A064216 -id:A064216 - cp's OEIS frontend

A246342 a(0) = 12, after which, if a(n-1) = product_{k >= 1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k >= 1} (p_{k+1})^(c_k)), where p_k indicates the k-th prime, A000040(k).

12, 23, 15, 18, 38, 35, 39, 43, 24, 68, 86, 71, 37, 21, 28, 50, 74, 62, 56, 149, 76, 104, 230, 305, 235, 186, 278, 224, 1337, 1062, 2288, 8951, 4482, 16688, 67271, 33637, 16821, 66688, 571901, 338059, 181516, 258260, 455900, 1180337, 1080207, 1817863, 1157487, 984558, 1230848, 53764115

Offset: 0

Antti Karttunen, Aug 24 2014

nonn

Iterates of A048673 starting from value 12.
All numbers 1 .. 11 are in finite cycles of A048673/A064216, thus 12 is the smallest number in this cycle, regardless of whether the cycle is infinite or finite.
This sequence soon reaches much larger values than the corresponding A246343 (iterating the same cycle in the other direction). However, with the corresponding sequences starting from 16 (A246344 & A246345), there is no such pronounced difference, and with them the bias is actually the other way.

			Start with a(0) = 12; thereafter each new term is obtained by replacing each prime factor of the previous term with the next prime, to whose product 1 is added before it is halved:
12 = 2^2 * 3 = p_1^2 * p_2 -> ((p_2^2 * p_3)+1)/2 = ((9*5)+1)/2 = 23, thus a(1) = 23.
23 = p_9 -> (p_10 + 1)/2 = (29+1)/2 = 15, thus a(2) = 15.

Antti Karttunen, Table of n, a(n) for n = 0..1001

A246343 gives the terms of the same cycle when going in the opposite direction from 12.
Cf. A048673, A064216, A246344, A246345.
Cf. also A246281, A246282, A246351, A246352, A246361, A246362, A246371, A246372.

default(primelimit, 2^30);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
k = 12; for(n=0, 1001, write("b246342.txt", n, " ", k) ; k = A048673(k));
(Scheme, with memoization-macro definec)
(definec (A246342 n) (if (zero? n) 12 (A048673 (A246342 (- n 1)))))

a(0) = 12, and for n >= 1, a(n) = A048673(a(n-1)).

A246343 a(0) = 12, after which, if (2*a(n-1)) - 1 = product_{k >= 1} (p_k)^(c_k) then a(n) = product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

12, 19, 31, 59, 44, 46, 55, 107, 134, 166, 317, 398, 282, 557, 470, 622, 763, 531, 1051, 1267, 1807, 3607, 7211, 4522, 9041, 3700, 3725, 3982, 7951, 15889, 30053, 24018, 24189, 34535, 14630, 12916, 21769, 27599, 24524, 32678, 26094, 43073, 34446, 68881, 116479, 143359, 275221, 550439, 667462, 1051489

Offset: 0

Antti Karttunen, Aug 24 2014

nonn

Iterates of A064216 starting from value 12.

All numbers from 1 to 11 are in finite cycles of A048673/A064216, thus 12 is the smallest number in this cycle, regardless of whether it is infinite or finite.

			Start with a(0) = 12; then after each new term is obtained by doubling the previous term, from which one is subtracted, after which each prime factor is replaced with the previous prime:
12 -> ((2*12)-1) = 23 = p_9, and p_8 = 19, thus a(1) = 19.
19 -> ((2*19)-1) = 37 = p_12, and p_11 = 31, thus a(2) = 31.
31 -> ((2*31)-1) = 61 = p_18, and p_17 = 59, thus a(3) = 59.
59 -> ((2*59)-1) = 117 = 3*3*13 = p_2 * p_2 * p_6, and p_1 * p_1 * p_5 = 2*2*11 = 44, thus a(4) = 44.

Antti Karttunen, Table of n, a(n) for n = 0..1001

A246342 gives the terms of the same cycle when going to the opposite direction from 12.
Cf. A048673, A064216, A246344, A246345.
Cf. also A246281, A246282, A246351, A246352, A246361, A246362, A246371, A246372.

default(primelimit, 2^30);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
k = 12; for(n=0, 1001, write("b246343.txt", n, " ", k); k = A064216(k));
(Scheme, with memoization-macro definec)
(definec (A246343 n) (if (zero? n) 12 (A064216 (A246343 (- n 1)))))

a(0) = 12, a(n) = A064216(a(n-1)).

A246344 a(0) = 16, after which, if a(n-1) = product_{k >= 1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k >= 1} (p_{k+1})^(c_k)), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

16, 41, 22, 20, 32, 122, 101, 52, 77, 72, 338, 434, 611, 451, 280, 1040, 4820, 7907, 3960, 30713, 15364, 22577, 12154, 9791, 4902, 8108, 9131, 5815, 4099, 2056, 3551, 2095, 1474, 1385, 984, 2903, 1455, 1768, 4361, 5869, 2940, 19058, 18845, 13227, 11053, 8707, 4357, 2182, 1640, 4064, 15917, 9432, 46238

Offset: 0

Antti Karttunen, Aug 24 2014

nonn

Iterates of A048673 starting from value 16.

Either this sequence is actually part of the cycle containing 12 (see A246342) or 16 is the smallest member of this cycle (regardless of whether this cycle is finite or infinite), which follows because all numbers 1 .. 11 are in finite cycles, and also 13 and 14 are in closed cycles and 15 is in the cycle of 12.

			Start with a(0) = 16; then after each new term is obtained by replacing each prime factor of the previous term with the next prime, to whose product is added one before it is halved:
16 = 2^4 = p_1^4 -> ((p_2^4)+1)/2 = (3^4 + 1)/2 = (81+1)/2 = 41, thus a(1) = 41.
41 = p_13 -> ((p_14)+1)/2 = (43+1)/2 = 22, thus a(2) = 22.

Antti Karttunen, Table of n, a(n) for n = 0..1001

A246345 gives the terms of the same cycle when going to the opposite direction from 16.
Cf. A048673, A064216, A246342, A246343.
Cf. also A246281, A246282, A246351, A246352, A246361, A246362, A246371, A246372.

default(primelimit, 2^30);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i,1] = nextprime(f[i,1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
k = 16; for(n=0, 1001, write("b246344.txt", n, " ", k) ; k = A048673(k));
(Scheme, with memoization-macro definec)
(definec (A246344 n) (if (zero? n) 16 (A048673 (A246344 (- n 1)))))

a(0) = 16, and for n >= 1, a(n) = A048673(a(n-1)).

A246345 a(0) = 16, after which, if (2*a(n-1)) - 1 = product_{k >= 1} (p_k)^(c_k) then a(n) = product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

16, 29, 34, 61, 49, 89, 106, 199, 389, 310, 617, 524, 694, 1207, 1921, 3097, 3899, 4142, 3374, 3674, 4234, 8461, 16903, 20211, 37841, 22408, 26853, 26391, 48031, 68605, 137201, 81272, 108334, 137809, 266737, 512627, 347932, 497005, 982081, 1942279, 3855031, 5292209

Offset: 0

Antti Karttunen, Aug 24 2014

nonn

Iterates of A064216 starting from value 16.

A246374 Primes p such that if 2p-1 = product_{k >= 1} A000040(k)^(c_k), then p > product_{k >= 1} A000040(k-1)^(c_k).

Original entry on oeis.org

5, 11, 13, 17, 23, 41, 43, 53, 59, 61, 73, 83, 113, 131, 137, 149, 163, 167, 173, 179, 193, 233, 239, 257, 263, 281, 293, 311, 313, 347, 353, 383, 389, 401, 419, 431, 443, 449, 463, 479, 491, 503, 509, 523, 557, 563, 587, 593, 599, 613, 617, 641, 653, 677, 683, 743, 761, 773, 787, 797

Offset: 1

Antti Karttunen, Aug 25 2014

nonn

Primes p such that A064216(p) < p, or equally, A064989(2p-1) < p.

For all primes p here, 2p-1 must be composite (a necessary but not sufficient condition).

			5 is present, as 2*5 - 1 = 9 = p_2 * p_2, and p_1 * p_1 = 4, and 5 > 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Intersection of A000040 and A246371.
A246373 gives the primes not here.
Cf. A000040, A064216, A064989.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
n = 0; forprime(p=2,2^31, if((A064989((2*p)-1) < p), n++; write("b246374.txt", n, " ", p); if(n > 9999, break)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246374 (MATCHING-POS 1 1 (lambda (n) (and (prime? n) (< (A064216 n) n)))))

A270432 a(n) = number of terms A270430 <= n; least monotonic left inverse of A270430.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 5, 6, 7, 8, 8, 9, 10, 10, 10, 11, 12, 12, 12, 13, 13, 13, 13, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 22, 23, 24, 24, 25, 26, 27, 28, 28, 28, 28, 28, 28, 29, 30, 31, 31, 32, 33, 33, 33, 33, 33, 34, 34, 34, 34, 35, 35, 36, 37, 37, 37, 38, 39, 39, 39, 39, 39, 40, 41, 41, 42, 42, 42, 43, 44, 45

Offset: 1

Antti Karttunen, Mar 17 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..35000

Cf. A270430, A270433, A270434.

f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; g[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; s = Select[Range@ 144, Xor[EvenQ@ f@ #, OddQ@ g@ #] &]; Table[Count[s, k_ /; k <= n], {n, 84}] (* Michael De Vlieger, Mar 17 2016 *)

a(1) = 1, for n > 1, a(n) = 1-(A048673(n)-A064216(n) reduced modulo 2) + a(n-1).
Other identities. For all n >= 1:
a(n) = n - A270433(n).
a(A270430(n)) = n.

A270433 a(n) = number of terms A270431 <= n; least monotonic left inverse of A270431.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 8, 9, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 14, 15, 16, 17, 18, 19, 19, 19, 19, 20, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 28, 28, 29, 30, 30, 30, 31, 32, 33, 34, 34, 34, 35, 35, 36, 37, 37, 37, 37, 38, 39, 39, 40, 41, 42

Offset: 1

Antti Karttunen, Mar 17 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..35000

Cf. A048673, A064216, A270431, A270432, A270434.

f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; g[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; s = Select[Range@ 200, Xor[EvenQ@ f@ #, EvenQ@ g@ #] &] ; Table[Count[s, k_ /; k <= n], {n, 88}] (* Michael De Vlieger, Mar 17 2016 *)

a(1) = 0, for n > 1, a(n) = (A048673(n)-A064216(n) reduced modulo 2) + a(n-1).
Other identities. For all n >= 1:
a(n) = n - A270432(n).
a(A270431(n)) = n.

A349573 a(n) = A048673(n) - n, where A048673(n) = (A003961(n)+1) / 2, and A003961(n) shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

0, 0, 0, 1, -1, 2, -1, 6, 4, 1, -4, 11, -4, 3, 3, 25, -7, 20, -7, 12, 7, -2, -8, 44, 0, 0, 36, 22, -13, 23, -12, 90, 0, -5, 4, 77, -16, -3, 4, 55, -19, 41, -19, 15, 43, -2, -20, 155, 12, 24, -3, 25, -23, 134, -9, 93, 1, -11, -28, 98, -27, -6, 75, 301, -5, 32, -31, 18, 4, 46, -34, 266, -33, -12, 48, 28, -5, 50, -37

Offset: 1

Antti Karttunen, Nov 23 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A048673, A064216, A349571.

Cf. A048674 (positions of zeros), A246351 (negative terms), A246281 (nonpositive terms), A246352 (nonnegative terms), A246282 (positive terms), A269860 (even terms), A269861 (odd terms).

f[p_, e_] := NextPrime[p]^e; a[1] = 0; a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2 - n; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A349573(n) = (A048673(n)-n);

a(n) = A048673(n) - n.

a(n) = Sum_{d|n, dA349571(n/d).

A247283 Positions of subrecords in A048673.

Original entry on oeis.org

5, 7, 9, 15, 18, 27, 36, 54, 72, 108, 144, 216, 288, 432, 576, 864, 1152, 1728, 2304, 3456, 4608, 6912, 9216, 13824, 18432, 27648, 36864, 55296, 73728, 110592, 147456, 221184, 294912, 442368, 589824, 884736, 1179648, 1769472, 2359296, 3538944, 4718592, 7077888

Offset: 1

Antti Karttunen, Sep 11 2014

nonn

Odd bisection seems to be A116453 (i.e. A005010, 9*2^n from a(3)=9 onward).

After terms 7 and 15, even bisection from a(6)=27 onward seems to be A175806 (27*2^n).

			The fourth (A246360(4) = 5) and the fifth (A246360(5) = 8) record of A048673 (1, 2, 3, 5, 4, 8, ...) occur at A029744(4) = 4 and A029744(5) = 6 respectively. In range between, the maximum must occur at 5, thus a(4-3) = a(1) = 5. (All the previous records of A048673 are in consecutive positions, 1, 2, 3, 4, thus there are no previous subrecords).
The ninth (A246360(9) = 68) and the tenth (A246360(10) = 122) record of A048673 occur at A029744(9) = 24 and A029744(10) = 32 respectively. For n in range 25 .. 31 the values of A048673 are: 25, 26, 63, 50, 16, 53, 19, of which 63 is the maximum, and because it occurs at n = 27, we have a(9-3) = a(6) = 27.

Antti Karttunen, Table of n, a(n) for n = 1..60

A247284 gives the corresponding values.

Cf. A005010 (A116453), A175806, A029744, A048673, A064216, A246360, A245449.

\\ Compute A245449, A246360, A247283 and A247284 at the same time:
default(primelimit,(2^31)+(2^30));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A048673(n) = (A003961(n)+1)/2;
n = 0; i2 = 0; i3 = 0; ir = 0; prevmax = 0; submax = 0; while(n < 2^32, n++; a_n = A048673(n); if((A048673(a_n) == n), i2++; write("b245449.txt", i2, " ", n)); if((a_n > prevmax), if(submax > 0, i3++; write("b247283.txt", i3, " ", submaxpt); write("b247284.txt", i3, " ", submax)); prevmax = a_n; submax = 0; ir++; write("b029744_empirical.txt", ir, " ", n); write("b246360_empirical.txt", ir, " ", a_n), if((a_n > submax), submax = a_n; submaxpt = n)));

(definec (A247283 n) (max_pt_in_range A048673 (+ (A029744 (+ n 3)) 1) (- (A029744 (+ n 4)) 1)))
(define (max_pt_in_range intfun lowlim uplim) (let loop ((i (+ 1 lowlim)) (maxnow (intfun lowlim)) (maxpt lowlim)) (cond ((> i uplim) maxpt) (else (let ((v (intfun i))) (if (> v maxnow) (loop (+ 1 i) v i) (loop (+ 1 i) maxnow maxpt)))))))

a(n) = A064216(A247284(n)).
Conjectures from Chai Wah Wu, Jul 30 2020: (Start)
a(n) = 2*a(n-2) for n > 6.
G.f.: x*(3*x^5 - x^3 + x^2 - 7*x - 5)/(2*x^2 - 1). (End)

A253893 a(1) = 0, for n > 1, a(n) = 1 + a(A253889(n)).

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 2, 3, 4, 3, 4, 3, 3, 4, 4, 3, 3, 4, 4, 5, 5, 3, 5, 4, 4, 5, 4, 5, 5, 5, 4, 5, 5, 5, 6, 6, 4, 5, 6, 4, 6, 5, 5, 6, 5, 5, 5, 6, 4, 6, 6, 4, 6, 6, 5, 6, 5, 5, 6, 5, 6, 4, 6, 6, 6, 6, 4, 7, 7, 6, 6, 6, 5, 7, 7, 5, 6, 7, 6, 6, 7, 5, 7, 6, 6, 7, 5, 6, 7, 7, 6, 6, 6, 5, 7, 7, 6, 7, 7, 6, 6, 6, 6, 6, 7, 7, 6, 7, 7, 7, 7, 5, 7, 7, 6, 7, 7, 7, 7, 7, 5

Offset: 1

Antti Karttunen, Jan 22 2015

nonn

When A048673 is represented as a binary tree, then a(n) gives the distance of node containing n from 1 at top.

Antti Karttunen, Table of n, a(n) for n = 1..8192

One less than A253894.

Cf. A000523, A007051, A064216, A253786, A253887, A253889.

a(1) = 0, for n > 1, a(n) = 1 + A253893(A253889(n)).
a(n) = A000523(A064216(n)).
a(n) = A253894(n) - 1.
Other identities:
a(A007051(n)) = n for all n >= 0.

cp's OEIS Frontend

A246342 a(0) = 12, after which, if a(n-1) = product_{k >= 1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k >= 1} (p_{k+1})^(c_k)), where p_k indicates the k-th prime, A000040(k).

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A246343 a(0) = 12, after which, if (2*a(n-1)) - 1 = product_{k >= 1} (p_k)^(c_k) then a(n) = product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

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A246344 a(0) = 16, after which, if a(n-1) = product_{k >= 1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k >= 1} (p_{k+1})^(c_k)), where p_k indicates the k-th prime, A000040(k).

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A246345 a(0) = 16, after which, if (2*a(n-1)) - 1 = product_{k >= 1} (p_k)^(c_k) then a(n) = product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

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A246374 Primes p such that if 2p-1 = product_{k >= 1} A000040(k)^(c_k), then p > product_{k >= 1} A000040(k-1)^(c_k).

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A270432 a(n) = number of terms A270430 <= n; least monotonic left inverse of A270430.

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A270433 a(n) = number of terms A270431 <= n; least monotonic left inverse of A270431.

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A349573 a(n) = A048673(n) - n, where A048673(n) = (A003961(n)+1) / 2, and A003961(n) shifts the prime factorization of n one step towards larger primes.

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