A082695 -id:A082695 - cp's OEIS frontend

A335851 Numbers that are powerful in Gaussian integers.

1, 2, 4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 81, 98, 100, 108, 121, 125, 128, 144, 162, 169, 196, 200, 216, 225, 242, 243, 250, 256, 288, 289, 324, 338, 343, 361, 392, 400, 432, 441, 450, 484, 486, 500, 512, 529, 576, 578, 625, 648, 675, 676, 686

Offset: 1

Amiram Eldar, Jun 26 2020

nonn

Numbers all of whose prime factors in Gaussian integers have multiplicity larger than 1.

The even powerful numbers divided by 4. - Amiram Eldar, May 28 2023

			2 is a term since 2 = -i * (1 + i)^2 in the ring of Gaussian integers. -i is a unit, and the multiplicity of its only Gaussian prime factor, 1 + i, is 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gaussian Integer.
Wikipedia, Gaussian integer.

Disjoint union of A001694 and 2 * A062739.

Cf. A082695.

gaussPowerQ[n_] := AllTrue[FactorInteger[n, GaussianIntegers -> True], Abs[First[#]] == 1 || Last[#] > 1 &]; Select[Range[1000], gaussPowerQ]

Sum_{n>=1} 1/a(n) = (4/3) * Sum_{n>=1} 1/A001694(n) = 4*zeta(2)*zeta(3)/(3*zeta(6)) = (4/3) * A082695 = 2.591461...

A049200 Euler totient function phi applied to the n-th squarefree number.

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 4, 10, 12, 6, 8, 16, 18, 12, 10, 22, 12, 28, 8, 30, 20, 16, 24, 36, 18, 24, 40, 12, 42, 22, 46, 32, 52, 40, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 88, 72, 60, 46, 72, 96, 100, 32, 102, 48, 52, 106, 108, 40, 72

Offset: 1

Labos Elemer

			The 12th squarefree number is 17 and phi(17) is 16, so a(12)=16.

Robert Israel, Table of n, a(n) for n = 1..10000
D. R. Ward, Some Series Involving Euler's Function, Journal of the London Mathematical Society, Vol. 1, No. 4 (1927), pp. 210-214.

Cf. A000010, A005117, A013929, A083343.
Cf. A001221, A082695, A265668.
Cf. A013661, A065464.

a049200 1 = 1
a049200 n = product $ map (subtract 1) $ a265668_row n
-- Reinhard Zumkeller, Dec 13 2015

[EulerPhi(n): n in [1..300] | IsSquarefree(n)]; // Vincenzo Librandi, Jul 13 2015

map(numtheory:-phi,select(numtheory:-issqrfree, [$1..1000])); # Robert Israel, Jul 12 2015

EulerPhi/@Select[Range[200],SquareFreeQ] (* Harvey P. Dale, Jan 13 2015 *)

lista(nn) = {for(n=1, nn, if (issquarefree(n), print1(eulerphi(n), ", ")));} \\ Michel Marcus, Jul 12 2015

a(n) = A000010(A005117(n)).
{phi(x) ; abs(mu(x)) = 1}.
a(n) = Product_{k = 1..A001221(n)} (A265668(n,k) + 1). - Reinhard Zumkeller, Dec 13 2015
Sum_{n>=1} 1/(A005117(n)*a(n)) = A082695. - Amiram Eldar, Oct 14 2020
Lim_{n->oo} Sum_{k=1..n} 1/a(k) - log(a(n)) = A083343 (Ward, 1927). - Amiram Eldar, Mar 05 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)^2/2) * Product_{p prime} (1 - 2/p^2 + 1/p^3) = A013661^2 * A065464 / 2 = 0.57938048727453660946... . - Amiram Eldar, Oct 09 2023

A183093 a(n) = number of divisors d of n such that d > 1 and if d = Product_(i) (p_i^e_i) then e_i = 1 for at least one i.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 4, 1, 4, 3, 3, 1, 5, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 5, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 6, 1, 4, 3, 4, 1, 5, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 6, 1, 3, 4, 4, 3, 7, 1, 6, 1, 3, 1, 10, 3, 3, 3, 5, 1, 10, 3, 4, 3, 3, 3, 7, 1, 4, 4, 5

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

a(n) = number of non-powerful divisors d of n where powerful numbers are numbers of the form A001694(m) for m >= 2.

Sequence is not the same as A183096: a(72) = 6, A183096(72) = 7.

			For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A001694, A005361, A183094, A183095, A183096.

Cf. A001620, A082695.

nonpowQ[n_] := Min[FactorInteger[n][[;;, 2]]] == 1; a[n_] := DivisorSum[n, 1 &, # > 1 && nonpowQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)

a(n) = sumdiv(n, d, d > 1 && vecmin(factor(d)[, 2]) == 1); \\ Amiram Eldar, Jan 30 2025

(define (A183093 n) (- (A000005 n) (A005361 n))) ;; Antti Karttunen, Nov 23 2017

a(n) = A000005(n) - A005361(n) = A183095(n) - 1.
a(1) = 0, a(p) = 1, a(pq) = 3, a(pq...z) = 2^k - 1, a(p^k) = 1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma - zeta(2)*zeta(3)/zeta(6) - 1), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 30 2025

Name corrected by Amiram Eldar, Jan 30 2025

A325240 Numbers whose minimum prime exponent is 2.

Original entry on oeis.org

4, 9, 25, 36, 49, 72, 100, 108, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 675, 676, 784, 800, 841, 900, 961, 968, 972, 1089, 1125, 1152, 1156, 1225, 1323, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1764, 1800

Offset: 1

Gus Wiseman, Apr 15 2019

nonn

Or barely powerful numbers, a subset of powerful numbers A001694.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose minimum multiplicity is 2 (counted by A244515).
Powerful numbers (A001694) that are not cubefull (A036966). - Amiram Eldar, Jan 30 2023

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    9: {2,2}
   25: {3,3}
   36: {1,1,2,2}
   49: {4,4}
   72: {1,1,1,2,2}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  121: {5,5}
  144: {1,1,1,1,2,2}
  169: {6,6}
  196: {1,1,4,4}
  200: {1,1,1,3,3}
  225: {2,2,3,3}
  288: {1,1,1,1,1,2,2}
  289: {7,7}
  324: {1,1,2,2,2,2}
  361: {8,8}
  392: {1,1,1,4,4}
  400: {1,1,1,1,3,3}

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

Positions of 2's in A051904.
Maximum instead of minimum gives A067259.
Cf. A001221, A001222, A001358, A001694, A007774, A036966, A051903, A052485, A118914, A244515, A325241.
Cf. A065483, A082695.

Select[Range[1000],Min@@FactorInteger[#][[All,2]]==2&]

is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) == 2; } \\ Amiram Eldar, Jan 30 2023

from math import isqrt, gcd
from sympy import integer_nthroot, factorint, mobius
def A325240(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
        for w in range(1,integer_nthroot(x,5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
                    if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c += integer_nthroot(z//y**4,3)[0]
        return c
    return bisection(f,n,n**2) # Chai Wah Wu, Oct 02 2024

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Product_{p prime} (1 + 1/(p^2*(p-1))) = A082695 - A065483 = 0.6038122832... . - Amiram Eldar, Jan 30 2023

A361132 Multiplicative with a(p^e) = e^4, p prime and e > 0.

Original entry on oeis.org

1, 1, 1, 16, 1, 1, 1, 81, 16, 1, 1, 16, 1, 1, 1, 256, 1, 16, 1, 16, 1, 1, 1, 81, 16, 1, 81, 16, 1, 1, 1, 625, 1, 1, 1, 256, 1, 1, 1, 81, 1, 1, 1, 16, 16, 1, 1, 256, 16, 16, 1, 16, 1, 81, 1, 81, 1, 1, 1, 16, 1, 1, 16, 1296, 1, 1, 1, 16, 1, 1, 1, 1296, 1, 1, 16, 16

Offset: 1

Vaclav Kotesovec, Mar 02 2023, following a suggestion from Amiram Eldar

In general, if the function is multiplicative with a(p^e) = e^k, where k>=1, then Sum_{m=1..n} a(m) ~ c(k) * n, where c(k) = Product_{primes p} (1 + PolyLog(-k, 1/p)) * (1 - 1/p).
Equivalently, c(k) = Product_{primes p} (1 + Sum_{j>=2} (j^k - (j-1)^k) / p^j).
Sum_{m=1..n} A005361(m)^k ~ c(k) * n.
Table of logarithms of the first twenty constants c(k):
log(c1)  =    0.6645400902595784780106197346845697376257107319484837534113838...
log(c2)  =    2.1027190979191945200514651557327047986978773488049101019457040...
log(c3)  =    4.6968549904993458045898305766669061238379561861949323835425304...
log(c4)  =    8.6865032221694100694964858752580123427478996289429265630701524...
log(c5)  =   14.2913129298819954890384122051888143114132125173972994127345117...
log(c6)  =   21.8135511355940060754244319875442802379763506456537810297977335...
log(c7)  =   31.6936244245134941047326145621097555406387768809071583785926496...
log(c8)  =   44.5357450879229051636129496942971942282070021854681649075237793...
log(c9)  =   61.1279313139359633940353674601273793850149492879803908371116076...
log(c10) =   82.5520903493060704390063479960346732401820956158379186266389560...
log(c11) =  110.2954981238150788264027780431082219466660734768697563026966486...
log(c12) =  146.3390378386537094475359791093275236623437203145309460650602987...
log(c13) =  193.3102629498150337396691694808577709247583271151043344733643302...
log(c14) =  254.7562108044458078036208253682699240853829328072028848109791635...
log(c15) =  335.5155584889434205169760027607421364026263435517505529418223175...
log(c16) =  442.1708823748701851244490135727342670822854621013078138839028927...
log(c17) =  583.6971600757633563987486782501478518757572163549653222049269791...
log(c18) =  772.3363960260522276224001927946529683262139600086441840227950538...
log(c19) = 1024.7789861796186438478485897805332932014500908873437888887485298...
log(c20) = 1363.8429394936892771815120584792965902670785987496833459129791344...
Conjecture: log(log(c(k)))/k converges to a constant (around 0.315).

Vaclav Kotesovec, Plot of log(log(c(k))) / k, for k = 1..40

Cf. A005361, A360969, A360970, A322328, A361148, A361179.

Cf. A082695 (c1).

g[p_, e_] := e^4; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1 - 4*X + 21*X^2 + X^3 + 6*X^4 - X^5)/(1-X)^5)[n], ", "))

a(n) = A005361(n)^4.
Dirichlet g.f.: Product_{primes p} (1 + p^s*(p^(3*s) + 11*p^(2*s) + 11*p^s + 1) / (p^s - 1)^5).
Sum_{k=1..n} a(k) ~ c * n, where c = Product_{primes p} (1 + (15*p^3 + 5*p^2 + 5*p - 1) / (p*(p-1)^4)) = 5922.43654748315227690838901234893132297258444672...

A368039 The product of exponents of prime factorization of the nonsquarefree numbers.

Original entry on oeis.org

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 4, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 6, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 4, 3, 6, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 8, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 4, 3, 2, 3, 6, 4, 2, 6, 2, 2, 4, 2, 9, 2, 5, 4, 2

Offset: 1

Amiram Eldar, Dec 09 2023

nonn

The terms of A005361 that are larger than 1, since A005361(k) = 1 if and only if k is squarefree (A005117).

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

Cf. A005117, A005361, A013929.
Cf. A084936, A174961, A275699, A368038, A368040.
Cf. A059956, A082695, A229099.

Select[Table[Times @@ FactorInteger[n][[;;, 2]], {n, 1, 250}], # > 1 &]

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p > 1, print1(p, ", ")));}

a(n) = A005361(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = ((zeta(2)*zeta(3)/zeta(6)) - 1/zeta(2))/(1-1/zeta(2)) = (A082695 - A059956)/A229099 = 3.406686208821... .

A045972 a(1)=9; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^{e_i+1}.

Original entry on oeis.org

9, 25, 49, 125, 121, 1225, 169, 625, 343, 3025, 289, 6125, 361, 4225, 5929, 3125, 529, 8575, 841, 15125, 8281, 7225, 961, 30625, 1331, 9025, 2401, 21125, 1369, 148225, 1681, 15625, 14161, 13225, 20449, 42875, 1849, 21025, 17689, 75625, 2209, 207025

Offset: 1

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045968, A045969, A045970, A045971, A045973, A082695.

f[p_, e_] := NextPrime[p, 2]^(e + 1); a[1] = 9; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Sum_{n>=1} 1/a(n) = (4/7) * (zeta(2)*zeta(3)/zeta(6)) - 8/9 = 0.221737646437... . - Amiram Eldar, Sep 19 2023

More terms from David W. Wilson

A076511 Numerator of cototient(n)/totient(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 7, 1, 1, 2, 1, 3, 3, 6, 1, 2, 1, 7, 1, 4, 1, 11, 1, 1, 13, 9, 11, 2, 1, 10, 5, 3, 1, 5, 1, 6, 7, 12, 1, 2, 1, 3, 19, 7, 1, 2, 3, 4, 7, 15, 1, 11, 1, 16, 3, 1, 17, 23, 1, 9, 25, 23, 1, 2, 1, 19, 7, 10, 17, 9, 1, 3, 1, 21, 1, 5, 21, 22, 31, 6, 1, 11, 19, 12, 11

Offset: 1

Reinhard Zumkeller, Oct 15 2002

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A076512 (denominators), A000010, A009195, A051953, A082695, A109395.

Table[Numerator[n/EulerPhi[n] - 1], {n, 1, 100}] (* Amiram Eldar, Nov 21 2022 *)

A076511(n) = numerator((n-eulerphi(n))/eulerphi(n)); \\ Antti Karttunen, Sep 07 2018

a(n) = A051953(n)/A009195(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A076512(k) = zeta(2)*zeta(3)/zeta(6) - 1 = A082695 - 1 = 0.943596... . Amiram Eldar, Nov 21 2022

A183095 a(n) = number of divisors d of n that are either 1 or of the form Product_(i) (p_i^e_i) where e_i = 1 for at least one i.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 5, 2, 4, 4, 2, 2, 5, 2, 5, 4, 4, 2, 6, 2, 4, 2, 5, 2, 8, 2, 2, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 2, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 11, 2, 4, 5, 2, 4, 8, 2, 5, 4, 8, 2, 7, 2, 4, 5, 5, 4, 8, 2, 7, 2, 4, 2, 11, 4, 4, 4, 6, 2, 11, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

a(n) = number of non-powerful divisors d of n where powerful numbers are numbers of the form A001694(m) for m >= 1.

			For n = 12, set of such divisors is {1, 2, 3, 6, 12}; a(12) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A001694, A183093, A183094.

Cf. A001620, A082695.

nonpowQ[n_] := Min[FactorInteger[n][[;;, 2]]] == 1;a[n_] := DivisorSum[n, 1 &, nonpowQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)

a(n) = sumdiv(n, d, d == 1 || vecmin(factor(d)[, 2]) == 1); \\ Amiram Eldar, Jan 30 2025

a(n) = A000005(n) - A183094(n) = A183093(n) + 1.
a(1) = 1, a(p) = 2, a(pq) = 4, a(pq...z) = 2^k, a(p^k) = 2, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma - zeta(2)*zeta(3)/zeta(6)), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 30 2025

Name corrected by Amiram Eldar, Jan 30 2025

A323332 The Dedekind psi function values of the powerful numbers, A001615(A001694(n)).

Original entry on oeis.org

1, 6, 12, 12, 24, 30, 36, 48, 72, 56, 96, 144, 108, 180, 216, 132, 150, 192, 288, 182, 336, 360, 432, 360, 324, 384, 576, 306, 648, 392, 380, 672, 720, 864, 672, 792, 900, 768, 552, 1152, 750, 1296, 1080, 1092, 972, 1344, 1440, 870, 1728, 2160, 992, 1584

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The sum of the reciprocals of all the terms of this sequence is Pi^2/6 (A013661).

The asymptotic density of a sequence S that possesses the property that an integer k is a term if and only if its powerful part, A057521(k) is a term, is (1/zeta(2)) * Sum_{n>=1, A001694(n) is a term of S} 1/a(n). Examples for such sequences are the e-perfect numbers (A054979), the exponential abundant numbers (A129575), and other sequences listed in the Crossrefs section. - Amiram Eldar, May 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, A property of Dedekind's psi-function, Proceedings of the American Mathematical Society, Vol. 12, No. 6 (1961), p. 996.

Cf. A001615, A001694, A013661, A082695, A112526.

Sequences whose density can be calculated using this sequence: A054979, A129575, A307958, A308053, A321147, A322858, A323310, A328135, A339936, A340109, A364990, A382061, A383693, A383695, A383697.

psi[1]=1; psi[n_] := n * Times@@(1+1/Transpose[FactorInteger[n]][[1]]); psi /@ Join[{1}, Select[Range@ 1200, Min@ FactorInteger[#][[All, 2]] > 1 &]] (* after T. D. Noe at A001615 and Harvey P. Dale at A001694 *)

from math import isqrt, prod
from sympy import mobius, integer_nthroot, primefactors
def A323332(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-squarefreepi(integer_nthroot(x,3)[0]), 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)
        return c+l
    a = primefactors(m:=bisection(f,n,n))
    return m*prod(p+1 for p in a)//prod(a) # Chai Wah Wu, Sep 14 2024

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A335851 Numbers that are powerful in Gaussian integers.

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A049200 Euler totient function phi applied to the n-th squarefree number.

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A183093 a(n) = number of divisors d of n such that d > 1 and if d = Product_(i) (p_i^e_i) then e_i = 1 for at least one i.

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A325240 Numbers whose minimum prime exponent is 2.

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A361132 Multiplicative with a(p^e) = e^4, p prime and e > 0.

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A368039 The product of exponents of prime factorization of the nonsquarefree numbers.

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A045972 a(1)=9; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^{e_i+1}.

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