A319686 -id:A319686 - 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.

A319685 Number of distinct values obtained when arithmetic derivative (A003415) is applied to proper divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 9, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 10, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 9, 2, 2, 2, 6, 1, 9, 2, 4, 2, 2, 2, 10, 1, 4, 4, 7, 1, 5, 1, 6, 5, 2, 1, 10, 1, 5, 2, 7, 1, 5, 2, 4, 4, 2, 2, 13

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Because for all d|n, dA003415(d) < A003415(n), it follows that the terms here are one less than in A319686.

Differs from A033273(n) = A000005(n) - A001221(n) at n = 1, 112, 156, 224, 280, 312, 336, 342, 380, 448, 468, 525, 560, 608, 624, 660, 672, 684, 756, 760, 780, 784, 840, 870, 896, 936, 984, 1008, ...

			The proper divisors of 112 are [1, 2, 4, 7, 8, 14, 16, 28, 56]. Applying arithmetic derivative A003415 to these, we obtain values [0, 1, 4, 1, 12, 9, 32, 32, 92], of which only 7 are distinct: 0, 1, 4, 9, 12, 32, and 92, thus a(112) = 7.

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

One less than A319686.
Cf. A003415.
Cf. also A304793, A305611, A316555, A316556, A319695 for similarly constructed sequences.

d[0] = d[1] = 0; d[n_] := d[n] = n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := CountDistinct[d /@ Most[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
A319685(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((dA003415(d)), mapput(m,s,s); k++)); (k); };

a(n) = A319686(n)-1.

A319357 Filter sequence combining A003415(d) from all proper divisors d of n, where A003415(d) = arithmetic derivative of d.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Restricted growth sequence transform of A319356.
The only duplicates in range 1..65537 with a(n) > 4 are the following six pairs: a(1445) = a(2783), a(4205) = a(11849), a(5819) = a(8381), a(6727) = a(15523), a(8405) = a(31211) and a(28577) = a(44573). All these have prime signature p^2 * q^1. If all the other duplicates respect the prime signature as well, then also the last implication given below is valid.
For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j),
  a(i) = a(j) => A319683(i) = A319683(j),
  a(i) = a(j) => A319686(i) = A319686(j),
  a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above]

			Proper divisors of 1445 are [1, 5, 17, 85, 289], while the proper divisors of 2783 are [1, 11, 23, 121, 253]. 1 contributes 0 and primes contribute 1, so only the last two matter in each set. We have A003415(85) = 22 = A003415(121) and A003415(289) = 34 = A003415(253), thus the value of arithmetic derivative coincides for all proper divisors, thus a(1445) = a(2783).

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

Cf. A003415, A319356.
Cf. A000041 (positions of 2's), A001248 (positions of 3's), A006881 (positions of 4's),
Cf. also A300245, A300249, A319353, A319693.

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; };
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
A319356(n) = { my(m=1); fordiv(n, d, if(dA003415(d)))); (m); };
v319357 = rgs_transform(vector(up_to,n,A319356(n)));
A319357(n) = v319357[n];

A322986 Number of distinct values obtained when the pi-based arithmetic derivative (A258851) is applied to the divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 5, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 7, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 11, 2, 4, 6, 6, 4, 8, 2, 9, 5, 4, 2, 11, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9, 2, 7, 2, 8, 8

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

			Divisors of 28 are [1, 2, 4, 7, 14, 28]. When A258851 is applied to them, we get five distinct values: [0, 1, 4, 4, 15, 44] (because A258851(4) = A258851(7) = 4), thus a(28) = 5, one less than A000005(28)=6.

Antti Karttunen, Table of n, a(n) for n = 1..20160
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A258851, A299701, A319686.

Differs from A000005 for the first time at n=28.

A258851(n) = n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i]); \\ From A258851
A322986(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s=A258851(d)), mapput(m,s,s); k++)); (k); };
\\ Or maybe more efficiently as, after David A. Corneth's Oct 02 2018 program in A319686:
A322986(n) = { my(d = divisors(n)); for(i=1, #d, d[i] = A258851(d[i])); #Set(d); };

a(n) <= A000005(n).

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

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A319685 Number of distinct values obtained when arithmetic derivative (A003415) is applied to proper divisors of n.

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A319357 Filter sequence combining A003415(d) from all proper divisors d of n, where A003415(d) = arithmetic derivative of d.

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A322986 Number of distinct values obtained when the pi-based arithmetic derivative (A258851) is applied to the divisors of n.

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