A057723 -id:A057723 - cp's OEIS frontend

A283052 Numbers k such that uphi(k)/phi(k) > uphi(m)/phi(m) for all m < k, where phi(k) is the Euler totient function (A000010) and uphi(k) is the unitary totient function (A047994).

1, 4, 8, 16, 32, 36, 72, 144, 216, 288, 432, 864, 1728, 2592, 3600, 5400, 7200, 10800, 21600, 43200, 64800, 108000, 129600, 216000, 259200, 324000, 529200, 1058400, 2116800, 3175200, 5292000, 6350400, 10584000, 12700800, 15876000, 31752000, 63504000, 95256000

Offset: 1

Amiram Eldar, May 19 2017

nonn

This sequence is infinite.
a(1) = 1, a(6) = 36, a(15) = 3600 and a(32) = 6350400 are the smallest numbers n such that uphi(n)/phi(n) = 1, 2, 3 and 4. They are squares of 1, 6, 60, and 2520.
Also, coreful superabundant numbers: numbers k with a record value of the coreful abundancy index, A057723(k)/k > A057723(m)/m for all m < k. The two sequences are equivalent since A057723(k)/k = A047994(k)/A000010(k) for all k. - Amiram Eldar, Dec 28 2020

			uphi(k)/phi(k) = 1, 1, 1, 3/2 for k = 1, 2, 3, 4, thus a(1) = 1 and a(2) = 4 since a(4) > a(m) for m < 4.

Amiram Eldar, Table of n, a(n) for n = 1..46

Cf. A000010, A047994, A057723, A285906.

Similar sequences: A002110, A004394, A037992, A061742, A292984, A329882, A333953.

uphi[n_] := If[n == 1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@
FactorInteger[n]))[[1]]]; a = {}; rmax = 0; For[k = 0, k < 10^9, k++; r = uphi[k]/EulerPhi[k]; If[r > rmax, rmax = r; a = AppendTo[a, k]]]; a

uphi(n) = my(f = factor(n)); prod(i=1, #f~, f[i,1]^f[i,2]-1);
lista(nn) = {my(rmax = 0); for (n=1, nn, if ((newr=uphi(n)/eulerphi(n)) > rmax, print1(n, ", "); rmax = newr););} \\ Michel Marcus, May 20 2017

A307986 Amicable pairs {x, y} such that y is the sum of the divisors of x that are not divided by every prime factor of x and vice versa.

Original entry on oeis.org

42, 54, 198, 204, 582, 594, 142310, 168730, 1077890, 1099390, 1156870, 1292570, 1511930, 1598470, 1669910, 2062570, 2236570, 2429030, 2728726, 3077354, 4246130, 4488910, 4532710, 5123090, 5385310, 5504110, 5812130, 6135962, 6993610, 7158710, 7288930, 8221598

Offset: 1

Paolo P. Lava, May 09 2019

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).
Here, only the non-coreful divisors of k are considered.
The non-coreful perfect numbers listed in A307888 are not considered here.
The first time a pair ordered by its first element is not adjacent is for x = 4532710 and y = 6135962, which correspond to a(23) and a(28), respectively.

			Divisors of x = 42 are 1, 2, 3, 6, 7, 14, 21, 42 and prime factors are 2, 3, 7. Among the divisors, 42 is the only one that is divisible by every prime factor, so we have 1 + 2 + 3 + 6 + 7 + 14 + 21 = 54 = y.
Divisors of y = 54 are 1, 2, 3, 6, 9, 18, 27, 54 and prime factors are 2, 3. Among the divisors, 6, 18, 54 are the only ones that are divisible by every prime factor, so we have 1 + 2 + 3 + 9 + 27 = 42 = x.

Amiram Eldar, Table of n, a(n) for n = 1..365 (terms below 10^10)
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy)

Cf. A000203, A007947, A057723, A307888, A307958, A307962, A307963, A308029.

with(numtheory): P:=proc(q) local a,b,c,k,n; for n from 2 to q do
a:=mul(k,k=factorset(n)); b:=sigma(n)-a*sigma(n/a);
a:=mul(k,k=factorset(b)); c:=sigma(b)-a*sigma(b/a);
if c=n and b<>c then print(n); fi; od; end: P(10^8);

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; ncs[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); seq = {}; Do[m = ncs[n]; If[m > 1 && m != n && n == ncs[m], AppendTo[seq, n]], {n, 2, 10^6}]; seq (* Amiram Eldar, May 11 2019 *)

A336567 Sum of proper divisors of {n divided by its largest squarefree divisor}.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0, 7, 0, 1, 0, 1, 0, 0, 0, 3, 1, 0, 4, 1, 0, 0, 0, 15, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 7, 1, 1, 0, 1, 0, 4, 0, 3, 0, 0, 0, 1, 0, 0, 1, 31, 0, 0, 0, 1, 0, 0, 0, 16, 0, 0, 1, 1, 0, 0, 0, 7, 13, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 15, 0, 1, 1, 8, 0, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jul 27 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A001065, A003557, A007947, A057723, A335341, A336563.

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A336563(n) = (A057723(n)-n);
A336567(n) = (A336563(n)/A007947(n));

a(n) = A001065(A003557(n)).

a(n) = A335341(n) - A003557(n) = A336563(n) / A007947(n).

A336647 a(n) = n - A336566(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 8, 8, 1, 10, 1, 10, 9, 15, 1, 15, 1, 18, 11, 14, 1, 18, 24, 16, 25, 14, 1, 18, 1, 31, 15, 20, 13, 35, 1, 22, 17, 30, 1, 30, 1, 42, 42, 26, 1, 34, 48, 49, 21, 50, 1, 42, 17, 54, 23, 32, 1, 54, 1, 34, 62, 63, 19, 54, 1, 66, 27, 66, 1, 69, 1, 40, 74, 74, 19, 66, 1, 78, 80, 44, 1, 70, 23, 46, 33, 86

Offset: 1

Antti Karttunen, Jul 30 2020

sign

Some terms, for example a(600) and a(6552), are negative. - Georg Fischer, Jul 31 2020

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A007947, A057723, A308135, A336563, A336564, A336566.
Cf. A336555 (positions where differs from A336646).
Cf. A336565 (positions where a(n) = 2*n - A057723(n) = n - A336563(n)).
Cf. also A336645.

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A308135(n) = (sigma(n)-A057723(n));
A336563(n) = (A057723(n)-n);
A336564(n) = (n - A308135(n));
A336566(n) = gcd(A336563(n), A336564(n));
A336647(n) = (n - A336566(n));

a(n) = n - A336566(n).

A339936 Odd coreful abundant numbers: the odd terms of A308053.

Original entry on oeis.org

99225, 165375, 231525, 297675, 496125, 694575, 826875, 893025, 1091475, 1157625, 1225125, 1289925, 1488375, 1620675, 1686825, 1819125, 1885275, 2083725, 2149875, 2282175, 2480625, 2546775, 2679075, 2811375, 2877525, 3009825, 3075975, 3142125, 3274425, 3472875

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

The asymptotic density of this sequence is Sum_{n>=1} f(A356871(n)) = 9.1348...*10^(-6), where f(n) = (4/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) if n is odd and 0 otherwise. - Amiram Eldar, Sep 02 2022

			99225 is a term since it is odd and the sum of its coreful divisors is A057723(99225) = 201600 > 2 * 99225.

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

Intersection of A005408 and A308053.
Subsequence of A321147.
Cf. A057723, A356871.
Similar sequences: A005231, A094889, A112643, A127666, A129485, A293186, A321147, A333950.

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

A376218 Odd exponentially odd numbers.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 149

Offset: 1

Amiram Eldar, Sep 16 2024

First differs from its subsequence A182318 at n = 8318: a(8318) = 19683 = 3^9 = 3^(3^2) is not a term of A182318.
Numbers whose prime factorization contains only odd primes and odd exponents.
Numbers whose sum of coreful divisors (A057723) is odd (a coreful divisor d of a number k is a divisor that is divisible by every prime that divides k, see also A307958).
The even exponentially odd numbers are numbers of the form 2^k * m, where k is odd and m is a term of this sequence.
The asymptotic density of this sequence is (3/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (3/5) * A065463 = 0.42266532... .

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

Intersection of A005408 and A268335.
Cf. A057723, A065463, A182318, A307958.
Other numbers with an odd sum of divisors: A000079 (unitary divisors), A028982 (all divisors), A069562 (non-unitary divisors), A357014 (exponential divisors).

Select[Range[1, 150, 2], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

is(k) = k % 2 && vecprod(factor(k)[,2]) % 2;

A308360 Product of positive divisors d of n that are divisible by every prime that divides n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 64, 27, 10, 11, 72, 13, 14, 15, 1024, 17, 108, 19, 200, 21, 22, 23, 1728, 125, 26, 729, 392, 29, 30, 31, 32768, 33, 34, 35, 46656, 37, 38, 39, 8000, 41, 42, 43, 968, 675, 46, 47, 82944, 343, 500, 51, 1352, 53, 5832, 55, 21952, 57, 58, 59

Offset: 1

Jaroslav Krizek, May 22 2019

nonn

			The divisors of 12 that are divisible by both 2 and 3 are 6 and 12. So a(12) = 6 * 12 = 72.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

See A005361 and A057723 for number and sum of such divisors.

Cf. A007955, A064549.

[1] cat [&*[d: d in Divisors(n) |  d gt 1  and #[c: c in Divisors(d) | IsPrime(c)] eq #[d: d in Divisors(n) | IsPrime(d)]]: n in [2..100]]

Table[Sqrt[(n*Product[If[PrimeQ[d], d, 1], {d, Divisors[n]}])^Product[ FactorInteger[n][[k, 2]], {k, 1, Length[FactorInteger[n]]}]], {n, 1, 100}] (* Vaclav Kotesovec, Jun 15 2019 *)

a(n) = n for squarefree numbers (A005117).

a(n) = A064549(n)^(A005361(n)/2). - Charlie Neder, Jun 03 2019

A339939 Coreful weird numbers: numbers k that are coreful abundant (A308053) but no subset of their aliquot coreful divisors sums to k.

Original entry on oeis.org

4900, 14700, 53900, 63700, 83300, 93100, 112700, 142100, 151900, 161700, 181300, 191100, 200900, 210700, 230300, 249900, 259700, 279300, 289100, 298900, 328300, 338100, 347900, 349448, 357700, 387100, 406700, 426300, 436100, 455700, 475300, 494900, 504700, 524300

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

First differs from A321146 at n = 24.

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

			4900 is a term since the sum of its aliquot coreful divisors, {70, 140, 350, 490, 700, 980, 2450}, is 5180 > 4900, and no subset of these divisors sums to 4900.

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

Subsequence of A308053.
Cf. A007947, A057723.
Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948.

corDiv[n_] := Module[{rad = Times @@ FactorInteger [n][[;;,1]]}, rad * Divisors[n/rad]]; corWeirdQ[n_] := Module[{d = Most@corDiv[n], x}, Plus @@ d > n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[10^5], corWeirdQ]

A339982 Coreful abundant numbers (A308053) with an odd sum of coreful divisors.

Original entry on oeis.org

1157625, 10418625, 12733875, 15049125, 19679625, 21994875, 26625375, 28940625, 33571125, 35886375, 40429125, 42832125, 47462625, 49777875, 54408375, 56723625, 61354125, 66733875, 68299875, 70615125, 77560875, 82191375, 84506625, 91452375, 93767625, 96082875

Offset: 1

Amiram Eldar, Dec 25 2020

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).
All the terms are odd numbers since the sum of coreful divisors (A057723) of an even number is even.
All the terms are exponentially odd numbers (A268335) since the sum of coreful divisors function is multiplicative and A057723(p^e) = p + p^2 + ... + p^e is even for a prime p and an even exponent e.
None of the terms are coreful Zumkeller numbers (A339979).

			1157625 is a term since A057723(1157625) = 2411955 > 2*1157625 and it is odd.

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

Intersection of A268335 and A339936.
Subsequence of A308053.
Cf. A007947, A057723, A339979.

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

A363331 a(n) is the sum of divisors of n that are both coreful and infinitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 14, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 42, 25, 26, 39, 28, 29, 30, 31, 50, 33, 34, 35, 36, 37, 38, 39, 70, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 78, 55, 98, 57, 58, 59, 60, 61, 62, 63, 84, 65, 66, 67, 68

Offset: 1

Amiram Eldar, May 28 2023

First differs from A363334 at n = 16.

The number of these divisors is A363329(n).

			a(8) = 14 since 8 has 3 divisors that are both infinitary and coreful, 2, 4 and 8, and 2 + 4 + 8 = 14.

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

Cf. A049417, A057723, A077609, A138302, A361810, A363329, A363334.

f[p_, e_] := Times @@ (1 + Flatten[p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], ?(# == 1 &)]))]) - 1; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), b); prod(i = 1, #f~, b = binary(f[i, 2]); prod(k = 1, #b, if(b[k], f[i, 1]^(2^(#b - k)) + 1, 1)) - 1);}

Multiplicative with a(p^e) = (Product_{k>=0} (p^(2^k*(b(k)+1)) - 1)/(p^(2^k) - 1)) - 1, where e = Sum_{k >= 0} b(k) * 2^k is the binary representation of e.
a(n) >= n, with equality if and only if n is in A138302.
a(n) >= A361810(n), with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p) * Sum_{k>=1} a(p^k)/p^(2*k)) = 0.53906337497505398777... .

cp's OEIS Frontend

A283052 Numbers k such that uphi(k)/phi(k) > uphi(m)/phi(m) for all m < k, where phi(k) is the Euler totient function (A000010) and uphi(k) is the unitary totient function (A047994).

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A307986 Amicable pairs {x, y} such that y is the sum of the divisors of x that are not divided by every prime factor of x and vice versa.

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A336567 Sum of proper divisors of {n divided by its largest squarefree divisor}.

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A336647 a(n) = n - A336566(n).

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A339936 Odd coreful abundant numbers: the odd terms of A308053.

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Mathematica

A376218 Odd exponentially odd numbers.

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A308360 Product of positive divisors d of n that are divisible by every prime that divides n.

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Magma

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A339939 Coreful weird numbers: numbers k that are coreful abundant (A308053) but no subset of their aliquot coreful divisors sums to k.

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