A247503 -id:A247503 - cp's OEIS frontend

A284563 a(n) = A247503(A277324(n)).

2, 2, 2, 10, 10, 10, 50, 10, 10, 250, 250, 50, 250, 250, 50, 110, 110, 250, 6250, 1250, 1250, 31250, 6250, 550, 2750, 6250, 6250, 13750, 2750, 2750, 6050, 110, 110, 30250, 68750, 13750, 343750, 781250, 156250, 151250, 151250, 781250, 19531250, 1718750, 343750, 8593750, 756250, 6050, 30250, 756250, 1718750, 3781250, 3781250, 8593750, 18906250, 151250

Offset: 0

Antti Karttunen, Mar 29 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4096

Odd bisection of A284553.

Cf. A001222, A247503, A277324, A284564, A284565.

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p, 2])^e) &@ a[2 n + 1], {n, 0, 55}] (* Michael De Vlieger, Apr 05 2017 *)

A284563(n) = A284553(n+n+1); \\ Other code as in A284553.

(define (A284563 n) (A247503 (A277324 n)))
(define (A284563 n) (A284553 (+ n n 1)))

a(n) = A247503(A277324(n)).
a(n) = A284553((2*n)+1).
Other identities. For all n >= 0:
A001222(a(n)) = A284565(n).

A284553 Prime factorization representation of Stern polynomials B(n,x) with only the even powers of x present: a(n) = A247503(A260443(n)).

Original entry on oeis.org

1, 2, 1, 2, 5, 2, 5, 10, 1, 10, 25, 10, 5, 50, 5, 10, 11, 10, 25, 250, 5, 250, 125, 50, 11, 250, 25, 250, 55, 50, 55, 110, 1, 110, 275, 250, 55, 6250, 125, 1250, 121, 1250, 625, 31250, 55, 6250, 1375, 550, 11, 2750, 275, 6250, 605, 6250, 1375, 13750, 11, 2750, 3025, 2750, 55, 6050, 55, 110, 17, 110, 275, 30250, 55, 68750, 15125, 13750, 121

Offset: 0

Antti Karttunen, Mar 29 2017

nonn

a(n) = Prime factorization representation of Stern polynomials B(n,x) where the coefficients of odd powers of x are replaced by zeros. In other words, only the constant term and other terms with even powers of x are present. See the examples.

Proof that A001222(a(1+n)) matches Ralf Stephan's formula for A000360(n): Consider functions A001222(a(n)) and A001222(A284554(n)) (= A284556(n)). They can be reduced to the following mutual recurrence pair: b(0) = 0, b(1) = 1, b(2n) = c(n), b(2n+1) = b(n) + b(n+1) and c(0) = c(1) = 0, c(2n) = b(n), c(2n+1) = c(n) + c(n+1). From the definitions it follows that the difference b(n) - c(n) for even n is b(2n) - c(2n) = -(b(n) - c(n)), and for odd n, b(2n+1) - c(2n+1) = (b(n)+b(n+1))-(c(n)+c(n+1)) = (b(n)-c(n)) + (b(n+1)-c(n+1)). Then by induction, if we assume that for 3n, 3n+1, 3n+2, ..., 6n, the value of difference b(n)-c(n) is always [0, +1, -1; repeated], it follows that from 6n to 12n the differences are [0, +1, -1; 0, +1, -1; repeated], which proves that b(n) - c(n) = A102283(n). As an implication, recurrence b can be defined without referring to c as: b(0) = 0, b(1) = 1, b(2n) = b(n) - A102283(n), b(2n+1) = b(n)+b(n+1), and this is equal to Ralf Stephan's Oct 05 2003 formula for A000360, but shifted once right, with prepended zero.

			n A260443(n)                      Stern            With odd powers
             prime factorization  polynomial       of x cleared  -> a(n)
------------------------------------------------------------------------
0       1    (empty)              B_0(x) = 0                    0  |  1
1       2    p_1                  B_1(x) = 1                    1  |  2
2       3    p_2                  B_2(x) = x                    0  |  1
3       6    p_2 * p_1            B_3(x) = x + 1                1  |  2
4       5    p_3                  B_4(x) = x^2                x^2  |  5
5      18    p_2^2 * p_1          B_5(x) = 2x + 1               1  |  2
6      15    p_3 * p_2            B_6(x) = x^2 + x            x^2  |  5
7      30    p_3 * p_2 * p_1      B_7(x) = x^2 + x + 1    x^2 + 1  | 10
8       7    p_4                  B_8(x) = x^3                  0  |  1
9      90    p_3 * p_2^2 * p_1    B_9(x) = x^2 + 2x + 1   x^2 + 1  | 10
10     75    p_3^2 * p_2          B_10(x) = 2x^2 + x         2x^2  | 25

Antti Karttunen, Table of n, a(n) for n = 0..8192
S. Klavzar, U. Milutinovic and C. Petr, Stern polynomials, Adv. Appl. Math. 39 (2007) 86-95.

Cf. A000360, A001222, A003961, A064989, A102283, A247503, A260443, A284554, A284556, A284563 (odd bisection).

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p, 2])^e) &@ a@ n, {n, 0, 72}] (* Michael De Vlieger, Apr 05 2017 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ Cf. Charles R Greathouse IV's code for "ps" in A186891 and A277013.
A247503(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 2] *= (primepi(f[i, 1]) % 2); ); factorback(f); } \\ After Michel Marcus
A284553(n) = A247503(A260443(n));

(define (A284553 n) (A247503 (A260443 n)))

a(0) = 1, a(1) = 2, a(2n) = A003961(A284554(n)), a(2n+1) = a(n)*a(n+1).
Other identities. For all n >= 0:
a(n) = A247503(A260443(n)).
a(n) = A260443(n) / A284554(n).
a(n) = A064989(A284554(2n)).
A001222(a(1+n)) = A000360(n). [Proof in Comments section.]

A277324 Odd bisection of A260443 (the even terms): a(n) = A260443((2*n)+1).

Original entry on oeis.org

2, 6, 18, 30, 90, 270, 450, 210, 630, 6750, 20250, 9450, 15750, 47250, 22050, 2310, 6930, 330750, 3543750, 1653750, 4961250, 53156250, 24806250, 727650, 1212750, 57881250, 173643750, 18191250, 8489250, 25467750, 2668050, 30030, 90090, 40020750, 1910081250, 891371250, 9550406250, 455814843750, 212713593750

Offset: 0

Antti Karttunen, Oct 10 2016

nonn

From David A. Corneth, Oct 22 2016: (Start)
The exponents of the prime factorization of a(n) are first nondecreasing, then nonincreasing.
The exponent of 2 in the prime factorization of a(n) is 1. (End)

			A method to find terms of this sequence, explained by an example to find a(7). To find k = a(7), we find k such that A048675(k) = 2*7+1 = 15. 7 has the binary partitions: {[7, 0, 0], [5, 1, 0], [3, 2, 0], [1, 3, 0], [3, 0, 1], [1, 1, 1]}. To each of those, we prepend a 1. This gives the binary partitions of 15 starting with a 1. For example, for the first we get [1, 7, 0, 0]. We see that only [1, 5, 1, 0], [1, 3, 2, 0] and [1, 1, 1, 1] start nondecreasing, then nonincreasing, so we only check those. These numbers will be the exponents in a prime factorization. [1, 5, 1, 0] corresponds to prime(1)^1 * prime(2)^5 * prime(3)^1 * prime(4)^0 = 2430. We find that [1, 1, 1, 1] gives k = 210 for which A048675(k) = 15 so a(7) = 210. - _David A. Corneth_, Oct 22 2016

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A005811, A005940, A007306, A007949, A156552, A260443, A277189, A277323, A283484, A283975, A284267, A284268, A284563, A284564, A284573.

Cf. A277200 (same sequence sorted into ascending order).

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[a[2 n + 1], {n, 0, 38}] (* Michael De Vlieger, Apr 05 2017 *)

from sympy import factorint, prime, primepi
from operator import mul
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
def a260443(n): return n + 1 if n<2 else a003961(a260443(n//2)) if n%2==0 else a260443((n - 1)//2)*a260443((n + 1)//2)
def a(n): return a260443(2*n + 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 21 2017

a(n) = A260443((2*n)+1).
a(0) = 2; for n >= 1, a(n) = A260443(n) * A260443(n+1).
Other identities. For all n >= 0:
A007949(a(n)) = A005811(n). [See comments in A125184.]
A156552(a(n)) = A277189(n), a(n) = A005940(1+A277189(n)).
A048675(a(n)) = 2n + 1. - David A. Corneth, Oct 22 2016
A001222(a(n)) = A007306(1+n).
A056169(a(n)) = A284267(n).
A275812(a(n)) = A284268(n).
A248663(a(n)) = A283975(n).
A000188(a(n)) = A283484(n).
A247503(a(n)) = A284563(n).
A248101(a(n)) = A284564(n).
A046523(a(n)) = A284573(n).
a(n) = A277198(n) * A284008(n).
a(n) = A284576(n) * A284578(n) = A284577(n) * A000290(A284578(n)).

More linking formulas added by Antti Karttunen, Apr 16 2017

A343430 Part of n composed of prime factors of the form 3k-1.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 11, 4, 1, 2, 5, 16, 17, 2, 1, 20, 1, 22, 23, 8, 25, 2, 1, 4, 29, 10, 1, 32, 11, 34, 5, 4, 1, 2, 1, 40, 41, 2, 1, 44, 5, 46, 47, 16, 1, 50, 17, 4, 53, 2, 55, 8, 1, 58, 59, 20, 1, 2, 1, 64, 5, 22, 1, 68, 23, 10, 71, 8, 1, 2, 25, 4, 11, 2, 1, 80, 1, 82, 83, 4, 85

Offset: 1

Peter Munn, Jun 08 2021

Largest term of A004612 that divides n.
Modulo 6, the prime numbers are partitioned into 4 nonempty sets: {2}, {3}, primes of the form 6k-1 (A007528) and primes of the form 6k+1 (A002476). The modulo 3 partition is nearly the same, but unites the only even prime, 2, with primes of the form 6k-1 in the set of primes we use here.
A positive integer m is a Loeschian number (a term of A003136) if and only if a(A007913(m)) = 1, that is the squarefree part of m has no prime factors of the form 3k-1.

			n = 60 has prime factorization 60 = 2 * 2 * 3 * 5. Factors 2 = 3*1 - 1 and 5 = 3*2 - 1 have form 3k-1, whereas 3 does not (having form 3k). Multiplying the factors of form 3k-1, we get 2 * 2 * 5 = 20. So a(60) = 20.

Equivalent sequences for prime factors of other forms: A006519 (2 only), A000265 (2k+1), A038500 (3 only), A038502 (3k+/-1), A170818 (4k+1), A097706 (4k-1), A248909 (6k+1), A343431 (6k-1).
Range of terms: A004612 (closure under multiplication of A003627).
Cf. A002476, A007528, squarefree part (A007913) of terms of A003136.
First 28 terms are the same as A247503.

f[p_, e_] := If[Mod[p, 3] == 2, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 11 2021 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 1] + 1) % 3, f[i, 1] = 1); ); factorback(f); } \\ after Michel Marcus at A248909

from math import prod
from sympy import factorint
def A343430(n): return prod(p**e for p, e in factorint(n).items() if p%3==2) # Chai Wah Wu, Dec 23 2022

Completely multiplicative with a(p) = p if p is of the form 3k-1, otherwise a(p) = 1.
For k >= 1, a(n) = a(k*n) / gcd(k, a(k*n)).
a(n) = A006519(n) * A343431(n).
a(n) = (n / A038500(n)) / A248909(n) = A038502(n) / A248909(n).
A006519(a(n)) = a(A006519(n)) = A006519(n).
A343431(a(n)) = a(A343431(n)) = A343431(n).
A038500(a(n)) = a(A038500(n)) = 1.
A248909(a(n)) = a(A248909(n)) = 1.

A248101 Completely multiplicative with a(prime(n)) = prime(n)^((n+1) mod 2).

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 1, 3, 13, 7, 3, 1, 1, 9, 19, 1, 21, 1, 1, 3, 1, 13, 27, 7, 29, 3, 1, 1, 3, 1, 7, 9, 37, 19, 39, 1, 1, 21, 43, 1, 9, 1, 1, 3, 49, 1, 3, 13, 53, 27, 1, 7, 57, 29, 1, 3, 61, 1, 63, 1, 13, 3, 1, 1, 3, 7, 71, 9, 1, 37, 3, 19, 7, 39, 79

Offset: 1

Tom Edgar, Mar 03 2015

To compute a(n) replace odd-indexed primes in the prime factorization of n by 1.
a(p) = p if p is in A031215.
a(p) = 1 if p is in A031368.

			Since 10 = 2*5, 2 = prime(1), and 5 = prime(3), a(10) = 1*1 = 1.
Since 9 = 3^2 and 3 is an even-indexed prime, 3 = prime(2), then a(9) = 3^2 = 9.
Since 35 = 5*7, 5 = prime(3), and 7 = prime(4), we see that a(35) = 1*7 = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index to divisibility sequences

Cf. A031368, A031215, A247503, A001222, A027746, A049084.

a248101 = product . filter (even . a049084) . a027746_row
-- Reinhard Zumkeller, Mar 06 2015

f[n_] := Block[{a, g, pf = FactorInteger@ n}, a = PrimePi[First /@ pf]; g[x_] := If[Or[OddQ@ x, x == 0], 1, Prime@ x]; Times @@ Power @@@ Transpose@ {g /@ a, Last /@ pf}]; Array[f, 120] (* Michael De Vlieger, Mar 03 2015 *)
Array[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p + 1, 2])^e) &, 79] (* Michael De Vlieger, Apr 05 2017 *)

a(n) = {f = factor(n); for (i=1, #f~, f[i,2] *= (primepi(f[i,1])+1) % 2;); factorback(f);} \\ Michel Marcus, Mar 03 2015

n=100; evenIndexPrimes=[primes_first_n(2*n+2)[2*i+1] for i in [0..n]]
[prod([(x[0]^(x[0] in evenIndexPrimes))^x[1] for x in factor(n)]) for n in [1..n]]

When n = Product_{k>=1} prime(k)^r_k, a(n) = Product_{k>=1} prime(k)^(r_k*((k+1) mod 2)).
a(n) = n / A247503(n).
a(n) = Product(A027746(n,k): k = 1 .. A001222(n) and A049084(A027746(n,k)) is even). - Reinhard Zumkeller, Mar 06 2015

A366244 The largest infinitary divisor of n that is a term of A366242.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 2, 1, 10, 11, 3, 13, 14, 15, 16, 17, 2, 19, 5, 21, 22, 23, 6, 1, 26, 3, 7, 29, 30, 31, 32, 33, 34, 35, 1, 37, 38, 39, 10, 41, 42, 43, 11, 5, 46, 47, 48, 1, 2, 51, 13, 53, 6, 55, 14, 57, 58, 59, 15, 61, 62, 7, 16, 65, 66, 67, 17, 69, 70, 71

Offset: 1

Amiram Eldar, Oct 05 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A063694, A366242, A366243, A366245.

See the formula section for the relationships with A007913, A046100, A059895, A059896, A059897, A225546, A247503, A352780.

f[p_, e_] := p^BitAnd[e, Sum[2^k, {k, 0, Floor@ Log2[e], 2}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = sum(k = 0, e, (-2)^k*floor(e/2^k));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}

Multiplicative with a(p^e) = p^A063694(e).
a(n) = n / A366245(n).
a(n) >= 1, with equality if and only if n is a term of A366243.
a(n) <= n, with equality if and only if n is a term of A366242.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1-1/p)*(Sum_{k>=1} p^(A063694(k)-2*k)) = 0.35319488024808595542... .
From Peter Munn, Jan 09 2025: (Start)
a(n) = max({k in A366242 : A059895(k, n) = k}).
a(n) = Product_{k >= 0} A352780(n, 2k).
Also defined by:
- for n in A046100, a(n) = A007913(n);
- a(n^4) = (a(n))^4;
- a(A059896(n,k)) = A059896(a(n), a(k)).
Other identities:
a(n) = sqrt(A366245(n^2)).
a(A059897(n,k)) = A059897(a(n), a(k)).
a(A225546(n)) = A225546(A247503(n)).
(End)

A360646 Square array A(n, k), n, k > 0, read by antidiagonals upwards; A(n, k) = A066208(n) * A066207(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 5, 12, 14, 9, 8, 15, 28, 18, 13, 10, 24, 35, 36, 26, 19, 11, 30, 56, 45, 52, 38, 21, 16, 33, 70, 72, 65, 76, 42, 27, 17, 48, 77, 90, 104, 95, 84, 54, 29, 20, 51, 112, 99, 130, 152, 105, 108, 58, 37, 22, 60, 119, 144, 143, 190, 168, 135, 116, 74, 39

Offset: 1

Rémy Sigrist, Feb 15 2023

Every positive integer occurs exactly once.

			Array A(n, k) begins:
  n\k |  1   2    3    4    5    6    7    8    9   10
  ----+-----------------------------------------------
    1 |  1   3    7    9   13   19   21   27   29   37
    2 |  2   6   14   18   26   38   42   54   58   74
    3 |  4  12   28   36   52   76   84  108  116  148
    4 |  5  15   35   45   65   95  105  135  145  185
    5 |  8  24   56   72  104  152  168  216  232  296
    6 | 10  30   70   90  130  190  210  270  290  370
    7 | 11  33   77   99  143  209  231  297  319  407
    8 | 16  48  112  144  208  304  336  432  464  592
    9 | 17  51  119  153  221  323  357  459  493  629
   10 | 20  60  140  180  260  380  420  540  580  740

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A249327, A066207, A066208, A247503, A248101.

PARI
```
See Links section.
```

A(n, 1) = A066208(n) = A247503(A(n,k)).

A(1, k) = A066207(k) = A248101(A(n,k)).

A374611 If n = A066207(i) * A066208(j) for some i, j > 0 then a(n) = A066207(j) * A066208(i).

Original entry on oeis.org

1, 3, 2, 7, 9, 6, 4, 13, 5, 19, 21, 14, 8, 12, 18, 27, 29, 15, 10, 37, 11, 39, 43, 26, 49, 24, 16, 28, 17, 38, 53, 57, 42, 61, 36, 35, 20, 30, 22, 63, 71, 33, 23, 79, 45, 81, 87, 54, 25, 89, 58, 56, 31, 48, 91, 52, 32, 51, 101, 74, 34, 107, 40, 111, 72, 78

Offset: 1

Rémy Sigrist, Jul 14 2024

nonn

This sequence is a self-inverse permutation of the positive integers with infinitely many fixed points.

			For n = 42: 42 = 21 * 2 = A066207(7) * A066208(2), so a(42) = A066207(2) * A066208(7) = 3 * 11 = 33.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A374600 for a similar sequence.

Cf. A066207, A066208, A247503, A248101, A360646.

PARI
```
\\ See Links section.
```

a(A360646(n, k)) = A360646(k, n).

a(n) = n iff n = A066207(k) * A066208(k) for some k > 0.

cp's OEIS Frontend

A284563 a(n) = A247503(A277324(n)).

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A284553 Prime factorization representation of Stern polynomials B(n,x) with only the even powers of x present: a(n) = A247503(A260443(n)).

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A277324 Odd bisection of A260443 (the even terms): a(n) = A260443((2*n)+1).

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A343430 Part of n composed of prime factors of the form 3k-1.

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A248101 Completely multiplicative with a(prime(n)) = prime(n)^((n+1) mod 2).

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A366244 The largest infinitary divisor of n that is a term of A366242.

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A360646 Square array A(n, k), n, k > 0, read by antidiagonals upwards; A(n, k) = A066208(n) * A066207(k).

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