A066503 -id:A066503 - cp's OEIS frontend

A255326 a(n) gives the number of steps needed to reach zero, when we start from x = n and repeatedly subtract x's squarefree kernel (A007947(x)) from it.

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

Offset: 0

Antti Karttunen, Mar 23 2015

nonn

In other words, number of iterations needed to reach zero with map x <- A066503(x), when starting from n.

Also, for n >= 1, a(n) = one more than the number of steps to reach a squarefree number (A005117) when we repeatedly subtract the largest squarefree number dividing x, starting from x <- n.

			Largest squarefree number dividing 27 is 3, and 27 - 3 = 24.
Largest squarefree number dividing 24 is 6, and 24 - 6 = 18.
Largest squarefree number dividing 18 is 6, and 18 - 6 = 12.
Largest squarefree number dividing 12 is 6, and 12 - 6 = 6.
Largest squarefree number dividing 6 is 6, and 6 - 6 = 0.
Thus a(6) = 1, a(12) = 2, a(18) = 3, a(24) = 4 and a(27) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A255409 (gives the positions of records, also the first positions where a(n) = n).

Cf. A005117, A007947, A066503.

a[n_] := -1 + Length@ NestWhileList[# - Times @@ FactorInteger[#][[;; , 1]] &, n, # > 0 &]; Array[a, 100, 0] (* Amiram Eldar, Mar 04 2024 *)

a(0) = 0; a(n) = 1 + a(A066503(n)).
Other identities:
a(k) = 1 iff k = A005117(n).

A336646 a(n) = n - A326144(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 8, 8, 1, 10, 1, 10, 9, 15, 1, 15, 1, 18, 11, 14, 1, 18, 24, 16, 25, 14, 1, 18, 1, 31, 15, 20, 13, 35, 1, 22, 17, 30, 1, 30, 1, 42, 39, 26, 1, 34, 48, 49, 21, 50, 1, 42, 17, 54, 23, 32, 1, 54, 1, 34, 61, 63, 19, 54, 1, 66, 27, 66, 1, 69, 1, 40, 73, 74, 19, 66, 1, 78, 80, 44, 1, 70, 23, 46, 33, 86

Offset: 1

Antti Karttunen, Jul 30 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007947, A066503, A326143, A326144.
Cf. A326145 (positions where coincides with A007947).
Cf. A336555 (positions where differs from A336647).
Cf. also A336645, A336647.

A007947(n) = factorback(factorint(n)[, 1]);
A066503(n) = (n - A007947(n));
A326143(n) = (sigma(n)-A007947(n)-n);
A326144(n) = gcd(A066503(n), A326143(n));
A336646(n) = (n - A326144(n));

A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 49, 63, 81, 99, 121, 125, 135, 169, 171, 245, 279, 289, 343, 361, 363, 369, 375, 387, 477, 529, 531, 575, 603, 625, 675, 711, 729, 747, 833, 841, 847, 873, 875, 891, 909, 961, 981, 1029, 1083, 1125, 1127, 1179, 1225, 1251, 1377, 1413, 1445, 1467

Offset: 1

Gus Wiseman, Feb 20 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A position of first appearance in a sequence q is an index k such that q(k) is different from q(j) for all j < k.

All terms are odd.

			The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    49: {4,4}
    63: {2,2,4}
    81: {2,2,2,2}
    99: {2,2,5}
   121: {5,5}
   125: {3,3,3}
   135: {2,2,2,3}
   169: {6,6}
   171: {2,2,8}
   245: {3,4,4}
   279: {2,2,11}

For length instead of product we have A151821, firsts of A046660.
For factors instead of indices we have A381076, sorted firsts of A066503.
For sum of factors instead of product of indices we have A381075 (unsorted A280286), A280292.
For quotient instead of difference we have A380988 (unsorted A380987), firsts of A290106.
For quotient and factors we have A001694 (unsorted A064549), firsts of A003557.
For sum instead of product we have A380957 (unsorted A380956), firsts of A380955.
Sorted firsts of A380986, which has nonzero terms at positions A038838.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists the squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000079, A000720, A001222, A081770, A075255, A116861, A178503, A374248, A379681, A380958.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Times@@prix[n]-Times@@Union[prix[n]],{n,10000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

cp's OEIS Frontend

A255326 a(n) gives the number of steps needed to reach zero, when we start from x = n and repeatedly subtract x's squarefree kernel (A007947(x)) from it.

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A336646 a(n) = n - A326144(n).

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A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

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Mathematica