A336702 -id:A336702 - cp's OEIS frontend

A347383 Odd composites k, not powers of primes, such that for all their nontrivial unitary divisors d it holds that A347381(d) > A347381(k).

189, 1271, 2125, 9261, 63767, 133907, 142859, 161257, 189209, 226967, 368063, 426373, 777923, 801727, 925101, 961193, 1003043, 4566661, 5244091, 5588327, 6031163, 6064439, 8135263, 8639879, 10074227, 10150571, 11234875, 12489107, 16016003, 19765547, 22635539

Offset: 1

Antti Karttunen, Sep 10 2021

Here nontrivial unitary divisor d of k means any divisor d|k, such that 1 < d < k and gcd(d, k/d) = 1.
Any hypothetical odd term x in A005820 (triperfect numbers) would also be a member of this sequence. Proof: such an odd number cannot be a prime power (although it must be a square), thus it must have at least two nontrivial unitary divisors (with A034444(x) >= 4). Because sigma(x) = 3*x, it must be a term of A347391. From the illustration given there, we see that any odd square y in that sequence (i.e. with A347381(y)=1) would have an abundancy index of at least three (sigma(y)/y >= 3). But because abundancy index is multiplicative and always > 1 for n > 1, any nontrivial unitary divisor d of an odd triperfect number x must have sigma(d)/d < 3, thus for all such d, A347381(d) <> 1. And neither such divisor d can be a term of A336702, because 3*x is odd, therefore we must have A347381(d) > 1 for all nontrivial unitary divisors d of such a hypothetical x.
Any odd term of A000396, i.e., an odd perfect number, if such a hypothetical number exists, must also be a term of this sequence, by reasoning similar to above. See also illustration in A347392.

			189 is a term, because A347381(189) = 1, and the only way to factor 189 into nontrivial unitary divisors is 7*27, and A347381(7) = A347381(27) = 3 > 1.
63767 = 11^2 * 17 * 31 is a term, as its nontrivial unitary divisors are [17, 31, 121, 527, 2057, 3751], at which points A347381 obtains values [6, 10, 5, 11, 6, 8], every one which is larger than A347381(63767) = 3.

Cf. A000396, A005820, A005940, A034444, A336702, A347381, A347391, A347392.

Subsequence of A347390, which is a subsequence of A347384.

isA347383(n) = if((1==n)||!(n%2)||isprimepower(n),0,my(w=A347381(n)); fordiv(n,d,if((d>1)&&(dA347381(d)<=w), return(0))); (1));

a(29)-a(31) from Jinyuan Wang, Jul 09 2025

A348738 Numbers k for which A326042(k) < k, where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Claim: If there is an odd term y of A336702 larger than one, and it is the least one of such terms, then it should satisfy condition that for all nontrivial unitary divisor pairs d and x/d of x = A064989(y) [with gcd(d,x/d) = 1, 1 < d < x], the other divisor should reside in this sequence, and the other divisor in A348739. Proof: Applying A064989 to the odd terms of A336702 gives the fixed points of A326042. Suppose there are other odd terms in A336702 in addition to its initial 1, and let y be the least of these odd terms > 1 and x = A064989(y). Because A326042 (from here on indicated with f) is multiplicative, it follows that if we take any two nontrivial unitary divisors a and b of x, with x = a*b, gcd(a,b) = 1, 1 < a,b < x, then f(a)*f(b) = f(x) = x. Because f(x)/x = 1, we must have f(a)/a * f(b)/b = 1, as also the ratio f(n)/n is multiplicative. But f(a)/a and f(b)/b cannot be equal to 1, because then a and b would also be fixed by f, which contradicts our assumption that x were the least such fixed point larger than one. Therefore f(a) < a and f(b) > b, or vice versa. See also the comments in A348930, A348933.
Moreover, all odd perfect numbers (a subsequence of A336702), if such numbers exist, should also satisfy the same condition, regardless of whether they are the least of such numbers or not, because having a non-deficient proper divisor will push the abundancy index (ratio sigma(n)/n) of any number over 2. That is, for any such pair of nontrivial unitary divisors d and x/d, both A003961(d) and A003961(x/d) should be deficient, i.e., neither one should be in A337386. See also the condition given in A347383.
Terms that occur also in A337386 are: 120, 240, 360, 420, 480, 504, 540, 600, 630, ...

Positions of positive terms in A348736, positions of 1's in A348737 (characteristic function).
Almost complement of A348739.
Cf. A000203, A003961, A003973, A006039, A027687, A064989, A326042, A336702, A337386, A347383, A348741, A348748 (corresponding odd numbers).
Subsequences: A000040, A374464 (after its initial 1).
Cf. also A348930, A348933.

f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := NextPrime[p]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; Select[Range[100], s1[DivisorSigma[1, s2[#]]] < # &] (* Amiram Eldar, Nov 04 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
isA348738(n) = (A326042(n)

A353750 a(n) = phi(sigma(n)) * A064989(sigma(n)), where A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

1, 4, 2, 30, 4, 8, 4, 48, 132, 24, 8, 60, 30, 16, 16, 870, 24, 528, 24, 120, 16, 48, 16, 96, 870, 120, 48, 120, 48, 96, 16, 720, 32, 144, 32, 3960, 306, 96, 120, 288, 120, 64, 140, 240, 528, 96, 32, 1740, 1224, 3480, 96, 1050, 144, 192, 96, 192, 96, 288, 96, 480, 870, 64, 528, 14238, 240, 192, 416, 720, 64, 192, 96

Offset: 1

Antti Karttunen, May 07 2022

nonn

In contrast to A353749, this is not multiplicative, except on positions given by A336547.

It seems that a(n) = A353749(n) only on n=1. This would then imply that the intersection of A006872 and A336702 = {1}.

Cf. A000010, A000203, A006872, A062401, A064989, A336547, A336548, A336549, A336702, A350073, A353747, A353748, A353749.
Cf. A353757, A353758 (where a(n) < A353749(n)), A353759 (where a(n) >= A353749(n)), A353760, A353790 [= a(A003961(n))].
Cf. also A353792.

A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353750(n) = { my(s=sigma(n)); (eulerphi(s)*A064989(s)); };

a(n) = A353749(A000203(n)) = A062401(n) * A350073(n).

a(n) = A353749(n) + A353757(n).

Dubious comment deleted by Antti Karttunen, Jan 26 2023

A065997 Numbers n such that sigma(n) / n is prime.

Original entry on oeis.org

6, 28, 120, 496, 672, 8128, 523776, 33550336, 459818240, 1476304896, 8589869056, 14182439040, 31998395520, 51001180160, 137438691328, 518666803200, 13661860101120, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 2305843008139952128

Offset: 1

Joseph L. Pe, Dec 10 2001

This is a subsequence of the sequence of multiply perfect numbers A007691.
The prime values of sigma(n) / n are A219545.
Numbers whose abundancy index is a prime. There are two visible bends (sudden changes in the growth rate) in the scatter plot. Compare also to the scatter plot of A336702. - Antti Karttunen, Feb 25 2022

Antti Karttunen, Table of n, a(n) for n = 1..597 (computed from the b-file of A007691 prepared by T. D. Noe, using Flammenkamp's data)

Subsequence of A007691 and of A342924.
Cf. A000396, A005820, A046060 (subsequences).
Cf. A219545, A336702.

isA065997(n) = { my(p=sigma(n)/n); (1==denominator(p) && isprime(p)); }; \\ Antti Karttunen, Feb 25 2022

Terms a(10) to a(14) from Jonathan Sondow, Nov 22 2012

Extended by T. D. Noe, Nov 26 2012

A347392 Numbers k such that nearest common ancestor of k and sigma(k) in Doudna tree (A347879) is the grandparent of k.

Original entry on oeis.org

8, 9, 12, 13, 24, 35, 160, 455, 42550, 127650, 8041950, 22469750, 58506250, 67409250, 175518750, 394055550, 4246782750

Offset: 1

Antti Karttunen, Aug 30 2021

Note how 13 * 35 = 455.
If there exists any odd perfect numbers x, with sigma(x) = 2x, then 2*x would be a term of this sequence, as then sigma(2*x) = 6*x would be situated as a descendant under the other branch of the grandparent of 2*x (a parent of x), which is m = A064989(x), with m in A005101. Opn x itself would be a term of A336702. Furthermore, if such x is not a multiple of 3 (in which case m is odd and in A005231), then also 3x would be a term of this sequence as sigma(3*x) = 4*sigma(x) = 8*x would be situated as a grandchild of 2x, with 2x being a first cousin of 3x. Also, in that case, 6*x would be located in A336702 (particularly, in A027687) because then sigma(6*x) = 12*sigma(x) = 24*x = 4*(6*x).
.
                <--A003961--  m  ---(*2)--->
              .............../ \...............
             /                                 \
            /                                   \
           /                                     \
          x                                       2m
  etc..../ \......2x = sigma(x)          3x....../ \......4m
                  / \                     / \            / \
               etc.  \                 etc.  \        etc.  etc.
                      \                       \
                       4x          sigma(2x) = 6x
                      / \                     / \
                  etc    \                 etc.  \
                          \                       \
                          8x = sigma(3x)          12x
                             if m odd               \
                                                     \
                                                    24x = sigma(6x) if m odd.
.
Furthermore, if there were any hypothetical odd terms y in A005820 (triperfect numbers), then 2y would be a term of this sequence. See the diagram in A347391.
If it exists, a(18) > 2^33.

			455 is included in the sequence as sigma(455) = 672, and the nearest common ancestor of 455 and 672 in Doudna tree is 42, which is the grandparent of 455 [as 455 = A003961(A003961(42))] and the grand-grand-grand-parent of 672 [as 672 = (2^4)*42].

Positions of 2's in A347381.
Cf. A000203, A005101, A005231, A005820, A005940, A027687, A336702, A347391, A347393, A347394, A347879, A348041.
Cf. also A336834, A353365.

PARI
```
isA347392(n) = (2==A347381(n));
```

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
isA347391(n) = if(1==n,0,my(m=A252463(n), s=sigma(n)); while(s>m, if(s==n, return(0)); s = A252463(s)); (s==m));
isA347391_or_A347392(n) = if(1==n,0,my(m=A252463(A252463(n)), s=sigma(n)); while(s>m, if(s==n, return(0)); s = A252463(s)); (s==m));
isA347392(n) = (isA347391_or_A347392(n) && !isA347391(n));

A351551 Numbers k such that the largest unitary divisor of sigma(k) that is coprime with A003961(k) is also a unitary divisor of k.

Original entry on oeis.org

1, 2, 10, 34, 106, 120, 216, 260, 340, 408, 440, 580, 672, 696, 820, 1060, 1272, 1666, 1780, 1940, 2136, 2340, 2464, 3320, 3576, 3960, 4280, 4536, 5280, 5380, 5860, 6456, 6960, 7520, 8746, 8840, 9120, 9632, 10040, 10776, 12528, 12640, 13464, 14560, 16180, 16660, 17400, 17620, 19040, 19416, 19992, 21320, 22176, 22968

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Numbers k for which A351546(k) is a unitary divisor of k.
The condition guarantees that A351555(k) = 0, therefore this is a subsequence of A351554.
The condition is also a necessary condition for A349745, therefore it is a subsequence of this sequence.
All six known 3-perfect numbers (A005820) are included in this sequence.
All 65 known 5-multiperfects (A046060) are included in this sequence.
Not all multiperfects (A007691) are present (only 587 of the first 1600 are), but all 23 known terms of A323653 are terms, while none of the (even) terms of A046061 or A336702 are.

			For n = 672 = 2^5 * 3^1 * 7^1, and the largest unitary divisor of the sigma(672) [= 2^5 * 3^2 * 7^1] coprime with A003961(672) [= 13365 = 3^5 * 5^1 * 11^1] is 2^5 * 7^1 = 224, therefore A351546(672) is a unitary divisor of 672, and 672 is included in this sequence.

Cf. A000203, A000396, A003961, A007691, A046061, A065997, A336702, A351546, A351555, A353633 (characteristic function).
Subsequence of A351552 and of A351554.
Cf. A349745, A351550 (subsequences), A005820, A046060, A323653 (very likely subsequences).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351546(n) = { my(f=factor(sigma(n)),u=A003961(n)); prod(k=1,#f~,f[k,1]^((0!=(u%f[k,1]))*f[k,2])); };
isA351551(n) =  { my(u=A351546(n)); (!(n%u) && 1==gcd(u,n/u)); };

A351446 Numbers k for which A003958(sigma(k)) = A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 6, 10, 26, 28, 49, 54, 74, 122, 126, 146, 294, 314, 386, 408, 490, 496, 554, 626, 680, 794, 842, 914, 1082, 1226, 1232, 1274, 1322, 1346, 1466, 1514, 1560, 1754, 1768, 1994, 2186, 2306, 2402, 2426, 2474, 2642, 2646, 2762, 2906, 3242, 3314, 3360, 3506, 3626, 3672, 3746, 3808, 3866, 3986, 4034

Offset: 1

Antti Karttunen, Feb 12 2022

nonn

Numbers k for which A351442(k) = A003958(k), or equally, for which k = A351444(k) = A322582(k) + A351442(k).

Index entries for sequences related to sigma(n)

Cf. A000203, A003958, A322582, A351442, A351447.
Fixed points of A351444, positions of zeros in A351445.
Subsequences: A000396, A351443 (odd terms), A351440, A336702 (numbers k for which A064989(sigma(k)) = A064989(k)).

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA351446(n) = (A003958(sigma(n))==A003958(n));

A348742 Odd numbers k for which A161942(k) >= k, where A161942 is the odd part of sigma.

Original entry on oeis.org

1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2205, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569, 7921, 8281, 8649, 9025, 9409, 9801, 10201, 10609, 11025, 11449, 11881, 12321

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

All odd squares (A016754) are present, but not all terms are squares. A348743 gives the nonsquare terms.

Odd terms of A336702 form a subsequence. Also all odd terms of A005820 would be present here, as well as any hypothetical quasi-perfect numbers (see comments and references in A332223, A336700), both in A016754. - Antti Karttunen, Nov 28 2024

Union of A016754 and A348743.
Cf. A161942, A162284 (subsequence), A336702, A348741 (complement among the odd numbers).
Cf. also A005820, A332223, A336700, A348739.

q:= n-> (t-> is(t/2^padic[ordp](t,2)>=n))(numtheory[sigma](n)):
select(q, [2*i-1$i=1..10000])[];  # Alois P. Heinz, Nov 28 2024

odd[n_] := n/2^IntegerExponent[n, 2]; Select[Range[1, 10^4, 2], odd[DivisorSigma[1, #]] >= # &] (* Amiram Eldar, Nov 02 2021, edited (because of the changed definition) by Antti Karttunen, Nov 28 2024 *)

A000265(n) = (n >> valuation(n, 2));
isA348742(n) = ((n%2)&&A000265(sigma(n))>=n); \\ revised by Antti Karttunen, Nov 28 2024

a(1) = 1 inserted as the initial term, because of the changed definition (from > to >=) - Antti Karttunen, Nov 28 2024

A349174 Odd numbers k for which gcd(k, A003961(k)) is equal to gcd(sigma(k), A003961(k)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 63, 67, 69, 71, 73, 79, 81, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 167, 169

Offset: 1

Antti Karttunen, Nov 10 2021

nonn

Odd numbers k for which A322361(k) = A342671(k).
Odd numbers k for which A348994(k) = A349161(k).
Odd numbers k such that A319626(k) = A349164(k).
Odd terms of A336702 form a subsequence of this sequence. See also A349169.
Ratio of odd numbers residing in this sequence, vs. in A349175 seems to slowly decrease, but still apparently stays > 2 for a long time. E.g., for range 2 .. 2^28, it is 95302074/38915653 = 2.4489...

Cf. A000203, A003961, A319626, A322361, A336702, A342671, A348994, A349161, A349164, A349169.
Cf. A349175 (complement among the odd numbers).
Union of A349176 and A349177.

Select[Range[1, 169, 2], GCD[#1, #3] == GCD[#2, #3] & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349174(n) = if(!(n%2),0,my(u=A003961(n)); gcd(u,sigma(n))==gcd(u,n));

A216782 Numbers such that numerator(sigma(n)/n) is even and denominator(sigma(n)/n) is odd.

Original entry on oeis.org

3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 21, 22, 23, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 51, 53, 54, 55, 57, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 99, 101, 102

Offset: 1

Michel Marcus, Sep 16 2012

a(n) contains odd primes (A065091), odd squarefree semiprimes (A046388), perfect numbers (A000396), and 2n-multiperfect (A027687, A046061).

			sigma(3)/3 = 4/3 (even/odd).

Vincenzo Librandi, Table of n, a(n) for n = 1..8000

Cf. A216780, A216781, A324903 (characteristic function).

Subsequences: A000396, A027687, A043305 (without its initial 1), A046061, A046388, A065091, A336702 (without its initial 1).

Select[Range[1000], EvenQ[Numerator[DivisorSigma[1, #] / # ]] && OddQ[Denominator[DivisorSigma[1, #]/#]]&] (* Vincenzo Librandi, Jun 24 2014 *)
nedoQ[n_]:=Module[{ds=DivisorSigma[1,n]/n},EvenQ[Numerator[ds]]&&OddQ[ Denominator[ ds]]]; Select[Range[200],nedoQ] (* Harvey P. Dale, Feb 28 2015 *)

eoab(n) = {for (i=1, n, ab = sigma(i)/i; if ((numerator(ab) % 2 == 0) && (denominator(ab) % 2 == 1), print1(i, ", ")););}

cp's OEIS Frontend

A347383 Odd composites k, not powers of primes, such that for all their nontrivial unitary divisors d it holds that A347381(d) > A347381(k).

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A348738 Numbers k for which A326042(k) < k, where A326042(n) = A064989(sigma(A003961(n))).

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A353750 a(n) = phi(sigma(n)) * A064989(sigma(n)), where A064989 shifts the prime factorization one step towards lower primes.

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A065997 Numbers n such that sigma(n) / n is prime.

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A347392 Numbers k such that nearest common ancestor of k and sigma(k) in Doudna tree (A347879) is the grandparent of k.

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A351551 Numbers k such that the largest unitary divisor of sigma(k) that is coprime with A003961(k) is also a unitary divisor of k.

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A351446 Numbers k for which A003958(sigma(k)) = A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

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A348742 Odd numbers k for which A161942(k) >= k, where A161942 is the odd part of sigma.

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