A277894 -id:A277894 - cp's OEIS frontend

A277892 a(n) = A001222(A048675(n)).

0, 1, 1, 2, 1, 3, 1, 2, 1, 4, 2, 5, 2, 2, 2, 6, 1, 7, 2, 2, 1, 8, 1, 3, 2, 2, 2, 9, 1, 10, 1, 3, 2, 3, 2, 11, 2, 2, 1, 12, 1, 13, 3, 3, 1, 14, 2, 4, 2, 3, 2, 15, 1, 3, 1, 3, 4, 16, 3, 17, 3, 3, 2, 4, 1, 18, 3, 3, 1, 19, 1, 20, 2, 2, 3, 4, 2, 21, 3, 3, 2, 22, 3, 3, 2, 2, 1, 23, 2, 4, 3, 5, 3, 4, 1, 24, 1, 3, 2, 25, 1, 26, 2, 2

Offset: 2

Antti Karttunen, Nov 08 2016

nonn

For n >= 3, a(n) = index of the row where n is located in array A277898.

Antti Karttunen (terms 2..3465) and Hans Havermann, Table of n, a(n) for n = 2..10000

Left inverse of A065091.
Cf. A001222, A048675.
Cf. A277319 (positions of ones).
Cf. A000040 (positions of records), A277900.
Cf. A277895 (ordinal transform from a(3) onward).
Cf. A277893, A277894, A277898.

A048675[n_] := If[n == 1, 0, Total[#[[2]]*2^(PrimePi[#[[1]]] - 1)& /@ FactorInteger[n]]];
a[n_] := PrimeOmega[A048675[n]];
Table[a[n], {n, 2, 105}] (* Jean-François Alcover, Jan 11 2022 *)

A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2;
A277892(n) = if(1==n,0,bigomega(A048675(n)));
for(n=1, 3465, write("b277892.txt", n, " ", A277892(n)));

from sympy import factorint, primepi, primefactors
def a001222(n): return 0 if n==1 else a001222(n//primefactors(n)[0]) + 1
def a048675(n):
    if n==1: return 0
    f=factorint(n)
    return sum(f[i]*2**(primepi(i) - 1) for i in f)
def a(n): return a001222(a048675(n))
print([a(n) for n in range(2, 101)]) # Indranil Ghosh, Jun 19 2017

(define (A277892 n) (if (= 1 n) 0 (A001222 (A048675 n))))

a(A019565(n)) = a(A260443(n)) = A001222(n).

For all n >= 2, a(A065091(n)) = n.

A277893 A277893(n) = the least k > n for which A277892(k) = A277892(n), 0 if no such number exists.

Original entry on oeis.org

0, 4, 6, 9, 8, 25, 10, 12, 18, 49, 14, 93, 15, 16, 20, 169, 22, 287, 21, 26, 24, 361, 30, 33, 27, 28, 34, 529, 32, 802, 40, 35, 36, 44, 38, 934, 39, 48, 42, 1333, 46, 1681, 45, 51, 54, 1754, 50, 58, 52, 55, 64, 2209, 56, 57, 66, 60, 65, 2809, 62, 2966, 63, 68, 74, 77, 70

Offset: 2

Antti Karttunen, Nov 08 2016

nonn

a(n) is the least k larger than n for which the number of divisors of A048675(k) is equal to the number of divisors of A048675(n) (counted with multiplicity), and 0 if no such number exists (which happens only for n=2).

Antti Karttunen, Table of n, a(n) for n = 2..102 (computed from the b-file provided by Hans Havermann for A277892)

Cf. A048675, A277892, A277894, A277898.

(define (A277893 n) (cond ((= 2 n) 0) (else (let ((v (A277892 n))) (let loop ((k (+ 1 n))) (if (= (A277892 k) v) k (loop (+ 1 k))))))))

For n >= 3, A277894(a(n)) = n.

A277895 a(n) is the index of the column where n is located in array A277898, a(2) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 2, 5, 1, 3, 1, 4, 5, 6, 1, 6, 1, 7, 8, 7, 1, 8, 2, 9, 10, 11, 1, 9, 1, 10, 3, 12, 4, 13, 1, 14, 15, 11, 1, 12, 1, 5, 6, 13, 1, 16, 2, 17, 7, 18, 1, 14, 8, 15, 9, 3, 1, 10, 1, 11, 12, 19, 4, 16, 1, 13, 14, 17, 1, 18, 1, 20, 21, 15, 5, 22, 1, 16, 17, 23, 1, 18, 19, 24, 25, 19, 1, 26, 6, 20, 2, 21, 7, 20, 1, 21, 22, 27, 1

Offset: 2

Antti Karttunen, Nov 08 2016

nonn

a(2) = 0 as 2 does not occur in the array A277898 proper.

From a(3) onward the ordinal transform of A277892 from its first nonzero term a(3) onward: 1, 1, 2, 1, 3, 1, 2, 1, 4, 2, 5, 2, 2, 2, 6, 1, 7, 2, ... The relation does not hold the other way, because not all columns of A277898 are monotonic, for example, 16 is located below 18 in the sixth column of that array. Already the array's second column (A277900) is nonmonotonic.

Antti Karttunen, Table of n, a(n) for n = 2..10000 (computed from the b-file provided by _Hans Havermann_ for A277892)

Cf. A010051, A277892, A277894, A277898, A277900.

A048675[n_] := If[n == 1, 0, Total[#[[2]]*2^(PrimePi[#[[1]]] - 1) & /@ FactorInteger[n]]];
A277892[n_] := PrimeOmega[A048675[n]];
Module[{b}, b[_] = 0;
a[n_] := If[n == 2, 0, With[{t = A277892[n]}, b[t] = b[t] + 1]]];
Table[a[n], {n, 2, 101}] (* Jean-François Alcover, Jan 11 2022 *)

(definec (A277895 n) (cond ((<= n 2) 0) ((= 1 (A010051 n)) 1) (else (+ 1 (A277895 (A277894 n))))))

a(2)=0, for n >= 3, if A010051(n) = 1 [when n is a prime], a(n) = 1, otherwise a(n) = 1 + a(A277894(n)).

cp's OEIS Frontend

A277892 a(n) = A001222(A048675(n)).

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A277893 A277893(n) = the least k > n for which A277892(k) = A277892(n), 0 if no such number exists.

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A277895 a(n) is the index of the column where n is located in array A277898, a(2) = 0.

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