A380845 -id:A380845 - cp's OEIS frontend

A383366 Smallest of a sociable triple i < j < k such that j = s(i), k = s(j), and i = s(k), where s(k) = A380845(k) - k is the sum of aliquot divisors of k that have the same binary weight as k.

4400700, 12963816, 29878920, 38353800, 44973480, 51894304, 52208520, 67849656, 73134432, 81685080, 100711656, 103759848, 105096096, 113044896, 113161320, 114608032, 128639034, 135465912, 135559080, 136786200, 139242740, 148758120, 156686088, 159628350, 171090416

Offset: 1

Amiram Eldar, Apr 24 2025

			4400700 is a term since s(4400700) = 4840770, s(4840770) = 5456868, and s(5456868) = 4400700.

Amiram Eldar, Table of n, a(n) for n = 1..4400
Amiram Eldar, Table of n, a(n), s(a(n)), s(s(a(n))) for n = 1..4400.

Cf. A380845, A380846, A380849, A380850.

f[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, # &, # < n && DigitCount[#, 2, 1] == h &]]; q[k_] := Module[{k1 = f[k], k2}, If[k1 <= k, False, k2 = f[k1]; k2 > k && f[k2] == k]]; Select[Range[13000000], q]

f(n) = {my(h = hammingweight(n)); sumdiv(n, d, d * (d < n && hammingweight(d) == h)); }
isok(k) = {my(k1 = f(k), k2); if(k1 <= k, 0, k2 = f(k1); k2 > k && f(k2) == k);}

A380844 The number of divisors of n that have the same binary weight as n.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 2, 2, 1, 3, 1, 2, 1, 5, 1, 4, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 2, 2, 2, 6, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 5, 2, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 2, 4, 1, 3, 1, 4, 1, 8, 1, 2, 2, 3, 1, 2, 1, 5, 1, 2, 1, 6, 1, 2, 1

Offset: 1

Amiram Eldar, Feb 05 2025

First differs from A325565 at n = 133: a(133) = 3 while A325565(133) = 2.

The sum of these divisors is A380845(n).

			a(6) = 2 because 6 = 110_2 has binary weight 2, and 2 of its divisors, 3 = 11_2 and 6, have the same binary weight.

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

Cf. A000005, A000043, A000120, A000265, A000396, A007814, A324392, A325565, A380845.

a[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, 1 &, DigitCount[#, 2, 1] == h &]]; Array[a, 100]

a(n) = {my(h = hammingweight(n)); sumdiv(n, d, hammingweight(d) == h);}

a(n) = Sum_{d|n} [A000120(d) = A000120(n)], where [ ] is the Iverson bracket.
a(2^n) = n+1.
a(n) <= A000005(n) with equality if and only if n is a power of 2.
a(n) = a(A000265(n)) * (A007814(n)+1), or equivalently, a(k*2^n) = a(k)*(n+1) for k odd and n >= 0.
In particular, since a(p) = 1 for an odd prime p, a(p*2^n) = n+1 for an odd prime p and n >= 0.
a(A000396(n)) = A000043(n), assuming that odd perfect numbers do no exist.

A381071 Numbers k such that the sum of the proper divisors of k that have the same binary weight as k is larger than k, and no subset of these divisors sums to k.

Original entry on oeis.org

1050, 3150, 4284, 4410, 5148, 6292, 6790, 7176, 8890, 10764, 17850, 18648, 19000, 19530, 32886, 33072, 33150, 35088, 35530, 35720, 35770, 38850, 41360, 43164, 45084, 49368, 49764, 50456, 50730, 52884, 54280, 54340, 58410, 58696, 59010, 59408, 63492, 66010, 68376

Offset: 1

Amiram Eldar, Feb 13 2025

Analogous to weird numbers (A006037), as A380846 is analogous to perfect numbers (A000396).

Amiram Eldar, Table of n, a(n) for n = 1..1724

Cf. A000396, A380845, A380846.
Subsequence of A380929.
A381072 is a subsequence.
Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948, A339939, A348525, A348631, A349285, A364862.

divs[n_] := Module[{hw = DigitCount[n, 2, 1]}, Select[Divisors[n], DigitCount[#, 2, 1] == hw &]];
weirdQ[n_, d_, s1_, m1_] :=  weirdQ[n, d, s1, m1] = Module[{s = s1, m = m1}, If[m == 0, False, While[m > 0 && d[[m]] > n, s -= d[[m]]; m--]; If[m == 0, True, d[[m]] < n && If[s > n, weirdQ[n - d[[m]], d, s - d[[m]], m - 1] && weirdQ[n, d, s - d[[m]], m - 1], s < n && m < Length[d] - 1]]]];
q[n_] := Module[{d = divs[n], s, m}, s = Total[d] - n; m = Length[d] - 1; weirdQ[n, d, s, m]]; Select[Range[70000], q] (* based on a Pari code by M. F. Hasler at A006037 *)

divs(n) = {my(h = hammingweight(n)); select(x -> hammingweight(x)==h, divisors(n));}
is(n, d = divs(n), s = vecsum(d)-n, m = #d-1) = {if(m == 0, return(0)); while(m > 0 && d[m] > n, s -= d[m]; m--); if(m==0, return(1)); (d[m] < n &&
if(s > n, is(n-d[m], d, s-d[m], m-1) && is(n, d, s-d[m], m-1), s < n && m < #d-1));} \\ based on a code by M. F. Hasler at A006037

A381073 Numbers k such that k and k+2 are both terms in A380846.

Original entry on oeis.org

8596, 9772, 10444, 17836, 19626, 21196, 23716, 27186, 35754, 36484, 38164, 42700, 45892, 54796, 56586, 85708, 91252, 98586, 100770, 104970, 112698, 132412, 136612, 139074, 140980, 141652, 144676, 149716, 152850, 165172, 166122, 171724, 182032, 182644, 184770, 190482

Offset: 1

Amiram Eldar, Feb 13 2025

Numbers k such that A380845(k) = 2*k and A380845(k+2) = 2*(k+2).

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

Subsequence of A380846.
A381074 is a subsequence.
Cf. A380845.

f[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, # &, DigitCount[#, 2, 1] == h &] == 2*n]; seq[lim_] := Module[{q = Table[False, {4}], s = {}}, q[[1 ;; 2]] = f /@ Range[2]; Do[q[[3 ;; 4]] = f /@ Range[k, k + 1]; If[q[[1]] && q[[3]], AppendTo[s, k - 2]]; If[q[[2]] && q[[4]], AppendTo[s, k - 1]]; q[[1 ;; 2]] = q[[3 ;; 4]], {k, 3, lim, 2}]; s]; seq[50000]

is1(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) == 2*k;}
list(lim) = {my(q1 = is1(1), q2 = is1(2), q3, q4); forstep(k = 3, lim, 2, q3 = is1(k); q4 = is1(k+1); if(q1 && q3, print1(k-2, ", ")); if(q2 && q4, print1(k-1, ", ")); q1 = q3; q2 = q4);}

A381074 Numbers k such that k, k+2 and k+4 are all terms in A380846.

Original entry on oeis.org

10820236, 24069388, 27802288, 39297580, 50717488, 56362960, 73070224, 97339504, 103605964, 112209580, 112526032, 140053564, 145315600, 155790124, 156415084, 158877232, 184667248, 185979664, 188913004, 189225484, 189541936, 224435536, 281740396, 292406380, 314388112

Offset: 1

Amiram Eldar, Feb 13 2025

Numbers k such that A380845(k) = 2*k, A380845(k+2) = 2*(k+2), and A380845(k+4) = 2*(k+4).

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

Subsequence of A380846 and A381073.

Cf. A380845.

f[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, # &, DigitCount[#, 2, 1] == h &] == 2*n]; seq[lim_] := Module[{q = Table[False, {6}], s = {}}, q[[1 ;; 4]] = f /@ Range[4]; Do[q[[5 ;; 6]] = f /@ Range[k, k + 1]; If[q[[1]] && q[[3]] && q[[5]], AppendTo[s, k - 4]]; If[q[[2]] && q[[4]] && q[[6]], AppendTo[s, k - 3]]; q[[1 ;; 4]] = q[[3 ;; 6]], {k, 5, lim, 2}]; s]; seq[11000000]

is1(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) == 2*k;}
list(lim) = {my(q1 = is1(1), q2 = is1(2), q3 = is1(3), q4 = is1(4), q5, q6); forstep(k = 5, lim, 2, q5 = is1(k); q6 = is1(k+1); if(q1 && q3 && q5, print1(k-4, ", ")); if(q2 && q4 && q6, print1(k-3, ", ")); q1 = q3; q2 = q4; q3 = q5; q4 = q6);}

A380933 Numbers k such that k and k+1 are both in A380929.

Original entry on oeis.org

121643775, 157390064, 161019495, 275734304, 584899875, 1493214975, 1614323655, 2043708975, 3081783375, 3118599224, 3426851295, 3902652495, 3947893424, 5849043375, 11731509855, 12138531615, 13008843224, 14598032624, 17588484584, 19782621495, 20191564575, 20759209064

Offset: 1

Amiram Eldar, Feb 08 2025

Numbers k such that A380845(k) > 2*k and A380845(k+1) > 2*(k+1).

			121643775 is a term since A380845(121643775) = 244722015 > 2 * 121643775 = 243287550, and A380845(121643776) = 256456081 > 2 * 121643776 = 243287552.

Amiram Eldar, Table of n, a(n) for n = 1..32

Subsequence of A096399 and A380929.
Cf. A380845, A380932.
Similar sequences: A283418, A318167, A327635, A327942, A331412, A333951, A357608, A364727, A364861.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k];
seq[lim_] := Module[{s = {}}, Do[If[q[k], If[q[k-1], AppendTo[s, k-1]]; If[q[k+1], AppendTo[s, k]]], {k, 3, lim, 2}]; s];
seq[3*10^8]

isab(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}
list(lim) = forstep(k = 3, lim, 2, if(isab(k), if(isab(k-1), print1(k-1, ", ")); if(isab(k+1), print1(k, ", "))));

cp's OEIS Frontend

A383366 Smallest of a sociable triple i < j < k such that j = s(i), k = s(j), and i = s(k), where s(k) = A380845(k) - k is the sum of aliquot divisors of k that have the same binary weight as k.

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A380844 The number of divisors of n that have the same binary weight as n.

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A381071 Numbers k such that the sum of the proper divisors of k that have the same binary weight as k is larger than k, and no subset of these divisors sums to k.

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A381073 Numbers k such that k and k+2 are both terms in A380846.

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A381074 Numbers k such that k, k+2 and k+4 are all terms in A380846.

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A380933 Numbers k such that k and k+1 are both in A380929.

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