A277900 -id:A277900 - 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.

A277898 Square array A(r,c), where each row r lists all numbers k for which A277892(k) = r, read by downwards antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

3, 4, 5, 6, 9, 7, 8, 12, 25, 11, 10, 14, 33, 49, 13, 18, 15, 35, 58, 93, 17, 22, 16, 44, 65, 119, 169, 19, 24, 20, 45, 77, 121, 185, 287, 23, 30, 21, 51, 91, 124, 209, 289, 361, 29, 32, 26, 55, 95, 143, 214, 299, 437, 529, 31, 40, 27, 57, 106, 161, 221, 323, 473, 589, 802, 37, 42, 28, 60, 111, 177, 247, 327, 493, 611, 841, 934, 41

Offset: 3

Antti Karttunen, Nov 08 2016

Permutation of natural numbers larger than 2.

			The top left corner of the array:
   3,    4,    6,    8,   10,   18,   22,   24,   30,   32
   5,    9,   12,   14,   15,   16,   20,   21,   26,   27
   7,   25,   33,   35,   44,   45,   51,   55,   57,   60
  11,   49,   58,   65,   77,   91,   95,  106,  111,  115
  13,   93,  119,  121,  124,  143,  161,  177,  187,  203
  17,  169,  185,  209,  214,  221,  247,  254,  301,  305
  19,  287,  289,  299,  323,  327,  391,  393,  398,  403
  23,  361,  437,  473,  493,  551,  565,  629,  633,  685
  29,  529,  589,  611,  667,  713,  779,  817,  889,  893
  31,  802,  841,  842,  851,  899,  901,  989, 1073, 1081
  37,  934,  961, 1121, 1147, 1154, 1189, 1227, 1271, 1293
  41, 1333, 1369, 1403, 1437, 1517, 1538, 1591, 1643, 1761
  43, 1681, 1739, 1763, 1927, 1943, 2183, 2257, 2263, 2302
  47, 1754, 1849, 2021, 2173, 2201, 2279, 2501, 2623, 2747
  53, 2209, 2491, 2537, 2594, 2643, 2701, 2773, 2881, 3053

Antti Karttunen, Table of n, a(n) for n = 3..353; the first 26 antidiagonals of array (computed from the b-file provided by _Hans Havermann_ for A277892)

Transpose: A277897.
Row 1: A277319.
Column 1: A065091, column 2: A277900.
Cf. A277892 (index of the row where n is located), A277895 (of the column).
Cf. A001222, A048675, A277893.

;; The terms for array proper start with A277898(3):
(define (A277898 n) (if (< n 3) n (A277898bi (A002260 (- n 2)) (A004736 (- n 2)))))
(define (A277898bi row col) (if (= 1 col) (A000040 (+ 1 row)) (A277893 (A277898bi row (- col 1)))))

A(r,1) = A065091(r); for c > 1, A(r,c) = A277893(A(r,c-1)).

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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A277898 Square array A(r,c), where each row r lists all numbers k for which A277892(k) = r, read by downwards antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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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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Mathematica

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Formula