1 of 341 = 11 * 31), which is quite a bit smaller than 178987. Can we somehow constrain the list of possible prime divisors to ease computation? (End)

David A. Corneth, Table of n, a(n) for n = 1..10324 (terms <= 10^18)
David A. Corneth, PARI program

Cf. A000203, A001694, A180090, A337045.

for(k=1, 60000000, if(ispowerful(k) && ispowerful(sigma(k)), print1(k, ", ")))

\\ See Corneth link \\ David A. Corneth, Aug 14 2020

A349109 Powerful numbers (A001694) whose sum of powerful divisors (including 1) is also powerful.

Original entry on oeis.org

1, 64, 243, 441, 1764, 9800, 15552, 28224, 41616, 60516, 82369, 88200, 189728, 226576, 329476, 336200, 648675, 741321, 968256, 1317904, 1428025, 1707552, 1943236, 2039184, 2056356, 2381400, 2446227, 2798929, 2965284, 2986568, 4372281, 5189400, 5271616, 6508832

Offset: 1

Amiram Eldar, Nov 08 2021

nonn

Numbers k such that A112526(k) = A112526(A183097(k)) = 1.

			64 = 2^6 is a term since it is powerful and the sum of its powerful divisors, A183097(64) =  1 + 4 + 8 + 16 + 32 + 64 = 125 = 5^3 is also powerful.

Amiram Eldar, Table of n, a(n) for n = 1..12154 (terms below 10^19)

Cf. A001694, A112526, A180090, A183097, A337044, A337045, A349110.

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;;,2]], # > 1 &]; f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := powQ[n] && powQ[s[n]]; Select[Range[7*10^6], q]

isok(n) = ispowerful(n) && ispowerful(sumdiv(n, d, d*ispowerful(d))); \\ Michel Marcus, Nov 08 2021

is(k) = {my(f = factor(k)); ispowerful(f) && ispowerful(prod(i = 1, #f~, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1) - f[i,1]));} \\ Amiram Eldar, Sep 14 2024

cp's OEIS Frontend

A180114 a(n) = sigma(A001694(n)), sum of divisors of the powerful number A001694(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Python

Formula

Extensions

A337045 Indecomposable sigma-powerful numbers: powerful numbers k such that sigma(k) is also powerful, but restricted to terms that are not the product of 2 terms > 1 of A337044.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A180097 Numbers n such that sigma(n) is powerful.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A180098 Sigma(A180097(n)), sum of divisors of A180097(n), numbers n such that sigma(n) is powerful.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A337044 Numbers k such that both k and sigma(k)=A000203(k) are powerful, i.e., are terms of A001694.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

A349109 Powerful numbers (A001694) whose sum of powerful divisors (including 1) is also powerful.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI