A020487 -id:A020487 - cp's OEIS frontend

A176799 a(n) = possible values of A176797(m) in increasing order, where A176797(m) = antiharmonic means of divisors of antiharmonic numbers A020487.

1, 3, 7, 11, 13, 21, 35, 43, 61, 63, 77, 85, 91, 111, 119, 129, 147, 157, 171, 183, 185, 231, 245, 255, 273, 301, 313, 333, 343, 425, 441, 455, 471, 473, 481, 507, 521, 547, 559, 629, 671, 683, 741, 765, 777, 793, 813, 819, 833, 841, 845, 903, 931, 935, 1015, 1029, 1099, 1105, 1183, 1197, 1221

Offset: 1

Jaroslav Krizek, Apr 26 2010

nonn

From Robert Israel, Sep 05 2024: (Start)
According to A000203, sigma_1(m) <= (6/Pi^2)*m^(3/2) for m >= 12. Thus since sigma_2(m) > m^2, sigma_2(m)/sigma_1(m) > (Pi^2/6) * m^(1/2).
This would suggest that to find all terms <= K of this sequence we should look at sigma_2(m)/sigma_1(m) for m <= 36 * K^2/Pi^4.  But using the b-file for A004394 we may get a good upper bound for sigma_1(m)/m for m in this interval, resulting in a much smaller search region. (End)

Robert Israel, Table of n, a(n) for n = 1..1009

Cf. A000203, A004394, A020487, A176797, A327054.

# This uses the b-file for A004394
K:= 10000: # to get terms <= K
M:= 36 * K^2/Pi^4:
for i from 1 while A004394[i] < M do od:
r:= numtheory:-sigma(A004394[i])/A004394[i]:
V:= Vector(K):
for m from 1 to r*K do
  F:= numtheory:-divisors(m);
v:= add(d^2, d=F)/add(d,d=F);
if v::integer and v <= K and V[v] = 0 then V[v]:= m fi;
od:
select(v -> V[v] > 0, [$1..K]); # Robert Israel, Sep 05 2024

More terms from Robert Israel, Sep 05 2024

A328952 Arithmetic numbers (A003601) that are not antiharmonic (A020487).

Original entry on oeis.org

3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102

Offset: 1

Jaroslav Krizek, Nov 17 2019

nonn

Numbers m such that the arithmetic mean of the divisors of m is an integer but the antiharmonic mean of the divisors of m is not an integer.
Numbers m such that A(m) = A000203(m) / A000005(m) is an integer but B(m) = A001157(m) / A000203(m) is not an integer.
Corresponding values of A(m): 2, 3, 3, 4, 6, 7, 6, 6, 9, 10, 8, 9, 12, 10, 15, 9, 16, 12, 12, 19, 15, 14, 21, 12, 22, ...
Corresponding values of B(m): 5/2, 13/3, 25/6, 25/4, 61/6, 85/7, 125/12, 65/6, 145/9, 181/10, 125/8, ...

Complement of A277553 with respect to A003601.

Cf. A000005, A000203, A001157, A328953, A328954.

[m: m in [1..10^5] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and not IsIntegral(&+[d^2: d in Divisors(m)] / SumOfDivisors(m))]

Select[Range[100], Divisible[(sigma = DivisorSigma[1, #]), DivisorSigma[0, #]] && !Divisible[DivisorSigma[2, #], sigma] &]  (* Amiram Eldar, Nov 17 2019 *)

isok(m) = !(sigma(m) % numdiv(m)) && (sigma(m,2) % sigma(m)); \\ Michel Marcus, Nov 18 2019

A328953 Antiharmonic numbers (A020487) that are not arithmetic (A003601).

Original entry on oeis.org

4, 9, 16, 25, 36, 50, 64, 81, 100, 117, 121, 144, 180, 196, 200, 225, 242, 256, 289, 324, 325, 400, 441, 450, 468, 484, 529, 576, 578, 625, 650, 676, 729, 784, 800, 841, 900, 968, 1024, 1058, 1089, 1156, 1225, 1280, 1296, 1300, 1444, 1476, 1521, 1600, 1620

Offset: 1

Jaroslav Krizek, Nov 17 2019

nonn

Numbers m such that the antiharmonic mean of the divisors of m is an integer but the arithmetic mean of the divisors of m is not an integer.
Numbers m such that B(m) = A001157(m) / A000203(m) is an integer but A(m) = A000203(m) / A000005(m) is not an integer.
Corresponding values of B(m): 3, 7, 11, 21, 21, 35, 43, 61, 63, 85, 111, 77, 91, 129, 119, 147, 185, 171, 273, 183, ...
Corresponding values of A(m): 7/3, 13/3, 31/5, 31/3, 91/9, 31/2, 127/7, 121/5, 217/9, 91/3, 133/3, ...

Complement of A277553 with respect to A020487.

Cf. A000005, A000203, A001157, A003601, A328952, A328954.

[m: m in [1..10^5] | not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(&+[d^2: d in Divisors(m)] / SumOfDivisors(m))]

Select[Range[1620], !Divisible[(sigma = DivisorSigma[1, #]), DivisorSigma[0, #]] && Divisible[DivisorSigma[2, #], sigma] &] (* Amiram Eldar, Nov 17 2019 *)

A328954 Numbers m that are neither arithmetic (A003601) nor antiharmonic (A020487).

Original entry on oeis.org

2, 8, 10, 12, 18, 24, 26, 28, 32, 34, 40, 48, 52, 58, 63, 72, 74, 75, 76, 80, 82, 84, 88, 90, 98, 104, 106, 108, 112, 120, 122, 124, 128, 130, 136, 146, 148, 152, 156, 160, 162, 170, 171, 172, 175, 176, 178, 192, 194, 202, 208, 216, 218, 226, 228, 232, 234

Offset: 1

Jaroslav Krizek, Dec 03 2019

nonn

Numbers m such that neither the arithmetic mean of the divisors of m nor the antiharmonic mean of the divisors of m is an integer.
Numbers m such that neither A(m) = A000203(m)/A000005(m) nor B(m) = A001157(m)/A000203(m) is an integer.
Corresponding values of A(m): 3/2, 15/4, 9/2, 14/3, 13/2, 15/2, 21/2, 28/3, 21/2, 27/2, 45/4, 62/5, ...
Corresponding values of B(m): 5/3, 17/3, 65/9, 15/2, 35/3, 85/6, 425/21, 75/4, 65/3, 725/27, 221/9, ...

Cf. A000005, A000203, A001157, A003601, A277553, A328952, A328953.

[m: m in [1..10^5] | not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and not IsIntegral(&+[d^2: d in Divisors(m)] / SumOfDivisors(m))]

Select[Range[235], !Divisible[DivisorSigma[2, #], (s = DivisorSigma[1, #])] && !Divisible[s, DivisorSigma[0, #]] &] (* Amiram Eldar, Dec 06 2019 *)

A176800 a(n) = all values of A176797(m) in increasing order, where A176797(m) = antiharmonic means of divisors of antiharmonic numbers A020487.

Original entry on oeis.org

1, 3, 7, 11, 13, 21, 21, 35, 43, 43, 61, 63, 77, 85, 91, 111, 119, 129, 147, 157, 171, 183, 185, 231, 245, 255, 255, 273, 301, 301

Offset: 1

Jaroslav Krizek, Apr 26 2010

nonn

A368215 a(n) is the smallest number k >= 1 that has exactly n divisors in A020487.

Original entry on oeis.org

1, 4, 16, 36, 256, 100, 200, 576, 400, 800, 2600, 900, 3200, 1800, 16900, 6400, 12800, 3600, 20800, 7200, 11700, 36000, 67600, 14400, 23400, 28800, 32400, 88200, 397800, 64800, 270400, 46800, 152100, 115200, 234000, 93600, 1258400, 230400, 259200, 352800, 1081600

Offset: 1

Marius A. Burtea, Dec 17 2023

nonn

a(n) exists for each n because 4^(n-1) has n antiharmonic divisors.

			a(1) = 1 because 1 has only one divisor 1 = A020487(1).
The numbers 2 and 3 have only the divisor 1 in A020487 and 4 has the divisors 1 = A020487(1) and 4 = A020487(2), so a(2) = 4.

Cf. A000005, A000203, A001157, A020487, A337326.

f:=func; a:=[]; for n in [1..41] do k:=1; while #[d:d in Divisors(k)|f(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

f[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[2, #], DivisorSigma[1, #]] &]; seq[len_] := Module[{s = Table[0, {len}], c = 0, n = 1}, While[c < len, If[(i = f[n]) <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[25] (* Amiram Eldar, Dec 18 2023 *)

a(n) = my(k=1); while(sumdiv(k, d, sigma(d, 2)%sigma(d)==0) != n, k++); k; \\ Michel Marcus, Dec 18 2023

A369384 The smallest number k that can be partitioned in n ways as the sum of two numbers from A020487.

Original entry on oeis.org

1, 2, 29, 181, 442, 425, 850, 1300, 2600, 3250, 5525, 11050, 17425, 16900, 44100, 18850, 72250, 44200, 122525, 75400, 55250, 110500, 237250, 188500, 266500, 397800, 375700, 377000, 187850, 221000, 469625, 718250, 640900, 1105000, 1812200, 2340900, 751400, 3591250

Offset: 0

Marius A. Burtea, Jan 25 2024

nonn

			a(0) = 1 because 1 cannot be written as the sum of two terms in A020487.
2 = 1 + 1 = A020487(1) + A020487(1), so a(1) = 2.
The numbers 3, 4, ..., 28 can be written as the sum of two terms in A020487 in at most one way and 29 = 4 + 25 = A020487(2) + A020487(6) and 29 = 9 + 20 = A020487(3) + A020487(5), so a(2) = 29.

Michael S. Branicky, Table of n, a(n) for n = 0..106

Cf. A000203, A001157, A020487.

ant:=func; b:=[n: n in [1..700000] |ant(n)]; a:=[]; for n in [0..30] do k:=1; while #RestrictedPartitions(k,2,Set(b)) ne n do k:=k+1; end while; Append(~a,k); end for; a;

a(16) corrected and more terms from Michael S. Branicky, Feb 24 2024

A335389 Numbers k such that k and k+1 are both antiharmonic numbers (A020487).

Original entry on oeis.org

49, 324, 1024, 1444, 1681, 2600, 9800, 265225, 332928, 379456, 421200, 1940449, 4198400, 4293184, 4739328, 8346320, 11309768, 27050400, 65918161, 203694425, 384199200, 418488849, 546717924, 2239277041, 2687489280, 4866742025, 5783450400, 6933892900, 7725003664

Offset: 1

Amiram Eldar, Jun 04 2020

nonn

Terms of this sequence k such that k and k+1 are both nonsquares (A227771) are 203694425, 4866742025, ...

Can two consecutive numbers be both primitive antiharmonic numbers (A228023)? Numbers k such that k and k+2 are both primitive antiharmonic numbers exist - the first two are 38246258 and 344321280.

			49 is a term since both 49 and 50 are antiharmonic: sigma_2(49)/sigma(49) = 43 and sigma_2(50)/sigma(50) = 35 are both integers.

Cf. A000203, A001157, A020487, A227771, A228023.

antihQ[n_] := Divisible[DivisorSigma[2, n], DivisorSigma[1, n]]; seq = {}; q1 = antihQ[1];  Do[q2 = antihQ[n]; If[q1 && q2, AppendTo[seq, n - 1]]; q1 = q2, {n, 2, 2 * 10^6}]; seq

A368216 Number of divisors of n that are antiharmonic numbers (A020487).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Marius A. Burtea, Jan 15 2024

nonn

Differs from A046951 for n = 20, 40, 50, 60, 80, ....

			a(1) = 1 because 1 has only one divisor 1 = A020487(1) antiharmonic number.
a(4) = 2 because 4 has divisors 1 = A020487(1) and 4 = A020487(2), antiharmonic numbers.

Cf. A005117, A020487, A046951, A368215.

f:=func; [#[d:d in Divisors(k)|f(d)]:k in [1..100]];

a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[2, #], DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, Jan 21 2024 *)

a(p^k) = floor((k + 2)/2), p prime, k >= 1.
a(p*q) = 1, for p, q prime, p <> q.
a(A005117(k)) = 1, k >= 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A020487(k) = 1.784... . - Amiram Eldar, Jan 26 2024

A377383 Numbers k in A020487 with arithmetic derivative k' (A003415) in A020487.

Original entry on oeis.org

4, 256, 500, 625, 2500, 4225, 11664, 12800, 14580, 81920, 250000, 262144, 364500, 531441, 800000, 2125764, 4734976, 11943936, 27541504, 64000000, 84050000, 107868672, 156250000, 162542848, 195312500, 253472000, 512635136, 544195584, 607642880, 701146368, 770786560

Offset: 1

Marius A. Burtea, Dec 05 2024

nonn

Numbers of the form m = 2^(2^(2*k - 1)) are terms. Indeed, m is a square, so it is a term in A020487, and m' = 2^(2*k - 1)*2^(2^(2*k - 1) - 1) = 2^(2^( 2*k - 1) +2*k- 2) is also a square, so it is in A020487.

			4' = 4 = A020487(2), so 4 is a term.
256 = A020487(22), 256' = 1024 = A020487(48), so 256 is a term.

Cf. A000203, A001157, A020487, A003415.

f:=func; ant:=func; [n:n in [2..100000]|ant(n) and ant(Floor(f(n)))];

ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ahQ[n_] := Divisible[DivisorSigma[2, n], DivisorSigma[1, n]]; Select[Range[2, 10^6], ahQ[#] && ahQ[ad[#]] &] (* Amiram Eldar, Dec 11 2024 *)

cp's OEIS Frontend

A176799 a(n) = possible values of A176797(m) in increasing order, where A176797(m) = antiharmonic means of divisors of antiharmonic numbers A020487.

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A328952 Arithmetic numbers (A003601) that are not antiharmonic (A020487).

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A328953 Antiharmonic numbers (A020487) that are not arithmetic (A003601).

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Magma

Mathematica

A328954 Numbers m that are neither arithmetic (A003601) nor antiharmonic (A020487).

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Magma

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A176800 a(n) = all values of A176797(m) in increasing order, where A176797(m) = antiharmonic means of divisors of antiharmonic numbers A020487.

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A368215 a(n) is the smallest number k >= 1 that has exactly n divisors in A020487.

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Magma

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PARI

A369384 The smallest number k that can be partitioned in n ways as the sum of two numbers from A020487.

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Magma

Extensions

A335389 Numbers k such that k and k+1 are both antiharmonic numbers (A020487).

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Mathematica

A368216 Number of divisors of n that are antiharmonic numbers (A020487).

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