A277811 -id:A277811 - cp's OEIS frontend

A302033 a(n) = A019565(A003188(n)).

1, 2, 6, 3, 15, 30, 10, 5, 35, 70, 210, 105, 21, 42, 14, 7, 77, 154, 462, 231, 1155, 2310, 770, 385, 55, 110, 330, 165, 33, 66, 22, 11, 143, 286, 858, 429, 2145, 4290, 1430, 715, 5005, 10010, 30030, 15015, 3003, 6006, 2002, 1001, 91, 182, 546, 273, 1365, 2730, 910, 455, 65, 130, 390, 195, 39, 78, 26, 13, 221, 442, 1326, 663, 3315, 6630, 2210, 1105

Offset: 0

Antti Karttunen & Peter Munn, Apr 16 2018

A squarefree analog of A207901 (and the subsequence consisting of its squarefree terms): Each term is either a divisor or a multiple of the next one, and the terms differ by a single prime factor. Compare also to A284003.

For all n >= 0, max(a(n + 1), a(n)) / min(a(n + 1), a(n)) = A094290(n + 1) = prime(valuation(n + 1, 2) + 1) = A000040(A001511(n + 1)). [See Russ Cox's Dec 04 2010 comment in A007814.] - David A. Corneth & Antti Karttunen, Apr 16 2018

Antti Karttunen, Table of n, a(n) for n = 0..8191

A permutation of A005117. Subsequence of A207901.
Cf. A302054 (gives the sum of prime divisors).
Cf. A000040, A001511, A003188, A019565, A034947, A059897, A064706, A094290.
Cf. also A277811, A283475, A284003.

Array[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[BitXor[#, Floor[#/2]], 2] &, 72, 0] (* Michael De Vlieger, Apr 27 2018 *)

A003188(n) = bitxor(n, n>>1);
A019565(n) = {my(j); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A302033(n) = A019565(A003188(n));

first(n) = {my(pr = primes(1 + logint(n, 2)), ex = vector(#pr, i, 1), res = vector(n)); res[1] = 1; for(i = 1, n-1, v = valuation(i, 2); res[i + 1] = res[i] * pr[v++] ^ ex[v]; ex[v]*=-1); res}

a(n) = A019565(A003188(n)).
a(n) = A284003(A064706(n)).
a(n+1) = A059897(a(n), A094290(n+1)). - Peter Munn, Apr 01 2019

A284003 a(n) = A007913(A283477(n)) = A019565(A006068(n)).

Original entry on oeis.org

1, 2, 6, 3, 30, 15, 5, 10, 210, 105, 35, 70, 7, 14, 42, 21, 2310, 1155, 385, 770, 77, 154, 462, 231, 11, 22, 66, 33, 330, 165, 55, 110, 30030, 15015, 5005, 10010, 1001, 2002, 6006, 3003, 143, 286, 858, 429, 4290, 2145, 715, 1430, 13, 26, 78, 39, 390, 195, 65, 130, 2730, 1365, 455, 910, 91, 182, 546, 273, 510510, 255255, 85085, 170170, 17017

Offset: 0

Antti Karttunen, Mar 18 2017

A squarefree analog of A302783. Each term is either a divisor or a multiple of the next one. In contrast to A302033 at each step the previous term can be multiplied (or divided), not just by a single prime, but possibly by a product of several distinct ones, A019565(A000975(k)). E.g., a(3) = 3, a(4) = 2*5*a(3) = 30. - Antti Karttunen, Apr 17 2018

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

A permutation of A005117.
Cf. A000975, A001222, A006068, A007913, A019565, A046523, A048675, A064707, A209281, A283477, A284004.
Cf. also A277811, A283475, A302033, A302783.

Table[Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Times @@ (p^Mod[e, 2])] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]]], {n, 0, 52}] (* Michael De Vlieger, Mar 18 2017 *)

A007913(n) = core(n);
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From Charles R Greathouse IV, Jun 28 2015
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A283477(n) = A108951(A019565(n));
A284003(n) = A007913(A283477(n));

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A284003(n) = A019565(A006068(n)); \\ (and use A019565 from above) - Antti Karttunen, Apr 16 2018

(define (A284003 n) (A007913 (A283477 n)))

a(n) = A007913(A283477(n)).
Other identities. For all n >= 0:
A048675(a(n)) = A006068(n).
A046523(a(n)) = A284004(n).
It seems that A001222(a(n)) = A209281(n).
a(n) = A019565(A006068(n)) = A302033(A064707(n)). - Antti Karttunen, Apr 16 2018

Name amended with a second formula by Antti Karttunen, Apr 16 2018

A277810 Square array A(r,c) = A019565(A277820(r,c)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

2, 6, 3, 10, 15, 30, 210, 21, 14, 5, 22, 1155, 462, 35, 70, 858, 39, 910, 55, 330, 105, 1870, 3315, 72930, 5005, 2002, 33, 42, 9699690, 5187, 2926, 85, 714, 2145, 770, 7, 46, 111546435, 238602, 11305, 248710, 3927, 390, 77, 154, 4002, 87, 93763670, 21505, 152490, 440895, 3094, 91, 546, 231, 7130, 13485, 620310, 1078282205, 2306486, 9867, 114114, 17017, 170170, 1365, 2310

Offset: 1

Antti Karttunen, Nov 01 2016

Permutation of squarefree numbers (A005117) after their initial term 1.

			The top left corner of the array:
    2,    6,     10,   210,      22,    858,      1870,    9699690
    3,   15,     21,  1155,      39,   3315,      5187,  111546435
   30,   14,    462,   910,   72930,   2926,    238602,   93763670
    5,   35,     55,  5005,      85,  11305,     21505, 1078282205
   70,  330,   2002,   714,  248710, 152490,   2306486,   60138078
  105,   33,   2145,  3927,  440895,   9867,   1870935,  691587897
   42,  770,    390,  3094,  114114, 520030,    162690,  581334754
    7,   77,     91, 17017,     133,  33649,     50141, 6685349671
  154,  546, 170170,   570,    6118, 254562, 357505330,   51269790
  231, 1365,   7293,  3135, 1312311, 983535,  11599797,  589602585

Antti Karttunen, Table of n, a(n) for n = 1..1540; the first 55 antidiagonals of the array

Transpose: A277809.
The topmost row: A123098, the leftmost column: A277811.
Cf. A005117, A019565, A277820.

(define (A277810 n) (A277810bi (A002260 n) (A004736 n)))
(define (A277810bi row col) (A019565 (A277820bi row col)))

A(r,c) = A019565(A277820(r,c)).

A277809 Transpose of square array A277810.

Original entry on oeis.org

2, 3, 6, 30, 15, 10, 5, 14, 21, 210, 70, 35, 462, 1155, 22, 105, 330, 55, 910, 39, 858, 42, 33, 2002, 5005, 72930, 3315, 1870, 7, 770, 2145, 714, 85, 2926, 5187, 9699690, 154, 77, 390, 3927, 248710, 11305, 238602, 111546435, 46, 231, 546, 91, 3094, 440895, 152490, 21505, 93763670, 87, 4002, 2310, 1365, 170170, 17017, 114114, 9867, 2306486, 1078282205, 620310, 13485, 7130

Offset: 1

Antti Karttunen, Nov 01 2016

See A277810.

			The top left corner of the array:
     2,    3,     30,     5,      70,     105,     42,     7,       154
     6,   15,     14,    35,     330,      33,    770,    77,       546
    10,   21,    462,    55,    2002,    2145,    390,    91,    170170
   210, 1155,    910,  5005,     714,    3927,   3094, 17017,       570
    22,   39,  72930,    85,  248710,  440895, 114114,   133,      6118
   858, 3315,   2926, 11305,  152490,    9867, 520030, 33649,    254562
  1870, 5187, 238602, 21505, 2306486, 1870935, 162690, 50141, 357505330

Antti Karttunen, Table of n, a(n) for n = 1..1275; the first 50 antidiagonals of the array

Transpose: A277810.
The topmost row: A277811, the leftmost column: A123098.
Permutation of squarefree numbers (A005117) after their initial term 1.
Cf. A019565, A277819.

(define (A277809 n) (A277810bi (A004736 n) (A002260 n))) ;; Code for A277810bi given in A277810.

A(r,c) = A019565(A277819(r,c)).

cp's OEIS Frontend

A302033 a(n) = A019565(A003188(n)).

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A284003 a(n) = A007913(A283477(n)) = A019565(A006068(n)).

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A277810 Square array A(r,c) = A019565(A277820(r,c)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A277809 Transpose of square array A277810.

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