A283477 -id:A283477 - cp's OEIS frontend

A324337 a(n) = A002487(A006068(n)).

0, 1, 2, 1, 3, 2, 1, 3, 4, 3, 2, 5, 1, 4, 5, 3, 5, 4, 3, 7, 2, 7, 8, 5, 1, 5, 7, 4, 7, 5, 3, 8, 6, 5, 4, 9, 3, 10, 11, 7, 2, 9, 12, 7, 11, 8, 5, 13, 1, 6, 9, 5, 10, 7, 4, 11, 9, 7, 5, 12, 3, 11, 13, 8, 7, 6, 5, 11, 4, 13, 14, 9, 3, 13, 17, 10, 15, 11, 7, 18, 2, 11, 16, 9, 17, 12, 7, 19, 14, 11, 8, 19, 5, 18, 21, 13, 1, 7, 11, 6, 13, 9, 5, 14, 13, 10

Offset: 0

Antti Karttunen, Feb 23 2019

nonn

Like in A324338, a few terms preceding each position n = 2^k seem to be a batch of nearby Fibonacci numbers in some order.

For all n > 0 A324338(n)/A324337(n) constitutes an enumeration system of all positive rationals. For all n > 0 A324338(n) + A324337(n) = A071585(n). - Yosu Yurramendi, Oct 22 2019

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A002487, A006068, A063946, A071585, A248663, A283477, A324287, A324338.

Block[{f}, f[m_] := Module[{a = 1, b = 0, n = m}, While[n > 0, If[OddQ[n], b = a + b, a = a + b]; n = Floor[n/2]]; b]; Array[f@ Fold[BitXor, #, Quotient[#, 2^Range[BitLength[#] - 1]]] &, 106, 0]] (* Michael De Vlieger, Dec 14 2019, after Jean-François Alcover at A002487 and Jan Mangaldan at A006068 *)

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
A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
A324337(n) = A002487(A006068(n));

maxlevel <- 6 # by choice
#
b <- 0; A324338 <- 1; A324337 <- 1
for(i in 1:2^maxlevel) {
  b[2*i  ] <-     b[i]
  b[2*i+1] <- 1 - b[i]
  A324338[2*i  ] <- A324338[i]          + A324337[i]*   b[i]
  A324338[2*i+1] <- A324338[i]          + A324337[i]*(1-b[i])
  A324337[2*i  ] <- A324338[i]*(1-b[i]) + A324337[i]
  A324337[2*i+1] <- A324338[i]*   b[i]  + A324337[i]}
#
A324338[1:127]; A324337[1:127]
# Yosu Yurramendi, Oct 22 2019

From Yosu Yurramendi, Oct 22 2019: (Start)
a(2^m+        k) = A324338(2^m+2^(m-1)+k), m > 0, 0 <= k < 2^(m-1)
a(2^m+2^(m-1)+k) = A324338(2^m+        k), m > 0, 0 <= k < 2^(m-1). (End)
a(n) = A324338(A063946(n)), n > 0. Yosu Yurramendi, Nov 04 2019
a(n) = A002487(A248663(A283477(n))). - Antti Karttunen, Nov 06 2019
a(n) = A002487(1+A233279(n)). - Yosu Yurramendi, Nov 08 2019
From Yosu Yurramendi, Nov 28 2019: (Start)
a(2^(m+1)+k) - a(2^m+k) = A324338(k), m >= 0, 0 <= k < 2^m.
a(A059893(2^(m+1)+A001969(k+1))) - a(A059893(2^m+A001969(k+1))) = A071585(k), m >= 0, 0 <= k < 2^(m-1).
a(A059893(2^(m+1)+ A000069(k+1))) = A071585(k), m >= 1, 0 <= k < 2^(m-1). (End)
From Yosu Yurramendi, Nov 29 2019: (Start)
For n > 0:
A324338(n) + A324337(n) = A071585(n).
A324338(2*A001969(n)  )-A324337(2*A001969(n)  ) =  A071585(n-1)
A324338(2*A001969(n)+1)-A324337(2*A001969(n)+1) = -A324337(n-1)
A324338(2*A000069(n)  )-A324337(2*A000069(n)  ) = -A071585(n-1)
A324338(2*A000069(n)+1)-A324337(2*A000069(n)+1) =  A324338(n-1) (End)
a(n) = A002487(1+A233279(n)). - Yosu Yurramendi, Dec 27 2019

A324377 a(0) = 0; for n > 0, a(n) = A000265(A005187(n)).

Original entry on oeis.org

0, 1, 3, 1, 7, 1, 5, 11, 15, 1, 9, 19, 11, 23, 25, 13, 31, 1, 17, 35, 19, 39, 41, 21, 23, 47, 49, 25, 53, 27, 7, 57, 63, 1, 33, 67, 35, 71, 73, 37, 39, 79, 81, 41, 85, 43, 11, 89, 47, 95, 97, 49, 101, 51, 13, 105, 109, 55, 7, 113, 29, 117, 119, 15, 127, 1, 65, 131, 67, 135, 137, 69, 71, 143, 145, 73, 149, 75, 19

Offset: 0

Antti Karttunen, Feb 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16385
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000265, A005187, A283477, A322821, A324287, A324378, A324379.

A000265(n) = (n/2^valuation(n, 2));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A324377(n) = if(!n,n,A000265(A005187(n)));

a(0) = 0; for n > 0, a(n) = A000265(A005187(n)) = A005187(n) / 2^A324379(n).

a(n) = A322821(A283477(n)).

A283984 Sums of distinct nonzero terms of A007489: a(n) = Sum_{k>=0} A030308(n,k)*A007489(1+k).

Original entry on oeis.org

0, 1, 3, 4, 9, 10, 12, 13, 33, 34, 36, 37, 42, 43, 45, 46, 153, 154, 156, 157, 162, 163, 165, 166, 186, 187, 189, 190, 195, 196, 198, 199, 873, 874, 876, 877, 882, 883, 885, 886, 906, 907, 909, 910, 915, 916, 918, 919, 1026, 1027, 1029, 1030, 1035, 1036, 1038, 1039, 1059, 1060, 1062, 1063, 1068, 1069, 1071, 1072, 5913

Offset: 0

Antti Karttunen, Mar 19 2017

nonn

Indexing starts from zero, with a(0) = 0.

Cf. A000142, A007489, A030308, A059590, A070939, A276075, A276091, A283477, A283985.

A007489(n) = sum(k=1, n, k!);
A030308(n,k) = bittest(n,k);
A283984(n) = sum(i=0,(#binary(n)-1),A030308(n,i)*A007489(1+i));

(define (A283984 n) (A276075 (A283477 n)))

a(n) = Sum_{i=0..A070939(n)} A030308(n,i)*A007489(1+i).
a(n) = A276075(A283477(n)).
Other identities. For all n >= 0:
a(2^n) = A007489(n+1).

A324286 a(n) = A002487(A048675(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 4, 2, 1, 1, 3, 1, 2, 3, 5, 1, 3, 1, 6, 2, 3, 1, 3, 1, 3, 4, 7, 2, 2, 1, 8, 5, 3, 1, 5, 1, 4, 1, 9, 1, 2, 1, 4, 6, 5, 1, 3, 3, 5, 7, 10, 1, 1, 1, 11, 2, 2, 4, 7, 1, 6, 8, 5, 1, 3, 1, 12, 3, 7, 2, 9, 1, 1, 1, 13, 1, 2, 5, 14, 9, 7, 1, 4, 3, 8, 10, 15, 6, 3, 1, 5, 3, 3, 1, 11, 1, 9, 3

Offset: 1

Antti Karttunen, Feb 22 2019

nonn

Like A323902 and A323903, this also has quite a moderate growth rate, even though some terms of A048675 soon grow quite big.

Cf. A002487, A048675, A283477, A322821, A323902, A323903, A324287.

A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ From A048675
A324286(n) = A002487(A048675(n));

a(n) = A002487(A048675(n)) = A002487(A322821(n)).

a(A283477(n)) = A324287(n).

A283483 Sums of distinct nonzero terms of A003462: a(n) = Sum_{k>=0} A030308(n,k)*A003462(1+k).

Original entry on oeis.org

0, 1, 4, 5, 13, 14, 17, 18, 40, 41, 44, 45, 53, 54, 57, 58, 121, 122, 125, 126, 134, 135, 138, 139, 161, 162, 165, 166, 174, 175, 178, 179, 364, 365, 368, 369, 377, 378, 381, 382, 404, 405, 408, 409, 417, 418, 421, 422, 485, 486, 489, 490, 498, 499, 502, 503, 525, 526, 529, 530, 538, 539, 542, 543, 1093, 1094, 1097, 1098, 1106, 1107, 1110, 1111

Offset: 0

Antti Karttunen, Mar 19 2017

nonn

Indexing starts from zero, with a(0) = 0.

Cf. A000244, A003462, A005187, A030308, A070939, A090880, A283477.

A003462(n) = (3^n-1)/2;
A030308(n,k) = bittest(n,k);
A283483(n) = sum(i=0,(#binary(n)-1),A030308(n,i)*A003462(1+i));

(define (A283483 n) (A090880 (A283477 n)))

a(n) = Sum_{i=0..A070939(n)} A030308(n,i)*A003462(1+i).
a(n) = A090880(A283477(n)).
Other identities. For all n >= 0:
a(2^n) = A003462(n+1).

A283981 a(n) = A029931(n) - A280700(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 3, 3, 3, 0, 4, 4, 4, 4, 4, 6, 7, 0, 5, 5, 5, 5, 5, 7, 8, 5, 5, 8, 9, 8, 9, 11, 11, 0, 6, 6, 6, 6, 6, 8, 9, 6, 6, 9, 10, 9, 10, 12, 12, 6, 6, 10, 11, 10, 11, 13, 13, 10, 11, 14, 14, 14, 14, 14, 17, 0, 7, 7, 7, 7, 7, 9, 10, 7, 7, 10, 11, 10, 11, 13, 13, 7, 7, 11, 12, 11, 12, 14, 14, 11, 12, 15, 15, 15, 15, 15, 18, 7, 7, 12, 13, 12, 13, 15, 15, 12

Offset: 0

Antti Karttunen, Mar 19 2017

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A005187, A029931, A124757, A280700, A283475, A283477, A283982.

Table[#.Reverse@ Range@ Length@ # &@ IntegerDigits[n, 2] - DigitCount[2 n - DigitCount[2 n, 2, 1], 2, 1], {n, 0, 120}] (* Michael De Vlieger, Mar 20 2017, after Jean-François Alcover at A029931 *)

a(n) = if(n<1, 0, a(n - 2^logint(n,2)) + logint(n,2) + 1);
b(n) = if(n<1, 0, b(n\2) + n%2);
A(n) = b(2*n - b(2*n));
for(n=0, 150, print1(a(n) - A(n),", ")) \\ Indranil Ghosh, Mar 21 2017

import math
def L(n): return int(math.floor(math.log(n,2)))
def a(n): return 0 if n<1 else a(n - 2**L(n)) + L(n) + 1
def A(n): return bin(2*n - bin(2*n)[2:].count("1"))[2:].count("1")
print([a(n) - A(n) for n in range(151)]) # Indranil Ghosh, Mar 21 2017

(define (A283981 n) (- (A029931 n) (A280700 n)))

a(n) = A029931(n) - A280700(n).

a(n) = A283982(n) + A124757(n).

A283982 a(0) = 0, and for n > 0, a(n) = A070939(n) - A280700(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 0, 0, 3, 2, 1, 1, 0, 1, 1, 0, 4, 3, 2, 2, 1, 2, 2, 1, 0, 2, 2, 1, 1, 2, 1, 0, 5, 4, 3, 3, 2, 3, 3, 2, 1, 3, 3, 2, 2, 3, 2, 1, 0, 3, 3, 2, 2, 3, 2, 1, 1, 3, 2, 2, 1, 0, 2, 0, 6, 5, 4, 4, 3, 4, 4, 3, 2, 4, 4, 3, 3, 4, 3, 2, 1, 4, 4, 3, 3, 4, 3, 2, 2, 4, 3, 3, 2, 1, 3, 1, 0, 4, 4, 3, 3, 4, 3, 2, 2, 4, 3, 3, 2, 1, 3, 1, 1, 4, 3, 3, 2, 1, 3, 2

Offset: 0

Antti Karttunen, Mar 19 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A005187, A070939, A124757, A280700, A283475, A283477, A283981.

(define (A283982 n) (if (zero? n) n (- (A070939 n) (A280700 n))))

a(0) = 0, for n > 0, a(n) = A070939(n) - A280700(n).

a(n) = A283981(n) - A124757(n).

A322804 Numbers that can be written as a product of one or more consecutive primorial numbers.

Original entry on oeis.org

1, 2, 6, 12, 30, 180, 210, 360, 2310, 6300, 30030, 37800, 75600, 485100, 510510, 9699690, 14553000, 69369300, 87318000, 174636000, 223092870, 6469693230, 14567553000, 15330615300, 200560490130, 437026590000, 2622159540000, 4951788741900, 5244319080000

Offset: 1

Ilya Gutkovskiy, Apr 14 2020

nonn

David A. Corneth, Table of n, a(n) for n = 1..1668 (terms < 10^1000)
Eric Weisstein's World of Mathematics, Primorial
Index entries for sequences related to primorial numbers

Cf. A002110, A006939, A034882, A073485, A129912, A283477.

A309841 If n = Sum (2^e_k) then a(n) = Product ((e_k + 2)!).

Original entry on oeis.org

1, 2, 6, 12, 24, 48, 144, 288, 120, 240, 720, 1440, 2880, 5760, 17280, 34560, 720, 1440, 4320, 8640, 17280, 34560, 103680, 207360, 86400, 172800, 518400, 1036800, 2073600, 4147200, 12441600, 24883200, 5040, 10080, 30240, 60480, 120960, 241920, 725760, 1451520, 604800

Offset: 0

Ilya Gutkovskiy, Aug 19 2019

			21 = 2^0 + 2^2 + 2^4 so a(21) = 2! * 4! * 6! = 34560.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000142, A000178, A019565, A029930, A058295, A059590, A121663, A283477.

a:= n-> (l-> mul((i+1)!^l[i], i=1..nops(l)))(convert(n, base, 2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2020

nmax = 40; CoefficientList[Series[Product[(1 + (k + 2)! x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := (Floor[Log[2, n]] + 2)! a[n - 2^Floor[Log[2, n]]]; Table[a[n], {n, 0, 40}]

a(n)={vecprod([(k+1)! | k<-Vec(select(b->b, Vecrev(digits(n, 2)), 1))])} \\ Andrew Howroyd, Aug 19 2019

G.f.: Product_{k>=0} (1 + (k + 2)! * x^(2^k)).
a(0) = 1; a(n) = (floor(log_2(n)) + 2)! * a(n - 2^floor(log_2(n))).
a(2^(k-1)-1) = A000178(k).

A361376 Rewrite A129912(n), a product of distinct primorials P(i) = A002110(i) instead as a sum of powers 2^(i-1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 16, 11, 17, 12, 13, 18, 19, 32, 14, 33, 20, 15, 21, 34, 35, 22, 24, 64, 23, 36, 25, 65, 37, 26, 66, 38, 27, 67, 40, 128, 39, 41, 28, 68, 129, 29, 69, 42, 130, 48, 43, 30, 70, 72, 131, 49, 31, 71, 44, 73, 256, 132, 45, 50, 257, 133, 74, 51, 46, 80, 75, 258, 134, 136

Offset: 1

Michael De Vlieger, Jun 08 2023

nonn

Permutation of nonnegative numbers.

			a(1) = 0 by convention.
a(8) = 8 comes before a(9) = 7, since we interpret 8 = 2^3 instead as P(4) = 210, while for a(9), 7 = 2^2 + 2^1 + 2^0 becomes P(3)*P(2)*P(1) = 30*6*2 = 360. Because 210 < 360, 8 appears before 7 in this sequence.
Table relating a(n), n=1..19 with the set S(n) of indices of distinct primorial factors of A129912(n):
   n A129912(n)  S(n)   a(n)  A272011(a(n))
  -----------------------------------------
   1         1            0
   2         2   1        1   0
   3         6   2        2   1
   4        12   2,1      3   1,0
   5        30   3        4   2
   6        60   3,1      5   2,0
   7       180   3,2      6   2,1
   8       210   4        8   3
   9       360   3,2,1    7   2,1,0
  10       420   4,1      9   3,0
  11      1260   4,2     10   3,1
  12      2310   5       16   4
  13      2520   4,2,1   11   3,1,0
  14      4620   5,1     17   4,0
  15      6300   4,3     12   3,2
  16     12600   4,3,1   13   3,2,0
  17     13860   5,2     18   4,1
  18     27720   5,2,1   19   4,1,0
  19     30030   6       32   5
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..15303 (a(15303) = 2^29.)
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^6.
Michael De Vlieger, Plot terms S(n) = A272011(a(n)) at (x,y) = (n,S(n,k)) for n = 1..2^11.

Cf. A002110, A129912, A272011, A283477.

a6939[n_] := Product[Prime[n + 1 - i]^i, {i, n}];
g[m_] := Block[{f, j = 1},
  f[n_, i_, e_] :=
   If[n < m, Block[{p = Prime[i + 1]}, If[e == 1, Sow@ n];
     f[n p^e, i + 1, e];
     If[e > 1, f[n p^(e - 1), i + 1, e - 1]]]];
  Sort@ Reap[While[a6939[j] < m, f[2^j, 1, j]; j++]][[-1, 1]] ];
Map[Total@
     Map[2^(# - 1) &,
      Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} ] &[
FactorInteger[#][[All, -1]]] &, g[2^31]] (* Michael De Vlieger, Jun 08 2023, after Giovanni Resta at A129929 *)

Let S(n) be the set of indices of primorials P(i), reverse sorted, such that A129912(n) = Product_{k=1..m} S(n,k), where m = | S(n) |. Then a(n) = Sum_{k=1..m} 2^(S(n,k)-1).

cp's OEIS Frontend

A324337 a(n) = A002487(A006068(n)).

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A324377 a(0) = 0; for n > 0, a(n) = A000265(A005187(n)).

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A283984 Sums of distinct nonzero terms of A007489: a(n) = Sum_{k>=0} A030308(n,k)*A007489(1+k).

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A324286 a(n) = A002487(A048675(n)).

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A283483 Sums of distinct nonzero terms of A003462: a(n) = Sum_{k>=0} A030308(n,k)*A003462(1+k).

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A283981 a(n) = A029931(n) - A280700(n).

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A283982 a(0) = 0, and for n > 0, a(n) = A070939(n) - A280700(n).

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A322804 Numbers that can be written as a product of one or more consecutive primorial numbers.

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