A277893 -id:A277893 - 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)).

A277894 a(n) = the largest k < n for which A277892(k) = A277892(n), 0 if no such number exists, a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 4, 0, 6, 5, 8, 0, 9, 0, 12, 14, 15, 0, 10, 0, 16, 20, 18, 0, 22, 7, 21, 26, 27, 0, 24, 0, 30, 25, 28, 33, 34, 0, 36, 38, 32, 0, 40, 0, 35, 44, 42, 0, 39, 11, 48, 45, 50, 0, 46, 51, 54, 55, 49, 0, 57, 0, 60, 62, 52, 58, 56, 0, 63, 68, 66, 0, 70, 0, 64, 74, 69, 65, 75, 0, 76, 80, 78, 0, 81, 84, 82, 86, 72, 0, 87, 77, 85, 13, 92, 91, 88, 0

Offset: 0

Antti Karttunen, Nov 08 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000 (computed from the 10000 term b-file provided by _Hans Havermann_ for A277892)

Cf. A277892, A277893, A277895.

Cf. A008578 (gives the positions of zeros after a(0)).

(define (A277894 n) (cond ((<= n 2) 0) (else (let ((v (A277892 n))) (let loop ((k (- n 1))) (cond ((= 1 k) 0) ((= (A277892 k) v) k) (else (loop (- k 1)))))))))

For n >= 3, a(A277893(n)) = n.

A277896 a(n) = the least k > n for which A048675(k) = A048675(n), 0 if no such number exists (when n is a power of 2).

Original entry on oeis.org

0, 0, 4, 0, 9, 8, 25, 0, 12, 18, 49, 16, 121, 50, 20, 0, 169, 24, 289, 27, 28, 98, 361, 32, 45, 242, 36, 75, 529, 40, 841, 0, 44, 338, 63, 48, 961, 578, 52, 54, 1369, 56, 1681, 147, 60, 722, 1849, 64, 175, 90, 68, 363, 2209, 72, 99, 150, 76, 1058, 2809, 80, 3481, 1682, 84, 0, 117, 88, 3721, 507, 92, 126, 4489, 96, 5041, 1922, 100, 867, 275

Offset: 1

Antti Karttunen, Nov 15 2016

nonn

Apart from zeros, a permutation of A013929.

Antti Karttunen, Table of n, a(n) for n = 1..192

Cf. A000079, A013929, A048675, A209229, A277886, A277893, A277905.

Numbers not in this sequence: A005117 (A019565).

(define (A277896 n) (if (= 1 (A209229 n)) 0 (let ((v (A048675 n))) (let loop ((k (+ 1 n))) (if (= (A048675 k) v) k (loop (+ 1 k)))))))

a(A000079(n)) = 0.

For all n, except powers of two, a(n) >= A277893(n).

A277900 Column 2 of A277898; position of the second occurrence of n in A277892.

Original entry on oeis.org

4, 9, 25, 49, 93, 169, 287, 361, 529, 802, 934, 1333, 1681, 1754, 2209, 2809, 2966, 3482, 4453, 5041, 5329, 6241, 5378, 6374, 9167

Offset: 1

Antti Karttunen, Nov 08 2016, terms a(18)-a(25) obtained from the 10000 term b-file of A277892 computed by Hans Havermann

a(n) = the second smallest number k for which A277892(k) = n, where A277892(n) = the number of prime divisors (counted with multiplicity) of A048675(n).

Note that the sequence is not monotonic: a(22) = 6241 and a(23) = 5378.

			A277892[2..4] = [0, 1, 1], thus as the second 1 occurs at A277892(4), a(1) = 4.
A277892[5..9] = [2, 1, 3, 1, 2], thus as the second 2 occurs at A277892(9), a(2) = 9.

Column 2 of A277898.

Cf. A001248 (an upper bound), A048675, A065091, A277892, A277893.

(define (A277900 n) (A277898bi n 2)) ;; Code for A277898bi given in A277898.
(define (A277900 n) (A277893 (A065091 n))) ;; Alternatively.

a(n) = A277893(A065091(n)).

a(n) <= A001248(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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A277894 a(n) = the largest k < n for which A277892(k) = A277892(n), 0 if no such number exists, a(0) = a(1) = 0.

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A277896 a(n) = the least k > n for which A048675(k) = A048675(n), 0 if no such number exists (when n is a power of 2).

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A277900 Column 2 of A277898; position of the second occurrence of n in A277892.

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