Waldemar Puszkarz - cp's OEIS frontend

A377134 Abundant numbers k such that k^2 + A033880(k)^2 is a perfect square.

336, 1080, 3078, 6048, 6552, 19845, 47616, 239760, 435708, 599400, 760320, 873180, 997920, 1468800, 1602300, 2004480, 4312440, 4612608, 4713984, 10181808, 10665984, 11554816, 12160512, 24149664, 31244850, 46431744, 56439504, 64995840, 116958492

Offset: 1

Waldemar Puszkarz, Oct 17 2024

nonn

These abundant numbers along with their abundances form the legs of an integral Pythagorean triangle.
Odd terms are very rare: 19845 is the only one up to 10^9.
19845 is the only odd term up to 4*10^10. - Amiram Eldar, Mar 11 2025

			336 is a term because its abundance is 320 and 320^2 + 336^2 = 464^2.

Amiram Eldar, Table of n, a(n) for n = 1..56

Cf. A005101, A033880, A000290.

l={}; Do[a=DivisorSigma[1,n]-2*n; If[a>0&&IntegerQ@Sqrt[n^2+a^2], AppendTo[l, n]], {n, 12, 2*10^8}]; l

for(n=12, 2*10^8, a=sigma(n)-2*n; a>0&&issquare(n^2+a^2)&&print1(n", "))

import sympy as sp
for i in range(12, 200000000):
    a=sp.ntheory.factor_.divisor_sigma(i) - 2*i
    if a>0 and sp.ntheory.primetest.is_square(i*i+a*a):
        print(i, end=", ")

A377121 Numbers whose totient is refactorable.

Original entry on oeis.org

1, 2, 3, 4, 6, 13, 15, 16, 19, 20, 21, 24, 26, 27, 28, 30, 35, 36, 37, 38, 39, 41, 42, 45, 52, 54, 55, 56, 57, 61, 63, 70, 72, 73, 74, 75, 76, 77, 78, 82, 84, 87, 88, 89, 90, 91, 93, 95, 97, 99, 100, 108, 109, 110, 111, 114, 115, 116, 117, 119, 122, 123, 124, 126, 129, 132, 133, 135, 137, 146, 147, 148, 150, 152, 153

Offset: 1

Waldemar Puszkarz, Oct 17 2024

nonn

If k is an odd term, 2*k is also a term due to the multiplicative nature of the totient function and their totients are equal.

			3, 4, and 6 are terms as their totients equal 2, which is the second refactorable number.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A033950.

filter:= proc(n) local p;
  p:= numtheory:-phi(n);
  p mod numtheory:-tau(p) = 0
end proc:
select(filter, [$1..200]); # Robert Israel, Dec 12 2024

Select[Range[200], Mod[EulerPhi[#], DivisorSigma[0,EulerPhi[#]]]==0&]

for(n=1, 200, a=eulerphi(n); a%numdiv(a)==0&&print1(n", "))

from sympy import divisor_sigma, totient
for i in range(1, 200):
    if totient(i)%divisor_sigma(totient(i), 0)==0:
        print(i, end=", ")

A376119 Abundant numbers that are perfect powers.

Original entry on oeis.org

36, 100, 144, 196, 216, 324, 400, 576, 784, 900, 1000, 1296, 1600, 1728, 1764, 1936, 2304, 2500, 2704, 2744, 2916, 3136, 3600, 4356, 4624, 4900, 5184, 5776, 5832, 6084, 6400, 7056, 7744, 7776, 8000, 8100, 8464, 9216, 9604, 10000, 10404, 10648, 10816, 11025, 11664, 12100, 12544

Offset: 1

Waldemar Puszkarz, Sep 11 2024

nonn

Intersection of A001597 and A005101.

			36 is a term being a power (36=6^2) and an abundant number as a multiple of 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001597, A005101.

N:= 20000: # to get terms <= N
isab:= proc(n) local F, t;
  F:= ifactors(n)[2];
  mul((t[1]^(t[2]+1)-1)/(t[1]-1), t = F) > 2*n
end proc:
S:= select(isab, {seq(seq(x^i,i=2..ilog[x](N)),x=2..isqrt(N))}):
sort(convert(S,list)); # Robert Israel, Sep 12 2024

Select[Range[2*10^4], DivisorSigma[1,#]-2#>0&&GCD@@FactorInteger[#][[All, 2]]>1&]

PARI
```
ok(n)=sigma(n)-2*n>0 && ispower(n)
```

A376120 Refactorable numbers that are perfect powers.

Original entry on oeis.org

1, 8, 9, 36, 128, 225, 441, 625, 1089, 1521, 2025, 2601, 3249, 3600, 4761, 5625, 6561, 7569, 7776, 8100, 8649, 10000, 12321, 15129, 16641, 19881, 21952, 22500, 25281, 26244, 28224, 31329, 32400, 32768, 33489, 35721, 40401, 45369, 47961, 50625, 56169, 62001, 64000, 71289, 84681, 90000

Offset: 1

Waldemar Puszkarz, Sep 11 2024

nonn

Intersection of A001597 and A033950.

			8 is a perfect power, as 8=2^3, and it is also a refactorable numbers, being divisible by its number of divisors (4).

Cf. A001597 (perfect powers), A033950 (refactorable numbers).

Join[{1}, Select[Range[10^5], Divisible[#, DivisorSigma[0,#]]&&GCD@@FactorInteger[#][[All, 2]]>1&]]

ok(n) = n==1 || (n%numdiv(n)==0&&ispower(n))

from itertools import count, islice
from math import prod
from sympy import mobius, integer_nthroot, factorint
def A376120_gen(): # generator of terms
    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): return int(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m = 1
    for n in count(1):
        m = bisection(lambda x:f(x)+n,m,m)
        if not m%prod(e+1 for e in factorint(m).values()): yield m
A376120_list = list(islice(A376120_gen(),40)) # Chai Wah Wu, Oct 04 2024

A376114 Refactorable numbers that are twice a square.

Original entry on oeis.org

2, 8, 18, 72, 128, 288, 450, 882, 1152, 1250, 1800, 2178, 3042, 3528, 4050, 5000, 5202, 6498, 8712, 9522, 11250, 12168, 13122, 15138, 16200, 17298, 18432, 20808, 24642, 25992, 28800, 30258, 32768, 33282, 38088, 39762, 45000, 50562, 52488, 56448, 60552, 62658, 64800, 66978, 69192, 71442, 80000

Offset: 1

Waldemar Puszkarz, Sep 10 2024

nonn

Intersection of A001105 and A033950.

			8 is a term because it is twice 4, which is square, and 8 is refactorable.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001105, A033950, A181795.

filter:= proc(n) n mod numtheory:-tau(n) = 0 end proc:
select(filter, [seq(2*i^2,i=1..1000)]); # Robert Israel, Oct 10 2024

Select[Range[2,10^5], IntegerQ@Sqrt[#/2]&&Divisible[#, DivisorSigma[0,#]]&]

PARI
```
ok(n)=n%numdiv(n)==0&&issquare(n/2)
```

from itertools import count, islice
from math import prod
from sympy import factorint
def A376114_gen(): # generator of terms
    for n in count(1):
        k = prod((e<<1|1)+(p==2) for p, e in factorint(n).items())
        if not (m:=n**2<<1)%k: yield m
A376114_list = list(islice(A376114_gen(),47)) # Chai Wah Wu, Oct 04 2024

Conjecture: a(n) = A181795(n)/2.

A361634 Integers whose number of square divisors is coprime to the number of their nonsquare divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94

Offset: 1

Waldemar Puszkarz, Mar 19 2023

nonn

Appears to be a supersequence of A210490, and so also of positive squares and squarefree numbers (A005117). The first term that belongs in here but not in A210490 is 48. The nonsquarefree terms that are not squares are of the form p^(4k)*a, where a is a squarefree number, p is prime, and k > 0. About half of perfect numbers are of this form; one example is 496 = 2^4*31. The sequence has an asymptotic density of about 0.6420.

			48 has 3 square divisors (1, 4, and 16) and 7 nonsquare ones. Consequently, gcd(3,7)=1, thus 48 is a term.

Index entries for sequences related to divisors of numbers

Cf. A210490, A005117 (subsequences), A046951 (number of square divisors), A056595 (number of nonsquare divisors).

Select[Range[100],CoprimeQ[Total@(Boole/@IntegerQ/@Sqrt/@Divisors[#]),DivisorSigma[0,#]-Total@(Boole/@IntegerQ/@Sqrt/@Divisors[#])]&]

for(n=1, 100, a=divisors(n); c=0; for(i=1, #a, issquare(a[i])&&c++); gcd(#a-c, c)==1&&print1(n, ", "))

isok(n) = gcd(numdiv(n), numdiv(sqrtint(n/core(n))))==1 \\ Andrew Howroyd, Mar 19 2023

A301975 Numbers whose abundance is divisible by its number of divisors.

Original entry on oeis.org

1, 3, 5, 6, 7, 11, 13, 14, 17, 19, 22, 23, 28, 29, 31, 37, 38, 41, 43, 45, 46, 47, 52, 53, 56, 59, 60, 61, 62, 67, 71, 73, 76, 79, 83, 86, 89, 94, 96, 97, 99, 101, 103, 107, 109, 113, 118, 124, 126, 127, 130, 131, 132, 134, 137, 139, 142, 147, 148, 149, 150, 151, 153, 157, 158, 163, 166, 167, 168, 170, 172, 173, 175, 176, 179

Offset: 1

Waldemar Puszkarz, Mar 29 2018

nonn

Numbers n such that f(n) = A033880(n)/A000005(n) is an integer.
Perfect numbers (A000396) and odd primes (A065091) are members, unified (along with 1) into a subsequence on which abs(f(n)) reaches record extrema. For perfect numbers, these are global minima, for the other terms, maxima.
Another notable subsequence is defined by f(n)=1: numbers whose abundance equals their number of divisors. They all belong to A056075. The first 3 terms are 56, 7192, 7232. There are 11 of them up to 10^9.

			11 is a term as its abundance is -10 and its number of divisors is 2, the former number being divisible by the latter.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A033880 (abundance), A000005 (number of divisors), A065091, A000396 (subsequences), A056075 (related sequence).

Select[Range[180], Divisible[DivisorSigma[1,#]-2#, DivisorSigma[0,#]]&]

for(n=1, 180, ((sigma(n)-2*n)%numdiv(n)==0) && print1(n ", "))

isok(n) = !((sigma(n)-2*n)%numdiv(n)); \\ Michel Marcus, Apr 09 2018

A301859 Abundant numbers whose abundance is a perfect number.

Original entry on oeis.org

48, 2002, 2632, 4540, 5170, 6952, 8925, 29056, 32445, 32980, 88330, 133042, 174856, 189472, 280228, 442365, 518368, 566752, 892552, 1266952, 2030368, 2052256, 2218450, 3959752, 4120672, 4558936, 5568448, 9071752, 15921112, 38551936, 65969536, 70114936, 88149352, 97364848

Offset: 1

Waldemar Puszkarz, Mar 27 2018

nonn

There are 34 terms up to 10^8. The abundance of odd terms (only 3 terms) is 6 (see also A087167). The abundance of even terms is 28, 496, 8128, and 33550336 (for 97364848). There exist deficient numbers whose abundance is a perfect number in absolute terms, e.g., 7, 29, 62.

			48 is a term as it is abundant and its abundance, sigma(48)-2*48 = 28, is the second perfect number.

Amiram Eldar, Table of n, a(n) for n = 1..68

Cf. A005101 (abundant numbers), A033880 (abundance), A000396 (perfect numbers), A087167, A088834, A088012, A077374 (sequences related to the odd terms of this sequence).

Select[Range[10^8], PerfectNumberQ[DivisorSigma[1,# ]-2#]&]

for(n=1,10^8, a=sigma(n)-2*n; a>0&&sigma(a)==2*a&&print1(n ","))

A302125 Numbers whose deficiency is a perfect number.

Original entry on oeis.org

7, 15, 29, 52, 62, 182, 230, 315, 344, 592, 944, 998, 1155, 2012, 2570, 4028, 6710, 15128, 19688, 20264, 30248, 36224, 38252, 40730, 43964, 52088, 90332, 96128, 168116, 195224, 258512, 262112, 451952, 538112, 991904, 1209632, 1237856, 1659128, 2080544, 2085710, 2102272, 2186132

Offset: 1

Robert G. Wilson v and Waldemar Puszkarz, Apr 01 2018

nonn

Subsequence of deficient numbers (A005100) whose deficiency (A033879) is a member of perfect numbers (A000396).

			52 is in the sequence since the divisors of 52 are {1, 2, 4, 13, 26 & 52} so d(52) = 98 and 2*52 - 98 = 6, a perfect number.

Cf. A000396 (perfect numbers), A033879 (deficiency), A005100 (deficient numbers), A141548 (subsequence), A301859 (related sequence).

fQ[n_] := PerfectNumberQ[2n - DivisorSigma[1, n]]; Select[ Range@ 2500000, fQ]

for(n=1,25*10^5, d=2*n-sigma(n); d>0&&sigma(d)==2*d&&print1(n ","))

A301561 Sphenic Fibonacci numbers.

Original entry on oeis.org

610, 987, 10946, 3524578, 9227465, 24157817, 39088169, 63245986, 1836311903, 7778742049, 20365011074, 591286729879, 4052739537881, 17167680177565, 44945570212853, 61305790721611591, 420196140727489673, 1500520536206896083277, 6356306993006846248183

Offset: 1

Waldemar Puszkarz, Mar 23 2018

nonn

Intersection of A000045 and A007304. There are 28 sphenic numbers among the first 200 positive Fibonacci numbers.

			610 is a term since it is a Fibonacci number that is a product of 3 distinct primes, 610=2*5*61, which makes it a sphenic number.

Cf. A000045, A007304, A061305 (squarefree Fibonaccis), A137563 (Fibonaccis with 3 distinct primes).

Select[Fibonacci@Range[120], SquareFreeQ[#]&&PrimeNu[#]==3&]

for(n=1, 120, fn=fibonacci(n); issquarefree(fn)&&omega(fn)==3&&print1(fn ","))

A000045 INTERSECT A007304.

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User: Waldemar Puszkarz

A377134 Abundant numbers k such that k^2 + A033880(k)^2 is a perfect square.

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A377121 Numbers whose totient is refactorable.

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A376119 Abundant numbers that are perfect powers.

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A376120 Refactorable numbers that are perfect powers.

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A376114 Refactorable numbers that are twice a square.

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Formula

A361634 Integers whose number of square divisors is coprime to the number of their nonsquare divisors.

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PARI

A301975 Numbers whose abundance is divisible by its number of divisors.

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