_Torlach Rush - cp's OEIS frontend

A383080 Numbers k such that sopf(k) does not divide evenly sopfr(k).

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 198

Offset: 1

Torlach Rush, Apr 15 2025

nonn

First differs from A059404 and A323055 at n = 59.
a(n) has a square factor, A008683(a(n)) = 0.
If p and q are distinct primes, p*q^k is in the sequence iff p + q does not divide k - 1. - Robert Israel, Apr 16 2025

			12 is a term because sopf(12)=5 does not evenly divide sopfr(12)=7.
18 is a term because sopf(18)=5 does not evenly divide sopfr(18)=8.
20 is a term because sopf(20)=7 does not evenly divide sopfr(20)=9.

Cf. A001414, A008472, A008683.

Cf. A059404, A323055.

filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  add(t[1]*t[2],t=F) mod add(t[1],t=F) <> 0
end proc:
select(filter, [$2..300]); # Robert Israel, Apr 16 2025

q[k_] := Module[{f = FactorInteger[k]}, !Divisible[Plus @@ Times @@@ f, Plus @@ f[[;; , 1]]]]; Select[Range[200], q] (* Amiram Eldar, Apr 16 2025 *)

isok(k) = if (k>1, my(f=factor(k)); sum(j=1, #f~, f[j,1]*f[j,2]) % sum(j=1, #f~, f[j,1])); \\ Michel Marcus, Apr 16 2025

def spf(k):
    fl = list(factor(k))
    sr = sum(p * e for p, e in fl)
    sd = sum(p for p, _ in fl)
    return sd, sr
def output(limit=198):
    results = []
    for k in range(2, limit + 1):
        sd, sr = spf(k)
        if 0 < sd and sr % sd != 0:
            results.append(k)
    return results
print(output())

A381844 Quotients of A380487.

Original entry on oeis.org

1, 3, 5, 7, 11, 19, 23, 34, 91, 105, 209, 221, 231, 385, 399, 429, 481, 609, 665, 715, 805, 897, 1001, 1105, 1430, 1729, 1870, 2046, 2233, 2261, 3094, 3230, 3553, 3565, 3774, 4278, 4862, 4921, 4945, 5270, 5358, 5365, 6409, 6670, 7429, 7462, 7657, 7990, 8041, 8569

Offset: 1

Torlach Rush, Mar 10 2025

nonn

Sequence is strictly increasing based on definition of A380487.

			1 is a term because rad(2)/sopf(2) = 2/2.
3 is a term because rad(30)/sopf(30) = 30/10.
5 is a term because rad(70)/sopf(70) = 70/14.

Robert Israel, Table of n, a(n) for n = 1..168

Cf. A007947, A008472, A380487.

f:= proc(n,m) local q,S;
   S:= numtheory:-factorset(n);
   q:= convert(S,`*`)/convert(S,`+`);
   if q::integer and q > m  then q else 0 fi;
end proc:
m:= 0: R:= NULL: count:= 0:
for n from 2 while count < 50 do
  v:= f(n,m);
  if v > 0 then
    m:= v; R:= R,v; count:= count+1;
  fi;
od:
R; # Robert Israel, Apr 30 2025

lista(nn) = my(m=0, list=List()); for (n=2, nn, my(f=factor(n)[,1], q=factorback(f)/vecsum(f)); if ((denominator(q) == 1) && (q>m), listput(list, q); m=q);); Vec(list); \\ Michel Marcus, Mar 29 2025

def sopf(n): return sum(set(prime_factors(n)))
def rad(n):
    rad = 1
    for p in set(prime_factors(n)): rad *= p
    return rad
def output(limit=39):
    results = []
    n = 2
    result = 0
    while len(results) < limit:
        sopf_n = sopf(n)
        rad_n = rad(n)
        if rad_n % sopf_n == 0 and result < rad_n / sopf_n:
            result = rad_n / sopf_n
            print(result, end=', ')
        n += 1
    return
output()

A380487 Numbers such that the sum of prime factors without repetition divides the product of prime factors without repetition and each division yields a greater quotient.

Original entry on oeis.org

2, 30, 70, 105, 231, 627, 805, 1122, 2730, 3570, 8778, 9282, 10626, 15015, 24738, 24882, 31746, 33495, 33915, 44330, 45885, 49335, 51051, 62985, 72930, 95095, 106590, 132990, 145145, 156009, 170170, 222870, 230945, 274505, 290598, 329406, 335478, 418285, 449995

Offset: 1

Torlach Rush, Jan 24 2025

nonn

Conjecture: There exists m, the smallest positive integer such that a(k)|a(k+m), for all k.

			2 is a term because sopf(2)|rad(2) = 2|2.
30 is a term because sopf(30)|rad(30) = 10|30.
70 is a term because sopf(70)|rad(70) = 14|70.

Robert Israel, Table of n, a(n) for n = 1..168

Cf. A007947, A008472.

Subsequence of A086486.

f:= proc(n,m) local q,S;
   S:= numtheory:-factorset(n);
   q:= convert(S,`*`)/convert(S,`+`);
   if q::integer and q > m  then q else 0 fi;
end proc:
m:= 0: R:= NULL: count:= 0:
for n from 2 while count < 50 do
  v:= f(n,m);
  if v > 0 then
    m:= v;  R:= R, n; count:= count+1;
  fi;
od:
R; # Robert Israel, Apr 30 2025

s={};q=0;Do[pf=Times@@First/@FactorInteger[n];sf=Total[First/@FactorInteger[n]];If[Divisible[pf,sf]&&pf/sf>q,AppendTo[s,n];q=pf/sf],{n,2,449995}];s (* James C. McMahon, Apr 03 2025 *)

lista(nn) = my(m=0, list=List()); for (n=2, nn, my(f=factor(n)[,1], q=factorback(f)/vecsum(f)); if ((denominator(q) == 1) && (q>m), listput(list, n); m=q);); Vec(list); \\ Michel Marcus, Mar 29 2025

def sopf(n): return sum(set(prime_factors(n)))
def rad(n):
    rad = 1
    for p in set(prime_factors(n)): rad *= p
    return rad
def output(limit=39):
    results = []
    n = 2
    result = 0
    while len(results) < limit:
        sopf_n = sopf(n)
        rad_n = rad(n)
        if rad_n % sopf_n == 0 and result < rad_n / sopf_n:
            results.append(n)
            result = rad_n / sopf_n
            print(n, end=', ')
        n += 1
    return results
output()

A378993 a(n) = n - omega(n), where omega = A001221.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 7, 8, 8, 10, 10, 12, 12, 13, 15, 16, 16, 18, 18, 19, 20, 22, 22, 24, 24, 26, 26, 28, 27, 30, 31, 31, 32, 33, 34, 36, 36, 37, 38, 40, 39, 42, 42, 43, 44, 46, 46, 48, 48, 49, 50, 52, 52, 53, 54, 55, 56, 58, 57, 60, 60, 61, 63, 63, 63, 66, 66

Offset: 1

Torlach Rush, Dec 17 2024

			a(40) = 38, since 40 has two distinct prime divisors (2 and 5), and so 40 - 2 = 38.
a(41) = 40 also, since 41 is prime and therefore 41 - 1 = 40.
a(42) = 39, since 42 has three distinct prime divisors (2, 3, 7), and so 42 - 3 = 39.

Cf. A001221, A229109.

a[n_]:=n-PrimeNu[n]; Array[a,68] (* Stefano Spezia, Dec 29 2024 *)

PARI
```
a(n) = n - omega(n);
```

a(n) = n - A001221(n).

A376927 Totients whose inverses can be separated into ordered pairs (x, 2*x) with no remainder.

Original entry on oeis.org

1, 6, 10, 18, 22, 28, 30, 42, 46, 52, 54, 58, 66, 70, 78, 82, 100, 102, 106, 110, 126, 130, 136, 138, 148, 150, 162, 166, 172, 178, 180, 190, 196, 198, 210, 222, 226, 228, 238, 250, 262, 268, 270, 282, 292, 294, 306, 310, 316, 330, 342, 346, 348, 358, 366, 372

Offset: 1

Torlach Rush, Oct 11 2024

nonn

Conjecture: Numbers having an even number of totient inverses, the odd inverses being confined to the lower half of the list.

			1 is a totient and has totient inverses 1 and 2, giving an ordered pair (1, 2*1)
6 is a totient and has totient inverses 7, 9, 14, 18 giving ordered pairs (7,2*7) and (9, 2*9).
18 is a totient and has totient inverses 19, 27, 38, 54 giving ordered pairs (19,2*19) and (27, 2*27).
348 is a totient and has totient inverses 349, 413, 531, 698, 826, 1062 giving ordered pairs (349, 2*349), (413, 2*413), and (531, 2*531).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Subset of A002202.

Cf. A000010, A007617, A032447.

q:= n-> (s-> s<>{} and ((x, y)-> x = map(t->t/2, y))(
    selectremove(x-> x::odd, s)))({numtheory[invphi](n)[]}):
select(q, [$1..543])[];  # Alois P. Heinz, Nov 21 2024

okQ[k_] := With[{s = Sort[invphi[k]]}, s != {} && Select[s, EvenQ]/2 == Select[s, OddQ]];
Reap[For[k = 1, k <= 1000, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Apr 04 2025, using Maxim Rytin's 'invphi' program (see A007617) *)

evencardinality(list)={e=0;o=0;if(0 < #list,if(0 == Mod(#list, 2),for(i=1,#list,if(1==Mod(list[i],2),o++,e++)),o++), o++);return(if(o==e, 1, 0))}
for(n = 1, 372, if(1 == evencardinality(invphi(n)), print1(n, ", ")));

A376639 Terms of A151999 which are not a term of A293928.

Original entry on oeis.org

10, 30, 34, 42, 50, 60, 68, 78, 90, 102, 110, 114, 126, 136, 150, 156, 170, 180, 204, 210, 220, 222, 228, 234, 250, 270, 294, 300, 306, 330, 340, 342, 378, 390, 408, 410, 420, 438, 444, 450, 456, 468, 510, 514, 540, 546, 550, 570, 578, 582, 612, 630, 654, 660, 666

Offset: 1

Torlach Rush, Sep 30 2024

nonn

Conjecture: For each a(n) there is no a(n) = A000010(a(k)), k > n.

Conjecture: Every term of A293928 exists in A151999.

			10 is a term because 2 divides 4 and 10 and 10 is not a term of A293928.
666 is a term because 666 is a term of A151999 and 666 is not a term of A293928 as it has no totient inverses.

Cf. A000010, A151999, A293928.

terms = []
for n in range(1, 10000): # Equivalent of A151999/b151999.txt
    if euler_phi(n)**2 == euler_phi(euler_phi(n) * n): terms.append(n)
displayTerms = []
for n in range(0,10000):
    searchTerms = terms[n+1::]
    found = False
    for k in range(0, len(searchTerms)):
        if terms[n] == euler_phi(searchTerms[k]):
            found = True
            break
    if False == found and n < len(terms):
        displayTerms.append(terms[n])
for n in range(0, 55):
    print(displayTerms[n], end=', ')

A370513 Complement of A323599.

Original entry on oeis.org

2, 5, 29, 39, 53, 59, 73, 95, 119, 123, 125, 129, 137, 145, 147, 149, 157, 159, 163, 173, 179, 191, 199, 207, 209, 213, 219, 221, 235, 251, 257, 263, 265, 269, 271, 279, 291, 293, 299, 303, 305, 325, 327, 329, 343, 345, 347, 359, 365, 367, 369, 375, 385, 395, 397

Offset: 1

Torlach Rush, Feb 20 2024

nonn

Terms of this sequence are not solutions of Sum_{d|k} A069359(d), k >= 1.
Proof that 2 is not a solution of Sum_{d|k} A069359(d), k >= 1: (Start)
If 2 is a solution then the only summands of the above are either (0,2) or (0,1,1).
(0,2) cannot be the only summands. If 2 is a summand then it is also a divisor of a(n) and A069359(2) = 1. If 2 is a summand then so must 1 be a summand.
(0,1,1) cannot be the only summands. There must exist an additional summand A069359(p_1*p_2) where p_1 and p_2 (primes) contribute to each 1 in (0,1,1).
(End)
To prove that 5 is not a solution of Sum_{d|k} A069359(d), k >= 1 we need to show that each of the following summands cannot exist: (0,5), (0,1,4), (0,1,2,2), (0,1,1,3), (0,1,1,1,2). (0,1,1,1,1,1). Following from the above proof this is elementary.

			2 is a term because it is not a solution of Sum_{d|k} A069359(d), k >= 1. See proof in Comments.

Cf. A069359, A323599.

A370238 a(n) = n(3n + 23)/2.

Original entry on oeis.org

0, 13, 29, 48, 70, 95, 123, 154, 188, 225, 265, 308, 354, 403, 455, 510, 568, 629, 693, 760, 830, 903, 979, 1058, 1140, 1225, 1313, 1404, 1498, 1595, 1695, 1798, 1904, 2013, 2125, 2240, 2358, 2479, 2603, 2730, 2860, 2993, 3129, 3268, 3410, 3555, 3703, 3854, 4008

Offset: 0

Torlach Rush, Feb 12 2024

a(a(1)) = A000566(a(1)). This is also true for each of the sequences provided in the formulae below; e.g., A151542(A151542(1)) = A000566(A151542(1)).

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A000566, A008594, A005449, A045943, A059845, A115067, A140090, A140091, A140672, A140673, A140674, A140675, A151542, A277976.

Table[n(3n+23)/2,{n,0,48}] (* James C. McMahon, Feb 20 2024 *)

Python
```
def a(n): return n*(3*n+23)//2
```

a(n) = n*(3*n + 23)/2 = A277976(n)/2.
G.f.: x*(13-10*x)/(1-x)^3.
a(n) = A151542(n) + n.
a(n) = A140675(n) + 2*n.
a(n) = A140674(n) + 3*n.
a(n) = A140673(n) + 4*n.
a(n) = A140672(n) + 5*n.
a(n) = A059845(n) + 6*n.
a(n) = A140091(n) + 7*n.
a(n) = A140090(n) + 8*n.
a(n) = A115067(n) + 9*n.
a(n) = A045943(n) + 10*n.
a(n) = A005449(n) + 11*n.
a(n) = A000326(n) + A008594(n).
Sum_{n>=1} 1/a(n) = 823467/2769844 + sqrt(3)*Pi/69 -3*log(3)/23 = 0.2328608... - R. J. Mathar, Apr 23 2024
E.g.f.: exp(x)*x*(26 + 3*x)/2. - Stefano Spezia, Apr 26 2024

A368880 Primes in A024528.

Original entry on oeis.org

3, 11, 61, 457, 5237, 1226677, 34543329507310391, 1636619248175258407, 5186576044693944076609, 742779051038516950393163206833793, 1506853388294906471801157206440769816406928024502711, 651879075122842895567706351814676957742356330143458665568047

Offset: 1

Torlach Rush, Jan 08 2024

nonn

Primes which are the sum of the numerator and the denominator of partial sums of the reciprocals of primes.
Each term of this sequence can be expressed as the sum of an expression with exactly one odd term and n even terms, where the odd term is  A002110(n)/A002110(1), and n > 0 (see Alexander Adamchuk comment in A024528).
a(2) = 11 = 3 + 2 + 6 contains the only prime odd term 3.

			3 is a term because 1/2 = 1/2 and 1 + 2 = 3 which is prime.
11 is a term because 1/2 + 1/3 = 5/6 and 5 + 6 = 11 which is prime.
61 is a term because 5/6 + 1/5 = 31/30 and 31 + 30 = 61 which is prime.
457 is a term because 31/30 + 1/7 = 247/210 and 247 + 210 = 457 which is prime.

Intersection of A000040 and A024528.

Cf. A002110.

A365649 Dirichlet convolution of sigma with Dedekind psi function.

Original entry on oeis.org

1, 6, 8, 22, 12, 48, 16, 66, 41, 72, 24, 176, 28, 96, 96, 178, 36, 246, 40, 264, 128, 144, 48, 528, 97, 168, 176, 352, 60, 576, 64, 450, 192, 216, 192, 902, 76, 240, 224, 792, 84, 768, 88, 528, 492, 288, 96, 1424, 177, 582, 288, 616, 108, 1056, 288, 1056, 320

Offset: 1

Torlach Rush, Sep 14 2023

Cf. A000040, A000203, A001615, A365647, A365648.

f[p_, e_] := (2 + ((e + 1)*p^2 - 2*p - e - 1)*p^e)/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1+X) / ((1-X) * (1 - p*X)^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 15 2023

from sympy import divisors, primefactors, prod, reduced_totient, divisor_sigma
def psi(n):
    return n*prod(p+1 for p in primefactors(n))//prod(primefactors(n))
def a(n): return sum(divisor_sigma(d, 1) * psi(n//d) for d in divisors(n))

a(n) = Sum{d|n} A000203(d) * A001615(n/d).
a(p) = A365647(p) + 2 = A365648(p) + 2 where p is a term of A000040.
Multiplicative with a(p^e) = (2 + ((e+1)*p^2 - 2*p - e - 1)*p^e)/(p-1)^2. - Amiram Eldar, Sep 15 2023
From Vaclav Kotesovec, Sep 15 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * zeta(s-1)^2 / zeta(2*s).
Sum_{k=1..n} a(k) ~ 5*n^2 * (log(n)/4 + gamma/2 - 1/8 + 3*zeta'(2)/Pi^2 - 45*zeta'(4)/Pi^4), where gamma is the Euler-Mascheroni constant A001620. (End)

cp's OEIS Frontend

User: _Torlach Rush

A383080 Numbers k such that sopf(k) does not divide evenly sopfr(k).

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Maple

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PARI

Sage

A381844 Quotients of A380487.

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Maple

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Sage

A380487 Numbers such that the sum of prime factors without repetition divides the product of prime factors without repetition and each division yields a greater quotient.

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Maple

Mathematica

PARI

Sage

A378993 a(n) = n - omega(n), where omega = A001221.

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Mathematica

PARI

Formula

A376927 Totients whose inverses can be separated into ordered pairs (x, 2*x) with no remainder.

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Maple

Mathematica

PARI

A376639 Terms of A151999 which are not a term of A293928.

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Sage

A370513 Complement of A323599.

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A370238 a(n) = n*(3*n + 23)/2.

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A370238 a(n) = n(3n + 23)/2.