A327502 -id:A327502 - cp's OEIS frontend

A327503 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or not a perfect power (A327501, A327502).

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 16 2019

nonn

First differs from A052409 and A158378 at a(216) = 2, A052409(216) = A158378(216) = 3.
A multiset is aperiodic if its multiplicities are relatively prime. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Heinz numbers of aperiodic multisets are numbers that are not perfect powers (A007916).
a(n) does not depend only on the prime signature of n. A351948 gives the positions where a(A046523(n)) <> a(n). n = 125000 is the first time this happens, see the examples. - Antti Karttunen, Apr 03 2022

			The transformation A327502 takes 144 -> 2 -> 1, so a(144) = 2.
From _Antti Karttunen_, Apr 03 2022: (Start)
For n = 1728 = 2^6 * 3^3, A327501(1728) = 864 = 2^5 * 3^3, and therefore A327502(1728) = 1728/864 = 2. A327501(2) = 2, thus A327502(2) = 2/2 = 1, so we reached 1 (= A327502(1)) in two steps, and therefore a(1728) = 2.
For n = 125000 = 2^3 * 5^6, A327501(125000) = 31250 = 2^1 * 5^6, and therefore A327502(125000) = 125000/31250 = 4. A327501(4) = 2, thus A327502(4) = 4/2 = 2, from which we reach 1 in one more step, therefore a(125000) = 3.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.
Cf. A000961, A001597, A007916 (positions of 1's), A052409, A158378, A303386, A327501, A327502, A351948.
Cf. also A327500.

Table[Length[FixedPointList[#/Max[Select[Divisors[#],GCD@@Last/@FactorInteger[#]==1&]]&,n]]-2,{n,100}]

A327502(n) = if(n==1, 1, n/vecmax(select(x->((x>1) && !ispower(x)), divisors(n))));
A327503(n) = { my(u=A327502(n)); if(u==n, 0, 1+A327503(u)); }; \\ Antti Karttunen, Apr 02 2022

a(2^n) = n.

Data section extended up to 105 terms by Antti Karttunen, Apr 02 2022

A327501 Maximum divisor of n that is 1 or not a perfect power.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 5, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 18, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 2, 65, 66, 67, 68, 69

Offset: 1

Gus Wiseman, Sep 16 2019

nonn

First differs from A052410 at a(36) = 18, A052410(36) = 6.
The number of divisors that are 1 or not a perfect power is given by A327502.
A multiset is aperiodic if its multiplicities are relatively prime. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Heinz numbers of aperiodic multisets are numbers that are not perfect powers (A007916).
a(n) = n iff n is in A175082. - Bernard Schott, Sep 20 2019

			The divisors of 36 that are not perfect powers are {1, 2, 3, 6, 12, 18}, so a(36) = 18.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A000961, A001597, A007916, A303386, A327502.

[1] cat [Max([d:d in Divisors(n)| d gt 1 and not IsPower(d)]):n in [2..70]]; // Marius A. Burtea, Sep 20 2019

Table[Max[Select[Divisors[n],GCD@@Last/@FactorInteger[#]==1&]],{n,100}]

isp(n) = !ispower(n) && (n>1); \\ A007916
a(n) = if (n==1, 1, vecmax(select(x->isp(x), divisors(n)))); \\ Michel Marcus, Sep 18 2019

A351948 Numbers k for which A327503(A046523(k)) differs from A327503(k).

Original entry on oeis.org

125000, 941192, 8000000, 14172488, 16000000, 28344976, 38614472, 47832147, 60236288, 77228944, 120472576, 130323843, 193100552, 312500000, 376367048, 386201104, 625000000, 651714363, 752734096, 907039232, 1184287112, 1270238787, 1814078464, 1953125000, 2368574224, 2471326208, 3996969003

Offset: 1

Antti Karttunen, Apr 03 2022

nonn

Each term is a perfect power (in A001597).

Cf. A046523, A327503.

Subsequence of A001597.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A327502(n) = if(n==1, 1, n/vecmax(select(x->((x>1) && !ispower(x)), divisors(n))));
A327503(n) = { my(u=A327502(n)); if(u==n, 0, 1+A327503(u)); };
isA351948(k) = if(!ispower(k),0,(A327503(A046523(k)) != A327503(k)));

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A327503 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or not a perfect power (A327501, A327502).

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A327501 Maximum divisor of n that is 1 or not a perfect power.

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A351948 Numbers k for which A327503(A046523(k)) differs from A327503(k).

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