Marius A. Burtea - cp's OEIS frontend

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

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 *)

A377382 a(n) is the smallest number k for which exactly n of its divisors are interprime numbers (A024675).

Original entry on oeis.org

1, 4, 52, 12, 162, 36, 60, 120, 240, 300, 180, 600, 360, 1560, 720, 1260, 1440, 1620, 2520, 2880, 3240, 5040, 10920, 6300, 9360, 10080, 12960, 12600, 15840, 20160, 22680, 25200, 31680, 39600, 27720, 59400, 50400, 70560, 56700, 79200, 55440, 65520, 83160, 100800

Offset: 0

Marius A. Burtea, Dec 05 2024

nonn

			Because A024675(1) = 4 it follows that a(0) = 1 and a(1) = 4.
a(2) = 52 because 52 has the divisors 4 = A024675(1), 26 = A024675(8) and no number from 1 to 51 has exactly two interprime divisors.

Cf. A024675, A377381.

ipr:=func; a:=[]; for n in [0..43] do k:=1; while  #[d:d in Divisors(k)|ipr(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

d[n_] := DivisorSum[n, 1 &, CompositeQ[#] && NextPrime[#] + NextPrime[#, -1] == 2*# &]; seq[len_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len, i = d[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[44] (* Amiram Eldar, Dec 11 2024 *)

A377381 a(n) is the number of divisors of n that are interprime numbers (A024675).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 3, 0, 1, 1, 1, 0, 3, 0, 1, 0, 1, 0, 5, 0, 0, 1, 1, 0, 3, 0, 1, 3, 0, 0, 3, 0, 1, 0, 2, 0, 3, 0, 2, 0, 0, 0, 6, 0, 0, 2, 2, 0, 1, 0, 2, 1, 0, 0, 6, 0, 0, 1, 2, 0, 3, 0, 1, 2, 0, 0, 5, 0, 1, 0

Offset: 1

Marius A. Burtea, Dec 05 2024

nonn

			Because A024675(1) = 4 it follows that a(1) = a(2) = a(3) = 0 and a(4) = 1.
a(12) = 3 because it has the divisors 4 = A024675(1), 6 = A024675(2) and 12 = A024675(4).

Cf. A024675.

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

a[n_] := DivisorSum[n, 1 &, CompositeQ[#] && NextPrime[#] + NextPrime[#, -1] == 2*# &]; Array[a, 100] (* Amiram Eldar, Dec 11 2024 *)

A375325 a(n) is the number of divisors of n which are Loeschian numbers (A003136).

Original entry on oeis.org

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

Offset: 1

Marius A. Burtea, Sep 15 2024

nonn

			a(1) = 1 because 1 has only one divisor 1 = A003136(2).
a(3) = 2 because 3 has divisors 1 = A003136(2) and 3 = A003136(3).
a(9) = 3 because 9 has the divisors 1 = A003136(2), 3 = A003136(3) and 9 = A003136(6).

Cf. A003136.

f:=func; [#[d:d in Divisors(n)|f(d)]:n in [1..150]];

A375324 Numbers of the form 3^k + 2 that admit at least one divisor of the form 3^m + 2 with 1 <= m < k.

Original entry on oeis.org

245, 2189, 19685, 531443, 1594325, 129140165, 10460353205, 31381059611, 847288609445, 7625597484989, 68630377364885, 617673396283949, 1853020188851843, 5559060566555525, 450283905890997365, 36472996377170786405, 109418989131512359211, 2954312706550833698645

Offset: 1

Marius A. Burtea, Sep 15 2024

nonn

The sequence is inspired by problem 3, Balkan Mathematical Olympiad 27 April - 2 May 2024, Varna, Bulgaria, (see link).

The sequence is infinite because numbers of the form m = 3^(4*k + 1) + 2 are divisible by 5 = 3^1 + 2.

			245 = 3^5 + 2  and 245 = 49*5 = 49 * (3^1 + 2), so 245 is a term.
2189 = 3^7 + 2  and 2189 = 11*199 = 199 * (3^2 + 2), so 2189 is a term.
129140165 = 3^17 + 2 and 129140165 = 5*25828033 = (3^1 + 2)*25828033 or 129140165 = 11*11740015 = (3^2 + 2)*11740015, so 129140165 is a term.

Balkan Mathematical Olympiad, 2024, Problems

Cf. A168607.

f:=func; [n:n in [3^a+2:a in [1..50]]|exists{d: d in Divisors(n)|d ne n and f(d-2) }];

A375323 Natural numbers k for which there exist distinct nonzero naturals a,b,c, such that k = a + b + c and (a + b)(b + c)(c + a) is a perfect cube.

Original entry on oeis.org

10, 19, 20, 30, 37, 38, 40, 46, 47, 50, 57, 60, 61, 66, 67, 68, 70, 74, 75, 76, 80, 90, 91, 92, 94, 95, 100, 101, 107, 109, 110, 111, 113, 114, 120, 122, 127, 129, 130, 131, 132, 133, 134, 136, 138, 139, 140, 141, 148, 150, 152, 160, 167, 169, 170, 171, 180, 182

Offset: 1

Marius A. Burtea, Sep 15 2024

nonn

The sequence is inspired by problem 1, Junior Balkan Team Selection Tests - Romania 2023, Brasov, 13.04.2023, (see link).

If k >= 1 is a term, then for any m >= 1 the number m*k is also a term.

			10 = 1 + 2 + 7 and (1 + 2)*(2 + 7)*(7 + 1) = 27*8 = 6^3, is a cube, so 10 is a term.
19 = 1 + 7 + 11 and (1 + 7)*(7 + 11)*(11 + 1) = 8*18*12 = 2^3*6^3 = 12^3, is a cube, so 19 is a term.
57 = 3 + 7 + 47 and (3 + 7)*(7 + 47)*(47 + 3) = 10*54*50 = 27*1000 = 30^3, is a cube. Also 57 = 3 + 21 + 33 and (3 + 21)*(21 + 33)*(33 + 3) = 24*54*36 = 36^2*36 = 36^3, is a cube.

Junior Balkan Team Selection Tests - Romania 2023, Problems

[n:n in [1..200]|exists(u){:a in [1..n-2],b in [1..n-2]|a lt b and #{a,b,n-a-b} eq 3 and n-a-b gt 0 and IsPower((n-a)*(n-b)*(a+b),3)}];

from itertools import count, islice
from sympy import integer_nthroot
def A375323_gen(startvalue=1): # generator of terms >= startvalue
    return (k for k in count(max(startvalue,1)) if any(integer_nthroot(a*(a*(m:=b-k)+b*(m-k)+k**2)-b*k*m,3)[1] for a in range(1,k//3) for b in range(a+1,k-a+1>>1)))
A375323_list = list(islice(A375323_gen(),58)) # Chai Wah Wu, Oct 10 2024

A373969 The smallest number k whose divisors include exactly n Duffinian numbers (A003624).

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 256, 512, 576, 1152, 1600, 2304, 4608, 3600, 6300, 7200, 18900, 20736, 32725, 14400, 28800, 50400, 56700, 108900, 57600, 100800, 111321, 176400, 129600, 226800, 229075, 360000, 630000, 435600, 333963, 518400, 1374450, 871200, 1001889

Offset: 0

Marius A. Burtea, Jul 12 2024

nonn

Numbers of the form m = 2^(k+1), k >= 0, have exactly k divisors that are Duffinian numbers.

			Since A003624(1) = 4, a(0) = 1.
The numbers 2 and 3 have no divisors that are Duffinian numbers and 4 = A003624(1), so a(1) = 4.

Cf. A003624, A373968.

f:=func; a:=[]; for n in [0..38] 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 &, CompositeQ[#] && CoprimeQ[#, DivisorSigma[1, #]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[30, 10^7] (* Amiram Eldar, Jul 19 2024 *)

A373892 a(n) is the smallest number that can be partitioned in exactly n ways as the sum of two Duffinian numbers (A003624).

Original entry on oeis.org

1, 8, 25, 43, 84, 71, 102, 160, 150, 219, 226, 196, 244, 350, 328, 300, 330, 354, 400, 386, 448, 408, 434, 390, 510, 536, 462, 546, 570, 624, 608, 740, 722, 690, 714, 770, 726, 660, 750, 804, 842, 858, 876, 870, 932, 914, 924, 840, 986, 1038, 966, 1108, 1050, 1056

Offset: 0

Marius A. Burtea, Jul 12 2024

nonn

			1 cannot be written as the sum of two Duffinian numbers, so a(0) = 1.
The numbers from 2 to 7 cannot be written as the sum of two Duffinian numbers and 8 = 4 + 4 = A003624(1) + A003624(1), so a(1) = 8.
25 = 4 + 21 = 9 + 16 and 4 = A003624(1), 9 = A003624(3), 16 = A003624(4), 21 = A003624(5) and the numbers 9 to 24 cannot be written in two ways as a sum of two Duffinian numbers. Thus a(2) = 25.

Cf. A003624.

f:=func; b:=[n: n in [1..2000] |f(n)]; a:=[]; for n in [0..60] do k:=1; while #RestrictedPartitions(k,2,Set(b)) ne n do k:=k+1; end while; Append(~a,k); end for; a;

dufQ[n_] := CompositeQ[n] && CoprimeQ[n, DivisorSigma[1, n]]; f[n_] := Sum[If[dufQ[k] && dufQ[n - k], 1, 0], {k, 1, Floor[n/2]}]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[54, 2000] (* Amiram Eldar, Jul 19 2024 *)

A373968 a(n) is the number of divisors of n that are Duffinian numbers (A003624).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 1, 3, 0, 0, 1, 2, 0, 1, 0, 1, 1, 0, 0, 3, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 0, 1, 0, 0, 3, 5, 1, 0, 0, 1, 0, 1, 0, 4, 0, 0, 2, 1, 1, 1, 0, 3, 3, 0, 0, 2, 1, 0, 0, 2

Offset: 1

Marius A. Burtea, Jul 12 2024

nonn

			Since A003624(1) = 4 then a(1) = a(2) = a(3) = 0 and a(4) = 1.
a(8) = 2 because 8 has the divisors 4 = A003624(1) and 8 = A003624(2).

Cf. A000203, A003624, A009194.

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

a[n_] := DivisorSum[n, 1 &, CompositeQ[#] && CoprimeQ[#, DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, Jul 19 2024 *)

a(p^k) = k - 1, for p prime and k >= 1.

A372146 The smallest number k for which exactly n of its divisors are digitally balanced numbers in base 3 (A049354).

Original entry on oeis.org

1, 11, 105, 420, 924, 2772, 6240, 4620, 18480, 13860, 55440, 69300, 120120, 180180, 240240, 360360, 514800, 720720, 1029600, 1801800, 2162160, 2522520, 2282280, 5045040, 7207200, 4564560, 6846840, 12612600, 15135120, 11411400, 20540520, 29343600, 22822800, 49729680

Offset: 0

Marius A. Burtea, May 23 2024

			Since A049354(1) = 11 it follows that a(0) = 1.
The numbers 2 through 10 have no divisors in A049354 and A049354(1) = 11 = 102_3, so a(1) = 11.
105 has only two divisors in A049354, 15 = A049354(2) and 21 = A049354(4) and is the smallest with exactly two divisors in A049354, so a(2) = 105.

Cf. A049354, A357035.

bal:=func; a:=[]; for n in [0..34] do k:=1; while #[d:d in Divisors(k)|bal(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

N:= 15: # for terms before the first term >= 3^(N+1)
db:= proc(n) option remember; local L,d,m;
  L:= convert(n,base,3);
  d:= nops(L);
  d mod 3 = 0 and 3*numboccur(0,L) = d and 3*numboccur(1,L) = d
end proc:
W:= Vector(3^(N+1),datatype=integer[4]):
for d from 3 to N by 3 do
  for t from 3^(d-1) to 3^d-1 do
    if db(t) then
          J:= [seq(i, i=t..3^(N+1), t)];
          W[J]:= W[J] +~ 1;
    fi
od od:
M:= max(W):
V:= Array(0..M): count:= 0:
for i from 1 to 3^(N+1) while count < M+1 do
  if V[W[i]] = 0 then V[W[i]]:= i; count:= count+1 fi;
od:
L:= convert(V,list):
if not member(0,L,'m') then m:= M+2 fi:
L[1..m-1]; # Robert Israel, Jun 03 2024

balQ[n_, b_] := balQ[n, b] = MinMax@ Differences@ DigitCount[n, b] == {0, 0}; f[n_] := DivisorSum[n, 1 &, balQ[#, 3] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[12, 10^5] (* Amiram Eldar, Jun 03 2024 *)

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User: Marius A. Burtea

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

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A377382 a(n) is the smallest number k for which exactly n of its divisors are interprime numbers (A024675).

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Mathematica

A377381 a(n) is the number of divisors of n that are interprime numbers (A024675).

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Magma

Mathematica

A375325 a(n) is the number of divisors of n which are Loeschian numbers (A003136).

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A375324 Numbers of the form 3^k + 2 that admit at least one divisor of the form 3^m + 2 with 1 <= m < k.

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Magma

A375323 Natural numbers k for which there exist distinct nonzero naturals a,b,c, such that k = a + b + c and (a + b)*(b + c)*(c + a) is a perfect cube.

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A373969 The smallest number k whose divisors include exactly n Duffinian numbers (A003624).

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Magma

Mathematica

A373892 a(n) is the smallest number that can be partitioned in exactly n ways as the sum of two Duffinian numbers (A003624).

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Magma

Mathematica

A373968 a(n) is the number of divisors of n that are Duffinian numbers (A003624).

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A375323 Natural numbers k for which there exist distinct nonzero naturals a,b,c, such that k = a + b + c and (a + b)(b + c)(c + a) is a perfect cube.