A253560 -id:A253560 - cp's OEIS frontend

A336322 a(n) = A225546(A122111(n)).

1, 2, 3, 4, 6, 8, 5, 16, 9, 12, 10, 32, 15, 24, 18, 256, 30, 64, 7, 48, 27, 20, 14, 512, 36, 40, 81, 96, 21, 128, 42, 65536, 54, 60, 72, 1024, 35, 120, 45, 768, 70, 192, 105, 80, 162, 28, 210, 131072, 25, 144, 90, 160, 11, 4096, 108, 1536, 135, 56, 22, 2048, 33, 84, 243, 4294967296, 216, 384, 66, 240, 270, 288, 55, 262144, 110, 168, 324, 480, 50

Offset: 1

Antti Karttunen and Peter Munn, Jul 17 2020

nonn

A225546 and A122111 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A225546 maps the k-th prime to 2^2^(k-1), whereas A122111 maps it to 2^k.

In composing these permutations, this sequence maps the list of prime numbers to the squarefree numbers, as listed in A019565; and the "normal" numbers (A055932), as listed in A057335, to ascending powers of 2.

Index entries for sequences that are permutations of the natural numbers

A225546 composed with A122111.
Sorted even bisection: A335738.
Sorted odd bisection (excluding 1): A335740.
Sequences used to express relationship between terms of this sequence: A001222, A003961, A253560, A331590, A350066.
Sequences of sequences (S_1, S_2, ... S_j) with the property a(S_i) = S_{i+1}, or essentially so: (A033844, A000040, A019565), (A057335, A000079, A001146), (A000244, A011764), (A001248, A334110), (A253563, A334866).
The inverse permutation, A336321, lists sequences where the property is weaker (between the sets of terms).

a(A033844(m)) = A000040(m+1). [Offset corrected Peter Munn, Feb 14 2022]
a(A000040(m)) = A019565(m).
a(A057335(m)) = 2^m.
For m >= 1, a(2^m) = A001146(m-1).
a(A253563(m)) = A334866(m).
From Peter Munn, Feb 14 2022: (Start)
a(A253560(n)) = a(n)^2.
For n >= 2, a(A003961(n)) = A331590(a(n), 2^2^(A001222(n)-1)).
a(A350066(n, k)) = A331590(a(n), a(k)).
(End)

A369028 Exponential of Mangoldt function permuted by A253563.

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 3, 5, 2, 1, 1, 1, 3, 1, 5, 7, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 7, 11, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 11, 13, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Jan 12 2024

nonn

Also LCM-transform of A253563 (when viewed as an offset-1 sequence), because A253563 has the S-property explained in the comments of A368900.

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

Cf. A014963, A054429, A368900, A369029, A369030, A369053.

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A253550(n) = if(1==n, 1, (n/prime(A061395(n)))*prime(1+A061395(n)));
A253560(n) = if(1==n, 1, (n*prime(A061395(n))));
A253563(n) = if(n<2,(1+n),if(!(n%2),A253560(A253563(n/2)),A253550(A253563((n-1)/2))));
A369028(n) = A014963(A253563(n));

a(n) = A014963(A253563(n)).

a(1) = 0, and for n > 0, a(n) = lcm {1..A253563(n)} / lcm {1..A253563(n-1)}. [See comments]

A369029 Exponential of Mangoldt function permuted by A253565.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 1, 2, 7, 5, 1, 3, 1, 1, 1, 2, 11, 7, 1, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 13, 11, 1, 7, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 17, 13, 1, 11, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Jan 12 2024

nonn

Also LCM-transform of A253565 (when viewed as an offset-1 sequence), because A253565 has the S-property explained in the comments of A368900.

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

Cf. A014963, A253565, A368900, A369028, A369030, A369053.

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A253550(n) = if(1==n, 1, (n/prime(A061395(n)))*prime(1+A061395(n)));
A253560(n) = if(1==n, 1, (n*prime(A061395(n))));
A253565(n) = if(n<2,(1+n),if(!(n%2),A253550(A253565(n/2)),A253560(A253565((n-1)/2))));
A369029(n) = A014963(A253565(n));

a(n) = A014963(A253565(n)).

a(0) = 1, and for n > 0, a(n) = lcm {1..A253565(n)} / lcm {1..A253565(n-1)}. [LCM-transform, see comments]

A129598 a(n) = n * A111089(n).

Original entry on oeis.org

2, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 36, 169, 98, 75, 32, 289, 54, 361, 100, 147, 242, 529, 72, 125, 338, 81, 196, 841, 150, 961, 64, 363, 578, 245, 108, 1369, 722, 507, 200, 1681, 294, 1849, 484, 225, 1058, 2209, 144, 343, 250, 867, 676, 2809, 162, 605

Offset: 1

Antti Karttunen, May 01 2007

nonn

Conjecture: differs from A050399 at the positions given by A089966. E.g., a(15)=75, instead of A050399(15)=225, a(30)=150, instead of A050399(30)=450, a(33)=363, instead of A050399(33)=1089, a(45)=225, instead of A050399(45)=1225. Conjecture 2: for all n > 1, a(n) divides A050399(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000
G. Guninski, List of automatically found conjectural relations between this and other sequences

Row 2 of A129595.
Essentially the same as A253560, except that here we have a(1) = 2.
Cf. also A050399, A089966, A111089, A072995.

Join[{2},Table[n*Last@(First/@FactorInteger[n]),{n,2,200}]] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)

a(1) = 2; for n >= 2, a(n) = A253560(n).

A277332 a(n) = A253565(A003714(n)).

Original entry on oeis.org

1, 2, 3, 5, 9, 7, 25, 15, 11, 49, 35, 21, 75, 13, 121, 77, 55, 245, 33, 147, 105, 17, 169, 143, 91, 847, 65, 605, 385, 39, 363, 231, 165, 735, 19, 289, 221, 187, 1859, 119, 1183, 1001, 85, 845, 715, 455, 4235, 51, 507, 429, 273, 2541, 195, 1815, 1155, 23, 361, 323, 247, 3757, 209, 3179, 2431, 133, 2023, 1547, 1309, 13013, 95

Offset: 0

Antti Karttunen, Oct 12 2016

nonn

After the initial terms 1, 2 and 3, all other terms can be inductively generated by applying any finite composition-combination of A253560 and A253550 to 3, but with such a restriction that A253560 may not be applied twice in succession.
A permutation of A277334.
Note how A253565(A022340(n)) = A253565(2*A003714(n)) yields a permutation of A056911, odd squarefree numbers.

			55 = A253550(A253550(A253560(A253550(3)))), 55 is in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..10946

Cf. A003714, A022340, A253550, A253560, A253565.
Cf. A277334 (same sequence sorted into ascending order).
Cf. also A056911, A277006, A277331.

(define (A277332 n) (A253565 (A003714 n)))

a(n) = A253565(A003714(n)).

A286533 Restricted growth sequence of A278533 (prime-signature of A253563).

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 3, 2, 6, 7, 7, 5, 4, 5, 3, 2, 8, 9, 10, 7, 9, 11, 7, 5, 6, 7, 7, 5, 4, 5, 3, 2, 12, 13, 14, 9, 14, 15, 10, 7, 13, 15, 15, 11, 9, 11, 7, 5, 8, 9, 10, 7, 9, 11, 7, 5, 6, 7, 7, 5, 4, 5, 3, 2, 16, 17, 18, 13, 19, 20, 14, 9, 18, 21, 21, 15, 14, 15, 10, 7, 17, 20, 21, 15, 20, 22, 15, 11, 13, 15, 15, 11, 9, 11, 7, 5, 12, 13, 14, 9, 14, 15, 10, 7

Offset: 0

Antti Karttunen, May 17 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A046523, A253563, A286531, A286535, A286622.

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A253550(n) = if(1==n, 1, (n/prime(A061395(n)))*prime(1+A061395(n)));
A253560(n) = if(1==n, 1, (n*prime(A061395(n))));
A253563(n) = if(n<2,(1+n),if(!(n%2),A253560(A253563(n/2)),A253550(A253563((n-1)/2)))); \\ Would be better if memoized!
A278533(n) = A046523(A253563(n));
write_to_bfile(0,rgs_transform(vector(65538,n,A278533(n-1))),"b286533.txt");

A286535 Restricted growth sequence of A278535 (prime-signature of A253565).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 17 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A046523, A253565, A286531, A286533, A286622.

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A253550(n) = if(1==n, 1, (n/prime(A061395(n)))*prime(1+A061395(n)));
A253560(n) = if(1==n, 1, (n*prime(A061395(n))));
A253565(n) = if(n<2,(1+n),if(!(n%2),A253550(A253565(n/2)),A253560(A253565((n-1)/2)))); \\ Would be better if memoized!
A278535(n) = A046523(A253565(n));
write_to_bfile(0,rgs_transform(vector(65538,n,A278535(n-1))),"b286535.txt");

A387405 a(n) is the least k that is a multiple of d=A240991(n), with abundancy ratio sigma(k)/k equal to (sigma(d)+1)/d), or -1 if no such k exists.

Original entry on oeis.org

18, 54, 196, 1521, 486, 1372, 15376, 24025, 4374, 1032256, 67228, 39366, 476656, 257049, 744775, 27237961, 354294, 3341637, 3294172, 23088025, 870899121, 131096512, 3188646, 4990433449, 18837288001, 28697814, 43647148561, 458066416, 161414428, 131785698529, 8362054027, 274810802176, 564736653, 508339054441, 258280326

Offset: 1

Antti Karttunen, Aug 30 2025

Antti Karttunen, Table of n, a(n) for n = 1..67 (based on the b-file of A240991 as computed by Michel Marcus and Giovanni Resta)
C. A. Holdener and J. A. Holdener, Characterizing Quasi-Friendly Divisors, Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.4.

Cf. A006530, A240991, A253560, A387406.

A387405(n) = { my(d=A240991(n), r=(sigma(d)+1)/d); forstep(k=d,oo,d,if(sigma(k)/k==r, return(k))); }

a(n) = A253560(A240991(n)) = A006530(A240991(n)) * A240991(n). - Conjectured, most likely true, especially if A240991 is a subsequence of A387406.

A277331 a(n) = A253563(A003714(n)).

Original entry on oeis.org

1, 2, 4, 8, 6, 16, 12, 18, 32, 24, 36, 54, 30, 64, 48, 72, 108, 60, 162, 90, 150, 128, 96, 144, 216, 120, 324, 180, 300, 486, 270, 450, 750, 210, 256, 192, 288, 432, 240, 648, 360, 600, 972, 540, 900, 1500, 420, 1458, 810, 1350, 2250, 630, 3750, 1050, 1470, 512, 384, 576, 864, 480, 1296, 720, 1200, 1944, 1080, 1800, 3000, 840

Offset: 0

Antti Karttunen, Oct 12 2016

nonn

After the initial terms 1, 2 and 4, all other terms can be inductively generated by applying any finite composition-combination of A253560 and A253550 to 4, but with such a restriction that A253550 may not be applied twice in succession.

A permutation of A055932.

Antti Karttunen, Table of n, a(n) for n = 0..10946

Cf. A003714, A055932 (same sequence sorted into ascending order), A253550, A253560, A253563, A122111.

Cf. also A277006, A277332.

(define (A277331 n) (A253563 (A003714 n)))

a(n) = A253563(A003714(n)).

a(n) = A122111(A277006(n)).

A363473 Triangle read by rows: T(n, k) = k * prime(n - k + A061395(k)) for 1 < k <= n, and T(n, 1) = A008578(n).

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 10, 15, 8, 7, 14, 21, 12, 25, 11, 22, 33, 20, 35, 18, 13, 26, 39, 28, 55, 30, 49, 17, 34, 51, 44, 65, 42, 77, 16, 19, 38, 57, 52, 85, 66, 91, 24, 27, 23, 46, 69, 68, 95, 78, 119, 40, 45, 50, 29, 58, 87, 76, 115, 102, 133, 56, 63, 70, 121, 31, 62, 93, 92, 145, 114, 161, 88, 99, 110, 143, 36

Offset: 1

Werner Schulte, Jan 05 2024

Conjecture: this is a permutation of the natural numbers.

Generalized conjecture: Let T(n, k) = b(k) * prime(n - k + A061395(b(k))) for 1 < k <= n, and T(n, 1) = A008578(n), where b(n), n > 0, is a permutation of the natural numbers with b(1) = 1, then T(n, k), read by rows, is a permutation of the natural numbers.

			Triangle begins:
n\k :   1    2    3    4    5    6    7    8    9   10   11   12   13
=====================================================================
 1  :   1
 2  :   2    4
 3  :   3    6    9
 4  :   5   10   15    8
 5  :   7   14   21   12   25
 6  :  11   22   33   20   35   18
 7  :  13   26   39   28   55   30   49
 8  :  17   34   51   44   65   42   77   16
 9  :  19   38   57   52   85   66   91   24   27
10  :  23   46   69   68   95   78  119   40   45   50
11  :  29   58   87   76  115  102  133   56   63   70  121
12  :  31   62   93   92  145  114  161   88   99  110  143   36
13  :  37   74  111  116  155  138  203  104  117  130  187   60  169
etc.

Cf. A000040, A001747, A001749, A001750, A008578, A112773, A138636, A253560, A272470, A061395.

T(n, k) = { if(k==1, if(n==1, 1, prime(n-1)), i=floor((k+1)/2);
            while(k % prime(i) != 0, i=i-1); k*prime(n-k+i)) }

def prime(n): return sloane.A000040(n)
def A061395(n): return prime_pi(factor(n)[-1][0]) if n > 1 else 0
def T(n, k):
     if k == 1: return prime(n - 1) if n > 1 else 1
     return k * prime(n - k + A061395(k))
for n in range(1, 11): print([T(n,k) for k in range(1, n+1)])
# Peter Luschny, Jan 07 2024

T(n, n) = A253560(n) for n > 0.
T(n, 1) = A008578(n) for n > 0.
T(n, 2) = A001747(n) for n > 1.
T(n, 3) = A112773(n) for n > 2.
T(n, 4) = A001749(n-3) for n > 3.
T(n, 5) = A001750(n-2) for n > 4.
T(n, 6) = A138636(n-4) for n > 5.
T(n, 7) = A272470(n-3) for n > 6.

cp's OEIS Frontend

A336322 a(n) = A225546(A122111(n)).

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A369028 Exponential of Mangoldt function permuted by A253563.

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A369029 Exponential of Mangoldt function permuted by A253565.

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A129598 a(n) = n * A111089(n).

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Mathematica

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A277332 a(n) = A253565(A003714(n)).

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A286533 Restricted growth sequence of A278533 (prime-signature of A253563).

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A286535 Restricted growth sequence of A278535 (prime-signature of A253565).

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A387405 a(n) is the least k that is a multiple of d=A240991(n), with abundancy ratio sigma(k)/k equal to (sigma(d)+1)/d), or -1 if no such k exists.

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A277331 a(n) = A253563(A003714(n)).

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