A048250 -id:A048250 - cp's OEIS frontend

A325317 a(n) = A048250(n) XOR A162296(n), where XOR is the bitwise-XOR, A003987.

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 23, 20, 10, 32, 36, 24, 60, 31, 42, 32, 56, 30, 72, 32, 63, 48, 54, 48, 67, 38, 60, 56, 90, 42, 96, 44, 20, 46, 72, 48, 124, 57, 89, 72, 18, 54, 96, 72, 120, 80, 90, 60, 40, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 187, 74, 114, 124, 108, 96, 168, 80

Offset: 1

Antti Karttunen, Apr 21 2019

nonn

Cf. A000203, A003987, A048250, A162296, A325316, A325318.

Array[BitXor @@ Map[Total, {#3, Complement[#2, #3]}] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 79] (* Michael De Vlieger, Apr 21 2019 *)

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325317(n) = bitxor(A048250(n),A162296(n));

a(n) = A003987(A048250(n), A162296(n)).

a(n) = A000203(n) - 2*A325318(n) = A325316(n) - A325318(n).

A325318 a(n) = A048250(n) AND A162296(n), where AND is the bitwise-AND, A004198.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 16, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 32, 16, 0, 0, 0, 0, 2, 0, 40, 0, 12, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 16, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 16, 32, 2, 0, 0, 0, 40, 0

Offset: 1

Antti Karttunen, Apr 21 2019

nonn

Cf. A000203, A004198, A048250, A162296, A325316, A325317.

Array[BitAnd @@ Map[Total, {#3, Complement[#2, #3]}] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 105] (* Michael De Vlieger, Apr 21 2019 *)

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325318(n) = bitand(A048250(n),A162296(n));

a(n) = A004198(A048250(n), A162296(n)).
a(n) = A000203(n) - A325316(n) = (A000203(n) - A325317(n))/2.
a(n) = A325316(n) - A325317(n).

A325381 Lexicographically earliest sequence such that a(i) = a(j) => A048250(i) = A048250(j) and A126795(i) = A126795(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 3, 7, 8, 9, 10, 11, 12, 2, 13, 5, 14, 7, 15, 16, 17, 9, 4, 18, 3, 19, 20, 21, 15, 2, 22, 23, 24, 9, 25, 26, 27, 7, 28, 29, 30, 31, 12, 32, 24, 9, 6, 7, 33, 18, 34, 5, 35, 36, 37, 38, 39, 40, 41, 42, 15, 2, 43, 44, 45, 23, 46, 47, 35, 9, 48, 49, 12, 50, 51, 52, 37, 7, 3, 53, 43, 54, 55, 56, 57, 31, 58, 59, 60, 61, 62, 47, 63, 9, 64, 11

Offset: 1

Antti Karttunen, May 08 2019

Restricted growth sequence transform of the ordered pair [A048250(n), A126795(n)].

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A048250, A126795, A291751, A325382, A325383.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A126795(n) = gcd(n,A048250(n));
v325381 = rgs_transform(vector(up_to,n,[A048250(n),A126795(n)]));
A325381(n) = v325381[n];

A325382 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A048250(n), A126795(n)] for all other numbers, except f(p) = -(p mod 2) for primes.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 4, 6, 7, 3, 8, 3, 9, 10, 4, 3, 5, 3, 7, 11, 12, 3, 8, 13, 14, 6, 15, 3, 16, 3, 4, 17, 18, 19, 8, 3, 20, 21, 7, 3, 22, 3, 23, 10, 24, 3, 8, 25, 7, 26, 14, 3, 5, 27, 28, 29, 30, 3, 31, 3, 32, 11, 4, 33, 34, 3, 18, 35, 36, 3, 8, 3, 37, 10, 38, 39, 40, 3, 7, 6, 41, 3, 42, 43, 44, 45, 23, 3, 46, 47, 48, 49, 36, 50, 8, 3, 9, 17, 7, 3, 51, 3, 14

Offset: 1

Antti Karttunen, May 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A048250, A126795, A305801, A325381, A325384.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A126795(n) = gcd(n,A048250(n));
Aux325382(n) = if(isprime(n),-(n%2),[A048250(n),A126795(n)]);
v325382 = rgs_transform(vector(up_to,n,Aux325382(n)));
A325382(n) = v325382[n];

A344996 a(n) = A048250(n) * A051709(n).

Original entry on oeis.org

0, 0, 0, 3, 0, 24, 0, 9, 4, 36, 0, 96, 0, 48, 48, 21, 0, 108, 0, 180, 64, 72, 0, 240, 6, 84, 16, 288, 0, 1440, 0, 45, 96, 108, 96, 372, 0, 120, 112, 468, 0, 2304, 0, 576, 288, 144, 0, 528, 8, 234, 144, 756, 0, 360, 144, 768, 160, 180, 0, 4608, 0, 192, 448, 93, 168, 4608, 0, 1188, 192, 4032, 0, 900, 0, 228, 336, 1440, 192

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A048250, A051709.
Cf. also A344997.
Cf. A002117, A059956.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A051709(n) = ((sigma(n) + eulerphi(n)) - (2*n));
A344996(n) = (A048250(n)*A051709(n));

a(n) = A048250(n) * A051709(n).
a(n) = -[Sum_{d|n} mu(d)^2*d] * [Sum_{d|n, dA001065(d)].
a(n) = -Product(p_i + 1) * [Sum_{d|n, dA008683(n/d)*A001065(d)], where p_i are distinct primes dividing n.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3) * Product_{p prime} (1 + 1/p^2 - 1/p^3) + 6/Pi^2 - 2 = 0.177775281124... . - Amiram Eldar, Dec 04 2023

A345052 a(n) = A003557(n) * A048250(n) * A173557(n).

Original entry on oeis.org

1, 3, 8, 6, 24, 24, 48, 12, 24, 72, 120, 48, 168, 144, 192, 24, 288, 72, 360, 144, 384, 360, 528, 96, 120, 504, 72, 288, 840, 576, 960, 48, 960, 864, 1152, 144, 1368, 1080, 1344, 288, 1680, 1152, 1848, 720, 576, 1584, 2208, 192, 336, 360, 2304, 1008, 2808, 216, 2880, 576, 2880, 2520, 3480, 1152, 3720, 2880, 1152, 96

Offset: 1

Antti Karttunen, Jun 08 2021

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

Cf. A000010, A003557, A007434, A048250, A065484, A173557, A185197.

f[p_, e_] := (p^2 - 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 16 2022 *)

A345052(n) = { my(f=factor(n)); prod(i=1, #f~, ((f[i,1]^2)-1)*(f[i,1]^(f[i, 2]-1))); };

Multiplicative with a(p^e) = (p^2 - 1) * p^(e-1).
a(n) = A007434(n) / A003557(n) = A003557(n) * A048250(n) * A173557(n).
From Amiram Eldar, Oct 16 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/Pi^2 = 0.202642... (A185197).
Sum_{n>=1} 1/a(n) = A065484.
a(n) = A000010(n) * A048250(n). (End)

A369259 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003557(i) = A003557(j), A048250(i) = A048250(j) and A342671(i) = A342671(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 15, 23, 24, 25, 26, 27, 28, 29, 21, 30, 31, 32, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 29, 31, 43, 44, 45, 46, 47, 48, 49, 46, 50, 51, 52, 53, 54, 55, 39, 56, 57, 58, 59, 60, 61, 62, 59, 46, 63, 64, 65, 66, 67, 62, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 59

Offset: 1

Antti Karttunen, Jan 25 2024

nonn

Restricted growth sequence transform of the triplet [A003557(j), A048250(i), A342671(n)].
For all i, j >= 1:
  a(i) = a(j) => A323368(i) = A323368(j) => A291751(i) = A291751(j),
  a(i) = a(j) => A369260(i) = A369260(j) => A286603(i) = A286603(j).

Cf. A000203, A001615, A003557, A003961, A048250, A286603, A291751, A369260.

Differs from related A296089 and A323368 for the first time at n=79, where a(79) = 69, while A296089(79) = A323368(79) = 51.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003557(n) = (n/factorback(factor(n)[, 1]));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A342671(n) = gcd(sigma(n), A003961(n));
Aux369259(n) = [A003557(n), A048250(n), A342671(n)];
v369259 = rgs_transform(vector(up_to, n, Aux369259(n)));
A369259(n) = v369259[n];

A063964 Numbers k such that k and k+1 have the same sum of squarefree divisors, or sqf(k) = sqf(k+1), where sqf(k) = A048250(k).

Original entry on oeis.org

11, 14, 224, 957, 1334, 1634, 2685, 9347, 13915, 16260, 20145, 20335, 33998, 37236, 42251, 42818, 51624, 55308, 56419, 56975, 71874, 74918, 77748, 79824, 79826, 79833, 84134, 93632, 106600, 111506, 120680, 122073, 138237, 142116, 147454

Offset: 1

Jason Earls, Sep 04 2001

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..170 from Harry J. Smith)

Cf. A048250.

sqs[n_] := Times@@(1 + FactorInteger[n][[;; , 1]]); seq={}; s1 = 1; Do[s2 = sqs[n]; If[s2 == s1, AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^5}]; seq (* Amiram Eldar, Aug 18 2019 *)

sqf(n) = sumdiv(n,d, if(issquarefree(d),d,0)); for(n=1,10^7, if(sqf(n)==sqf(n+1),print(n)))

{ n=0; s=0; for (m=1, 10^9, t=sumdiv(m + 1, d, if(issquarefree(d), d, 0)); if(s==t, write("b063964.txt", n++, " ", m); if (n==170, break)); s=t ) } \\ Harry J. Smith, Sep 04 2009

A332230 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(2^k) = 0 for k >= 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 8, 9, 10, 11, 12, 13, 13, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 4, 29, 30, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 41, 46, 47, 48, 41, 49, 50, 51, 52, 53, 54, 55, 56, 4, 57, 58, 59, 60, 55, 58, 61, 62, 63, 64, 65, 66, 55, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 76

Offset: 1

Antti Karttunen, Feb 22 2020

nonn

For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  a(i) = a(j) => A048250(i) = A048250(j),
  a(i) = a(j) => A332455(i) = A332455(j),
  a(i) = a(j) => A332459(i) = A332459(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A046523, A048250, A295300, A332455, A332459.

Cf. also A326199.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A209229(n) = (n && !bitand(n,n-1));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux332230(n) = if((n>2)&&A209229(n),0,(1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n)));
v332230 = rgs_transform(vector(up_to,n,Aux332230(n)));
A332230(n) = v332230[n];

A071375 Smallest m such that sum of squarefree divisors of m equals n; a(n) = 0 if no solution to A048250(x) = n exists.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 6, 0, 13, 0, 0, 0, 10, 0, 19, 0, 0, 0, 14, 0, 0, 0, 0, 0, 29, 0, 21, 0, 0, 0, 22, 0, 37, 0, 0, 0, 26, 0, 43, 0, 0, 0, 33, 0, 0, 0, 0, 0, 34, 0, 39, 0, 0, 0, 38, 0, 61, 0, 0, 0, 0, 0, 67, 0, 0, 0, 30, 0, 73, 0, 0, 0, 0, 0, 57, 0, 0, 0, 65, 0, 0, 0, 0, 0, 58, 0, 0, 0, 0

Offset: 1

Labos Elemer, May 22 2002

nonn

			n=256: a(256)=217=7.31, all divisors are squarefree and 1+7+31+217=256=n.

Cf. A048250.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2 * w - 1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] t = Table[0, {256}]; Do[s = DivisorSigma[1, cor[n]]; If[s < 257 && t[[s]] == 0, t[[s]] = n], {n, 10^6}]; t

a(n)=Min{x; A048250[x]=n}, a(n)=0 if no solutions.

Definition corrected by Jaroslav Krizek, May 28 2014

cp's OEIS Frontend

A325317 a(n) = A048250(n) XOR A162296(n), where XOR is the bitwise-XOR, A003987.

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A325318 a(n) = A048250(n) AND A162296(n), where AND is the bitwise-AND, A004198.

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A325381 Lexicographically earliest sequence such that a(i) = a(j) => A048250(i) = A048250(j) and A126795(i) = A126795(j) for all i, j.

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A325382 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A048250(n), A126795(n)] for all other numbers, except f(p) = -(p mod 2) for primes.

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A344996 a(n) = A048250(n) * A051709(n).

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A345052 a(n) = A003557(n) * A048250(n) * A173557(n).

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A369259 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003557(i) = A003557(j), A048250(i) = A048250(j) and A342671(i) = A342671(j), for all i, j >= 1.

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A063964 Numbers k such that k and k+1 have the same sum of squarefree divisors, or sqf(k) = sqf(k+1), where sqf(k) = A048250(k).

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A332230 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(2^k) = 0 for k >= 2.

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