A184395 -id:A184395 - cp's OEIS frontend

A319696 Number of distinct values obtained when Euler phi (A000010) is applied to the divisors of n.

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

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

			For n = 6, it has four divisors: 1, 2, 3 and 6, and applying A000010 to these gives 1, 1, 2, 2, with just two distinct values, thus a(6) = 2.

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

Cf. A000010, A319695.

Cf. also A184395, A319686.

A319696(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s=eulerphi(d)), mapput(m,s,s); k++)); (k); };

a(n) = A319695(n) + [n (mod 4) != 2], where [ ] is the Iverson bracket, resulting 0 when n = 2 mod 4, and 1 otherwise.

A319686 Number of distinct values obtained when arithmetic derivative (A003415) is applied to the divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 4, 5, 2, 6, 2, 6, 3, 3, 3, 8, 2, 3, 3, 7, 2, 6, 2, 5, 5, 3, 2, 9, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 10, 2, 3, 5, 7, 3, 6, 2, 5, 3, 6, 2, 11, 2, 3, 5, 5, 3, 6, 2, 9, 5, 3, 2, 10, 3, 3, 3, 7, 2, 10, 3, 5, 3, 3, 3, 11, 2, 5, 5, 8, 2, 6, 2, 7, 6, 3, 2, 11, 2, 6, 3, 8, 2, 6, 3, 5, 5, 3, 3, 14

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

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

One more than A319685.
Cf. A003415.
Cf. also A184395, A319696.

d[0] = d[1] = 0; d[n_] := d[n] = n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := CountDistinct[d /@ Divisors[n]]; Array[a, 100] (* Amiram Eldar, Apr 17 2024 *)

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A319686(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s=A003415(d)), mapput(m,s,s); k++)); (k); };

a(n) = my(d = divisors(n)); for(i = 1, #d, d[i] = A003415(d[i])); #Set(d) \\ uses A003415 listed at Antti's programs. David A. Corneth, Oct 02 2018

a(n) = 1+A319685(n).

A184396 a(n) = number of numbers k <= sigma(n) such that k is not equal to sigma(d) for any divisor d of n, where sigma = A000203.

Original entry on oeis.org

0, 1, 2, 4, 4, 8, 6, 11, 10, 14, 10, 22, 12, 20, 20, 26, 16, 33, 18, 36, 28, 32, 22, 52, 28, 38, 36, 50, 28, 64, 30, 57, 44, 50, 44, 82, 36, 56, 52, 82, 40, 88, 42, 78, 72, 68, 46, 114, 54, 87, 68, 92, 52, 112, 68, 112, 76, 86, 58, 156, 60, 92, 98, 120, 80, 137, 66, 120, 92, 136, 70, 183, 72, 110, 118, 134, 92, 160, 78, 176, 116

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

Sequence is not the same as A065608(n): a(66) = 137, A065608(66) = 136.

			For n = 4, sigma(4) = 7, from numbers 1 - 7 there are four numbers k such that k is not equal to sigma(d) for any divisor d of n: 2, 4, 5, 6; a(4) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A065608, A184395.

f[n_] := Block[{c = 0, k = 1, lmt = DivisorSigma[1, n] + 1, sd = DivisorSigma[1, #] & /@ Divisors@ n}, While[k < lmt, If[! MemberQ[sd, k], c++]; k++]; c]; Array[f, 67]

A184395(n) = length(vecsort(apply(d->sigma(d),divisors(n)), , 8));
A184396(n) = (sigma(n) - A184395(n)); \\ Antti Karttunen, Aug 24 2017

a(n) = A000203(n) - A184395(n).

More terms from Antti Karttunen, Aug 24 2017

A206031 a(n) = product of numbers k <= sigma(n) such that k = sigma(d) for any divisor d of n where sigma = A000203.

Original entry on oeis.org

1, 3, 4, 21, 6, 144, 8, 315, 52, 324, 12, 28224, 14, 576, 576, 9765, 18, 73008, 20, 95256, 1024, 1296, 24, 25401600, 186, 1764, 2080, 225792, 30, 26873856, 32, 615195, 2304, 2916, 2304, 1302170688, 38, 3600, 3136, 128595600, 42, 84934656, 44, 762048, 584064

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Sequence is not the same as A206032(n): a(66) = 35831808, A206032(66) = 429981696.

In sequence A206032 are multiplied all values of sigma(d) of all divisors d of numbers n, in sequence a(n) are multiplied only distinct values of sigma(d) of all divisors d of numbers n.

			For n=6 -> divisors d of 6: 1,2,3,6; corresponding values of sigma(d): 1,3,4,12; a(6) = Product of k = 1*3*4*12 = 144. For n=66 -> divisors d of 66: 1,2,3,6,11,22,33,66; corresponding values of sigma(d): 1,3,4,12,12,36,48,144; a(66) = Product of k = 1*3*4*12*36*48*144 = 35831808.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A184388, A184395, A206032, A206033.

Table[Times @@ Union[DivisorSigma[1, Divisors[n]]], {n, 100}] (* T. D. Noe, Feb 10 2012 *)

a(n)=my(d=vecsort(apply(sigma,divisors(n)),,8));prod(i=2,#d,d[i]) \\ Charles R Greathouse IV, Feb 19 2013

a(p) = p+1, a(pq) = ((p+1)*(q+1))^2 for p, q = distinct primes.

cp's OEIS Frontend

A319696 Number of distinct values obtained when Euler phi (A000010) is applied to the divisors of n.

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A319686 Number of distinct values obtained when arithmetic derivative (A003415) is applied to the divisors of n.

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A184396 a(n) = number of numbers k <= sigma(n) such that k is not equal to sigma(d) for any divisor d of n, where sigma = A000203.

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A206031 a(n) = product of numbers k <= sigma(n) such that k = sigma(d) for any divisor d of n where sigma = A000203.

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