A277898 -id:A277898 - cp's OEIS frontend

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

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)).

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).

A277897 Transpose of square array A277898.

Original entry on oeis.org

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

Offset: 3

Antti Karttunen, Nov 08 2016

See A277898.

Antti Karttunen, Table of n, a(n) for n = 3..353; the first 26 antidiagonals of array

Transpose: A277898.

(define (A277897 n) (if (< n 3) n (A277898bi (A004736 (- n 2)) (A002260 (- n 2))))) ;; Code for A277898bi given in A277898.

A277319 Numbers k such that A048675(k) is a prime.

Original entry on oeis.org

3, 4, 6, 8, 10, 18, 22, 24, 30, 32, 40, 42, 46, 54, 56, 66, 70, 72, 88, 96, 98, 102, 114, 118, 126, 128, 130, 136, 150, 152, 168, 182, 200, 224, 234, 238, 246, 250, 266, 270, 294, 312, 318, 328, 330, 350, 354, 360, 370, 392, 402, 406, 416, 424, 434, 440, 442, 450, 472, 480, 486, 510, 536, 546, 594, 600, 630, 640, 646, 648, 650, 654, 666, 680, 690, 722

Offset: 1

Antti Karttunen, Oct 11 2016

nonn

After 3 and 4 each term is an even number with an odd exponent of 2. - David A. Corneth and Antti Karttunen, Oct 11 2016

Antti Karttunen (terms 1..4994) & Hans Havermann, Table of n, a(n) for n = 1..25000
Hans Havermann, 70000 terms with their associated primes

Row 1 of A277898. Positions of ones in A277892.
Cf. A048675 and A277321 for the primes themselves.
Cf. A277317 (a subsequence).
After two initial terms a subsequence of A036554.

allocatemem(2^30);
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From Michel Marcus, Oct 10 2016
isA277319 = n -> isprime(A048675(n));
i=0; n=1; while(i < 10000, n++; if(isA277319(n), i++; write("b277319.txt", i, " ", n)));

from sympy import factorint, primepi, isprime
def a048675(n):
    if n==1: return 0
    f=factorint(n)
    return sum([f[i]*2**(primepi(i) - 1) for i in f])
print([n for n in range(1, 1001) if isprime(a048675(n))]) # Indranil Ghosh, Jun 19 2017

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

Original entry on oeis.org

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.

cp's OEIS Frontend

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

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

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A277897 Transpose of square array A277898.

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A277319 Numbers k such that A048675(k) is a prime.

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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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