A005941 -id:A005941 - cp's OEIS frontend

A103969 Positions n such that A005941(n) = A005940(n).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 24, 28, 32, 36, 40, 48, 56, 64, 72, 80, 96, 112, 128, 144, 160, 192, 224, 256, 288, 320, 384, 448, 512, 576, 640, 768, 896, 1024, 1152, 1280, 1536, 1792, 2048, 2304, 2560, 3072, 3584, 4096, 4608, 5120, 6144, 7168

Offset: 1

Robert G. Wilson v, Feb 22 2005

nonn

Sequence with n>=5 appears to be quintisected with the quintisections multiples of A000079 (powers of two): a(5m) = 5,10,20,40... = 5*2^(m-1) for m>0; a(5m+1) = 6,12,24,48,... = 6*2^(m-1); likewise a(5m+2) = 7*2^(m-1); a(5m+3) = 8*2^(m-1); a(5m+4) = 9*2^(m-1). - Ralf Stephan, Nov 13 2010
From Antti Karttunen, Aug 01 2023: (Start)
Numbers k for which A005940(A005940(k)) = k, or equally, for which A005941(A005941(k)) = k, i.e., numbers that are either fixed points of permutation A005940/A005941, or elements of its 2-cycles.
If n is a term then also 2*n is present, and vice versa.
Question: Are 1, 3, 5, 7 and 9 the only odd terms of this sequence?
(End)

			56 is in the sequence since A005940(56) = A005941(56) = 72.
7 is in the sequence since A005940(7) = 9, and A005940(9) = 7, thus also A005941(7) = 9, and A005941(9) = 7. - _Antti Karttunen_, Aug 01 2023

R. J. Mathar, Table of n, a(n) for n = 1..60.

Cf. A005940, A005941, A029747 (subsequence).

f[n_] := Block[{p = Partition[ Split[ Join[ IntegerDigits[n - 1, 2], {2}]], 2]}, Times @@ Flatten[ Table[q = Take[p, -i]; Prime[ Count[ Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}] ]]; t = Table[ f[n], {n, 10^4}]; u = Flatten[ Table[ Position[t, n, 1, 1], {n, 10^4}]]; Do[ If[ u[[n]] == {}, u[[n]] = {0}], {n, 10^4}]; Flatten[ Position[ Take[t, 10^4] - Flatten[u], 0]]

from math import prod
from itertools import accumulate, count, islice
from sympy import prime, primepi, factorint
from collections import Counter
def A103969_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:sum((1<A103969_list = list(islice(A103969_gen(),20)) # Chai Wah Wu, Mar 11 2023

Empirical g.f.: x*(1 +x +x^2 +x^3 +x^4)^2 / (1-2*x^5). - Colin Barker, Nov 18 2016

Definition corrected and example updated by R. J. Mathar, Mar 06 2010

A364551 Odd numbers k such that k is a multiple of A005941(k).

Original entry on oeis.org

1, 3, 5, 3125, 7875, 12005, 13365, 22869, 23595, 46475, 703395, 985439, 2084775, 2675673, 13619125, 19144125

Offset: 1

Antti Karttunen, Jul 28 2023

Odd numbers k such that k is a multiple of 1+A156552(k).
Sequence A005940(A364545(n)) sorted into ascending order.
This is a subsequence of A364561, so the comments given in A364564 apply also here (see also the example section).

			In all these cases, the right hand side is a divisor of the left hand side:
      Term   (and its factorization)             A005941(term)
         1   (unity)                         ->    1
         3   (prime)                         ->    3
         5   (prime)                         ->    5
      3125 = 5^5                             ->    125 = 5^3
      7875 = 3^2 * 5^3 * 7                   ->    375 = 3 * 5^3
     12005 = 5 * 7^4                         ->    245 = 5 * 7^2
     13365 = 3^5 * 5 * 11                    ->    1215 = 3^5 * 5
     22869 = 3^3 * 7 * 11^2                  ->    847 = 7 * 11^2
     23595 = 3 * 5 * 11^2 * 13               ->    715 = 5 * 11 * 13
     46475 = 5^2 * 11 * 13^2                 ->    845 = 5 * 13^2
    703395 = 3^2 * 5 * 7^2 * 11 * 29         ->    33495 = 3 * 5 * 7 * 11 * 29
    985439 = 7^3 * 13^2 * 17                 ->    2873 = 13^2 * 17
   2084775 = 3 * 5^2 * 7 * 11 * 19^2         ->    12635 = 5 * 7 * 19^2
   2675673 = 3^5 * 7 * 11^2 * 13             ->    11583 = 3^4 * 11 * 13
  13619125 = 5^3 * 13 * 17^2 * 29            ->    36125 = 5^3 * 17^2
  19144125 = 3^2 * 5^3 * 7 * 11 * 13 * 17    ->    21879 = 3^2 * 11 * 13 * 17.

Cf. A005940, A005941, A156552, A364545, A364549, A364564.

Subsequence of A364561, which is a subsequence of A364560.

A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
isA364551(n) = ((n%2)&&!(n%A005941(n)));

A364557 Möbius transform of A005941.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 8, 4, 4, 4, 16, 4, 32, 8, 4, 8, 64, 4, 128, 8, 8, 16, 256, 8, 8, 32, 8, 16, 512, 4, 1024, 16, 16, 64, 8, 8, 2048, 128, 32, 16, 4096, 8, 8192, 32, 8, 256, 16384, 16, 16, 8, 64, 64, 32768, 8, 16, 32, 128, 512, 65536, 8, 131072, 1024, 16, 32, 32, 16, 262144, 128, 256, 8, 524288, 16, 1048576, 2048

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

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

Cf. A005941, A008683, A297112, A297113, A297167, A364558.

A364557(n) = if(1==n, 1, 2^(primepi(vecmax(factor(n)[, 1]))+(bigomega(n)-omega(n))-1));

A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
A364557(n) = sumdiv(n,d,moebius(n/d)*A005941(d));

from sympy import factorint, primepi
def A364557(n): return 1<1 else 1 # Chai Wah Wu, Jul 29 2023

a(n) = Sum_{d|n} A008683(n/d) * A005941(d).

a(1) = 1; for n > 1, a(n) = A297112(n) = 2^(A297113(n)-1) = 2^A297167(n).

A246368 Permutation of natural numbers: a(n) = A227413(A005941(n)).

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 8, 5, 9, 13, 10, 17, 20, 19, 12, 11, 46, 23, 166, 41, 15, 29, 858, 59, 14, 71, 16, 67, 6186, 37, 58645, 31, 18, 199, 22, 83, 705348, 983, 32, 179, 10428487, 47, 184718194, 109, 21, 6659, 3840230006, 277, 27, 43, 65, 353

Offset: 1

Antti Karttunen, Aug 26 2014

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse: A246367.
Similar or related permutations: A005941, A156552, A227413, A246364, A246366.
Cf. A000035, A010051.

default(primelimit,(2^31)+(2^30));
A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n }; \\ This function from M. F. Hasler
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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A005941(n) = A156552(n)+1;
A227413(n) = if(1==n, 1, if(!(n%2), prime(A227413(n/2)), A002808(A227413((n-1)/2))));
A246368(n) = A227413(A005941(n));
for(n=1, 52, write("b246368.txt", n, " ", A246368(n)));

(define (A246368 n) (A227413 (A005941 n)))

a(n) = A227413(A005941(n)) = A227413(1+A156552(n)).
Other identities:
For all n >= 1,  A010051(a(n)) = 1 - A000035(n). [This permutation maps even numbers to primes and odd numbers to nonprimes, in some order, because the permutation A227413 has the same property and A005941 preserves the parity].

A364558 a(n) = A364557(n) - A000010(n), where A364557 is the Möbius transform of A005941, and A000010 (Euler phi) is the Möbius transform of A000027.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 6, 0, 20, 2, -4, 0, 48, -2, 110, 0, -4, 6, 234, 0, -12, 20, -10, 4, 484, -4, 994, 0, -4, 48, -16, -4, 2012, 110, 8, 0, 4056, -4, 8150, 12, -16, 234, 16338, 0, -26, -12, 32, 40, 32716, -10, -24, 8, 92, 484, 65478, -8, 131012, 994, -20, 0, -16, -4, 262078, 96, 212, -16, 524218, -8, 1048504

Offset: 1

Antti Karttunen, Jul 28 2023

sign

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

Cf. A000010, A005941, A364557, A364559 (inverse Möbius transform), A364565 (positions of 0's), A364566 (of terms < 0).

PARI
```
A364558(n) = (A364557(n)-eulerphi(n));
```

from math import prod
from sympy import factorint, primepi
def A364558(n): return (1<1 else 0 # Chai Wah Wu, Jul 29 2023

A364559 a(n) = A005941(n) - n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 6, 0, 20, 4, -4, 0, 48, -4, 110, 0, -2, 12, 234, 0, -12, 40, -12, 8, 484, -8, 994, 0, 2, 96, -14, -8, 2012, 220, 28, 0, 4056, -4, 8150, 24, -22, 468, 16338, 0, -24, -24, 80, 80, 32716, -24, -18, 16, 202, 968, 65478, -16, 131012, 1988, -24, 0, 4, 4, 262078, 192, 446, -28, 524218, -16

Offset: 1

Antti Karttunen, Jul 28 2023

sign

			a(528581) = -4 as A005941(528581) = 528577 = 528581-4. Notably, 528581 = 17^2 * 31 * 59, with divisors [1, 17, 31, 59, 289, 527, 1003, 1829, 8959, 17051, 31093, 528581]. Applying A364557 to these divisors gives [1, 64, 1024, 65536, 128, 1024, 65536, 65536, 2048, 131072, 65536, 131072], while applying Euler totient phi (A000010) to them gives [1, 16, 30, 58, 272, 480, 928, 1740, 8160, 15776, 27840, 473280], their differences being [0, 48, 994, 65478, -144, 544, 64608, 63796, -6112, 115296, 37696, -342208], whose sum is -4.

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

Cf. A005941, A364499, A364557, A364558 (Möbius transform).
Cf. A029747 (known positions of 0's), A364560 (of terms <= 0), A364562 (of terms > 0), A364576.
Cf. also A364288.

A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
A364559(n) = (A005941(n)-n);

from sympy import factorint, primepi
def A364559(n): return sum(1<Chai Wah Wu, Jul 29 2023

a(n) = -A364499(A005941(n)).

a(n) = Sum_{d|n} A364558(d).

A246366 Permutation of natural numbers: a(n) = A005941(A227413(n)).

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 7, 5, 8, 33, 12, 65, 18, 257, 16, 17, 10, 129, 11, 4097, 34, 2049, 19, 65537, 15, 8193, 24, 4194305, 21, 32769, 66, 1025, 20, 513, 14, 262145, 22, 16385, 13, 1099511627777, 1026, 2097153, 130, 68719476737, 30, 1048577, 35, 288230376151711745, 8194, 67108865, 40, 4398046511105, 2050, 8388609, 28

Offset: 1

Antti Karttunen, Aug 26 2014

nonn

Maps even numbers to terms of A000051 (2^n + 1) in some order.

Antti Karttunen, Table of n, a(n) for n = 1..95
Index entries for sequences that are permutations of the natural numbers

Inverse: A246365.
Related or similar permutations: A005941, A156552, A227413, A246364, A246368.
Cf. A000051.

default(primelimit,(2^31)+(2^30));
A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n }; \\ This function from M. F. Hasler
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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A005941(n) = A156552(n)+1;
A227413(n) = if(1==n, 1, if(!(n%2), prime(A227413(n/2)), A002808(A227413((n-1)/2))));
A246366(n) = A005941(A227413(n));
for(n=1, 95, write("b246366.txt", n, " ", A246366(n)));

(define (A246368 n) (A227413 (A005941 n)))

a(n) = A005941(A227413(n)) = 1 + A156552(A227413(n)).

A364548 Numbers k such that k divides A005941(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 97, 128, 160, 192, 194, 256, 320, 345, 384, 388, 512, 549, 640, 690, 768, 776, 1024, 1093, 1098, 1280, 1380, 1536, 1552, 2048, 2186, 2196, 2560, 2760, 3072, 3104, 4096, 4372, 4392, 5120, 5520, 6144, 6208, 8192, 8744, 8784, 10240, 11040, 12288, 12416, 16384

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Numbers k such that k divides 1+A156552(k).
Sequence A005940(A364546(.)) sorted into ascending order.
If k is a term, then also 2*k is present in this sequence, and vice versa.

Cf. A005940, A005941, A156552, A364546.
Subsequences: A029747, A364549 (odd terms).
Cf. also A364497.

A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
isA364548(n) = !(A005941(n)%n);

A364549 Odd numbers k that divide A005941(k).

Original entry on oeis.org

1, 3, 5, 97, 345, 549, 1093, 64621, 671515, 3280317, 8957089

Offset: 1

Antti Karttunen, Jul 28 2023

Sequence A005940(A364547(.)) sorted into ascending order.
Odd numbers k such that k divides 1+A156552(k).
The first ten terms factored:
        1   (unity)
        3   (prime)
        5   (prime)
       97   (prime)
      345 = 3*5*23
      549 = 3^2 * 61
     1093   (prime)
    64621   (prime)
   671515 = 5*13*10331
  3280317 = 3*79*13841.
Primes p present are those that occur as factors of 1 + 2^(A000720(p)-1).

Odd terms in A364548.
Cf. A000720, A005940, A005941, A156552.
Cf. also A364498, A364547, A364551.

A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
isA364549(n) = ((n%2)&&!(A005941(n)%n));

from itertools import count, islice
from sympy import primepi, factorint
def A364549_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue+(startvalue&1^1),1),2):
        if not (sum(pow(2,i+int(primepi(p))-1,n) for i, p in enumerate(factorint(n, multiple=True)))+1) % n:
            yield n
A364549_list = list(islice(A364549_gen(),8)) # Chai Wah Wu, Jul 28 2023

a(11) from Chai Wah Wu, Jul 28 2023

A364550 Numbers k such that k is a multiple of A005941(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 128, 160, 192, 256, 320, 384, 512, 640, 768, 1024, 1280, 1536, 2048, 2560, 3072, 3125, 4096, 5120, 6144, 6250, 7875, 8192, 10240, 12005, 12288, 12500, 13365, 15750, 16384, 20480, 22869, 23595, 24010, 24576, 25000, 26730, 31500, 32768, 40960, 45738, 46475

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Numbers k such that k is a multiple of 1+A156552(k).

If k is a term, then also 2*k is present in this sequence, and vice versa.

Cf. A005941, A156552, A364548.
Subsequence of A364560.
Subsequences: A029747, A364551 (odd terms).
Cf. also

A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552) - Antti Karttunen, Jul 28 2023
isA364550(n) = !(n%A005941(n));

cp's OEIS Frontend

A103969 Positions n such that A005941(n) = A005940(n).

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A364551 Odd numbers k such that k is a multiple of A005941(k).

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A364557 Möbius transform of A005941.

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A246368 Permutation of natural numbers: a(n) = A227413(A005941(n)).

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A364558 a(n) = A364557(n) - A000010(n), where A364557 is the Möbius transform of A005941, and A000010 (Euler phi) is the Möbius transform of A000027.

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A364559 a(n) = A005941(n) - n.

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A246366 Permutation of natural numbers: a(n) = A005941(A227413(n)).

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A364548 Numbers k such that k divides A005941(k).

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