A277892 -id:A277892 - cp's OEIS frontend

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.

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)).

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.

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.

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).

A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 3, 4, 5, 16, 4, 32, 9, 6, 4, 64, 5, 128, 6, 10, 17, 256, 5, 8, 33, 6, 10, 512, 7, 1024, 5, 18, 65, 12, 6, 2048, 129, 34, 7, 4096, 11, 8192, 18, 8, 257, 16384, 6, 16, 9, 66, 34, 32768, 7, 20, 11, 130, 513, 65536, 8, 131072, 1025, 12, 6, 36, 19

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

The original motivation for this sequence was to encode the prime factorization of n in the binary representation of a(n), each such representation being unique as long as this map is restricted to A005117 (squarefree numbers, resulting a permutation of nonnegative integers A048672) or any of its subsequence, resulting an injective function like A048623 and A048639.
However, also the restriction to A260443 (not all terms of which are squarefree) results a permutation of nonnegative integers, namely A001477, the identity permutation.
When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443), then a(n) gives the evaluation of that polynomial at x=2.
The primitive completely additive integer sequence that satisfies a(n) = a(A225546(n)), n >= 1. By primitive, we mean that if b is another such sequence, then there is an integer k such that b(n) = k * a(n) for all n >= 1. - Peter Munn, Feb 03 2020
If the binary rank of an integer partition y is given by Sum_i 2^(y_i-1), and the Heinz number is Product_i prime(y_i), then a(n) is the binary rank of the integer partition with Heinz number n. Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices), and the function taking a multiset m to Product_i prime(m_i) is the inverse of A112798 (prime indices). - Gus Wiseman, May 22 2024

			From _Gus Wiseman_, May 22 2024: (Start)
The A018819(7) = 6 cases of binary rank 7 are the following, together with their prime indices:
   30: {1,2,3}
   40: {1,1,1,3}
   54: {1,2,2,2}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1024
Hans Havermann, Factorization of the first 10000 terms, in format [[primes], [exponents]]

Row 2 of A104244.
Similar logarithmic functions: A001414, A056239, A090880, A289506, A293447.
Left inverse of the following sequences: A000079, A019565, A038754, A068911, A134683, A260443, A332824.
A003961, A028234, A032742, A055396, A064989, A067029, A225546, A297845 are used to express relationship between terms of this sequence.
Cf. also A048623, A048676, A099884, A277896 and tables A277905, A285325.
Cf. A297108 (Möbius transform), A332813 and A332823 [= a(n) mod 3].
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000203,A331750), (A005940,A087808), (A007913,A248663), (A007947,A087207), (A097248,A048675), (A206296,A000129), (A248692,A056239), (A283477,A005187), (A284003,A006068), (A285101,A028362), (A285102,A068052), (A293214,A001065), (A318834,A051953), (A319991,A293897), (A319992,A293898), (A320017,A318674), (A329352,A069359), (A332461,A156552), (A332462,A156552), (A332825,A000010) and apparently (A163511,A135529).
See comments/formulas in A277333, A331591, A331740 giving their relationship to this sequence.
The formula section details how the sequence maps the terms of A329050, A329332.
A277892, A322812, A322869, A324573, A324575 give properties of the n-th term of this sequence.
The term k appears A018819(k) times.
The inverse transformation is A019565 (Heinz number of binary indices).
The version for distinct prime indices is A087207.
Numbers k such that a(k) is prime are A277319, counts A372688.
Grouping by image gives A277905.
A014499 lists binary indices of prime numbers.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000720, A035100, A071814, A096111, A225620, A243055, A304818, A355536, A358136, A372890.
Cf. A087207, A097248.

nthprime := proc(n) local i; if(isprime(n)) then for i from 1 to 1000000 do if(ithprime(i) = n) then RETURN(i); fi; od; else RETURN(0); fi; end; # nthprime(2) = 1, nthprime(3) = 2, nthprime(5) = 3, etc. - this is also A049084.
A048675 := proc(n) local s,d; s := 0; for d in ifactors(n)[ 2 ] do s := s + d[ 2 ]*(2^(nthprime(d[ 1 ])-1)); od; RETURN(s); end;
# simpler alternative
f:= n -> add(2^(numtheory:-pi(t[1])-1)*t[2], t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Oct 10 2016

a[1] = 0; a[n_] := Total[ #[[2]]*2^(PrimePi[#[[1]]]-1)& /@ FactorInteger[n] ]; Array[a, 100] (* Jean-François Alcover, Mar 15 2016 *)

a(n) = my(f = factor(n)); sum(k=1, #f~, f[k,2]*2^primepi(f[k,1]))/2; \\ Michel Marcus, Oct 10 2016

\\ The following program reconstructs terms (e.g. for checking purposes) from the factorization file prepared by Hans Havermann:
v048675sigs = readvec("a048675.txt");
A048675(n) = if(n<=2,n-1,my(prsig=v048675sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,ps[i]^es[i])); \\ Antti Karttunen, Feb 02 2020

from sympy import factorint, primepi
def a(n):
    if n==1: return 0
    f=factorint(n)
    return sum([f[i]*2**(primepi(i) - 1) for i in f])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 19 2017

a(1) = 0, a(n) = 1/2 * (e1*2^i1 + e2*2^i2 + ... + ez*2^iz) if n = p_{i1}^e1*p_{i2}^e2*...*p_{iz}^ez, where p_i is the i-th prime. (e.g. p_1 = 2, p_2 = 3).
Totally additive with a(p^e) = e * 2^(PrimePi(p)-1), where PrimePi(n) = A000720(n). [Missing factor e added to the comment by Antti Karttunen, Jul 29 2015]
From Antti Karttunen, Jul 29 2015: (Start)
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n.]
a(1) = 0; for n > 1, a(n) = (A067029(n) * (2^(A055396(n)-1))) + a(A028234(n)).
Other identities. For all n >= 0:
a(A019565(n)) = n.
a(A260443(n)) = n.
a(A206296(n)) = A000129(n).
a(A005940(n+1)) = A087808(n).
a(A007913(n)) = A248663(n).
a(A007947(n)) = A087207(n).
a(A283477(n)) = A005187(n).
a(A284003(n)) = A006068(n).
a(A285101(n)) = A028362(1+n).
a(A285102(n)) = A068052(n).
Also, it seems that a(A163511(n)) = A135529(n) for n >= 1. (End)
a(1) = 0, a(2n) = 1+a(n), a(2n+1) = 2*a(A064989(2n+1)). - Antti Karttunen, Oct 11 2016
From Peter Munn, Jan 31 2020: (Start)
a(n^2) = a(A003961(n)) = 2 * a(n).
a(A297845(n,k)) = a(n) * a(k).
a(n) = a(A225546(n)).
a(A329332(n,k)) = n * k.
a(A329050(n,k)) = 2^(n+k).
(End)
From Antti Karttunen, Feb 02-25 2020, Feb 01 2021: (Start)
a(n) = Sum_{d|n} A297108(d) = Sum_{d|A225546(n)} A297108(d).
a(n) = a(A097248(n)).
For n >= 2:
A001221(a(n)) = A322812(n), A001222(a(n)) = A277892(n).
A000203(a(n)) = A324573(n), A033879(a(n)) = A324575(n).
For n >= 1, A331750(n) = a(A000203(n)).
For n >= 1, the following chains hold:
A293447(n) >= a(n) >= A331740(n) >= A331591(n).
a(n) >= A087207(n) >= A248663(n).
(End)
a(n) = A087207(A097248(n)). - Flávio V. Fernandes, Jul 16 2025

Entry revised by Antti Karttunen, Jul 29 2015

More linking formulas added by Antti Karttunen, Apr 18 2017

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

A322812 a(n) = A001221(A048675(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 3, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 3, 3, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 3, 3, 3, 3, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2

Offset: 2

Antti Karttunen, Dec 27 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 2..10000

Cf. A001221, A048675, A277892, A322811, A322815, A322816.

A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A322812(n) = omega(A048675(n));

a(n) = A001221(A048675(n)).

a(n) <= A277892(n).

A324573 a(1) = 0; for n > 1, a(n) = sigma(A048675(n)).

Original entry on oeis.org

0, 1, 3, 3, 7, 4, 15, 4, 7, 6, 31, 7, 63, 13, 12, 7, 127, 6, 255, 12, 18, 18, 511, 6, 15, 48, 12, 18, 1023, 8, 2047, 6, 39, 84, 28, 12, 4095, 176, 54, 8, 8191, 12, 16383, 39, 15, 258, 32767, 12, 31, 13, 144, 54, 65535, 8, 42, 12, 252, 800, 131071, 15, 262143, 1302, 28, 12, 91, 20, 524287, 144, 528, 14, 1048575, 8, 2097151, 2736, 18

Offset: 1

Antti Karttunen, Mar 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's prime signature data)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A048675, A324574, A324575.

Cf. also A277892, A322812, A323243.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A324573(n) = if(1==n,0,sigma(A048675(n)));

a(1) = 0; for n > 1, a(n) = A000203(A048675(n)).

A322815 Lexicographically earliest such sequence a that a(i) = a(j) => A048675(i) = A048675(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 5, 4, 7, 8, 4, 9, 10, 11, 4, 12, 7, 13, 11, 14, 15, 16, 7, 6, 17, 11, 14, 18, 19, 20, 7, 21, 22, 23, 11, 24, 25, 26, 19, 27, 28, 29, 21, 6, 30, 31, 11, 8, 10, 32, 26, 33, 19, 34, 28, 35, 36, 37, 6, 38, 39, 23, 11, 40, 41, 42, 32, 43, 44, 45, 19, 46, 47, 14, 35, 48, 49, 50, 6, 6, 51, 52, 23, 53, 54, 55, 41, 56, 10, 57, 43, 58, 59, 60, 19, 61, 15, 34, 14

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

Restricted growth sequence transform of A048675.
For all i, j > 1:
  a(i) = a(j) => A277892(i) = A277892(j),
  a(i) = a(j) => A322812(i) = A322812(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A046523, A048675, A277892, A322812, A322816.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
v322815 = rgs_transform(vector(up_to,n,A048675(n)));
A322815(n) = v322815[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

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

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

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A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

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

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A322812 a(n) = A001221(A048675(n)).

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A324573 a(1) = 0; for n > 1, a(n) = sigma(A048675(n)).

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