A029747 -id:A029747 - cp's OEIS frontend

A364576 Starting from k=1, each subsequent term is the next larger odd k such that A156552(k) < k and the ratio A156552(k)/k is nearer to 1.0 than for any previous k in the sequence.

1, 3, 5, 21, 323, 66297, 139965, 263375, 264845, 528581

Offset: 1

Antti Karttunen, Aug 06 2023

All the odd fixed points of map n -> A005940(n) [and its inverse, map n -> A005941(n)] are included in this sequence. This includes both the known odd fixed points, 1, 3 and 5 (see A029747), and any additional hypothetical odd composites that would satisfy the condition n == A005940(n).

This is a subsequence of A364561, so the comments given in A364564 apply also here.

			       k  A156552(k)    A156552(k)/k  k-(1+A156552(k)) factorization of k
       1:       0         0                0
       3:       2         0.6666667        0
       5:       4         0.8              0
      21:      18         0.8571429        2           (3 * 7)
     323:     320         0.9907121        2           (17 * 19)
   66297:   65714         0.9912062      582           (3 * 7^2 * 11 * 41)
  139965:  139306         0.9952917      658           (3 * 5 * 7 * 31 * 43)
  263375:  262364         0.9961614     1010           (5^3 * 7^2 * 43)
  264845:  264244         0.9977307      600           (5 * 7^2 * 23 * 47)
  528581:  528576         0.9999905        4           (17^2 * 31 * 59).

Subsequence of A364561.
Cf. A005940, A005941, A029747, A156552.
Cf. also A364551, A364564, A364572.

A245709 Fixed points of A245705 and A245706.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 43, 48, 53, 64, 80, 86, 96, 106, 128, 160, 172, 192, 212, 249, 256, 320, 344, 384, 417, 424, 498, 512, 640, 688, 768, 834, 848, 996, 1024, 1280, 1321, 1376, 1536, 1668, 1696, 1992, 2048, 2560, 2642, 2752, 3072, 3336, 3392, 3984, 4096, 5120, 5284, 5504, 5545, 6144, 6672, 6784, 6827, 7081, 7968, 8192

Offset: 1

Antti Karttunen, Jul 30 2014

nonn

The odd terms less than 2^25 are: 1, 3, 5, 43, 53, 249, 417, 1321, 5545, 6827, 7081, 8535, 1485465, 1876261, 3298409, 13937375.

Contains also all such numbers k that A245608(k) = A245708(k), because that condition implies that A245607(A245708(k)) = k = A245707(A245608(k)). Conjecture: contains no numbers outside of that set, that is, for all n, A245608(a(n)) = A245708(a(n)).

Antti Karttunen, Table of n, a(n) for n = 1..230

A000079 is a subsequence.

Cf. A245705, A245706, A245608, A245708, A029747, A244154.

PARI
```
isA245709(n) = (A245705(n) == n);
```

\\ Alternative implementation, based on given conjecture:
isA245709v2(n) = (A245608(n) == A245708(n));

;; With Antti Karttunen's IntSeq-library.
(define A245709 (FIXED-POINTS 1 1 A245705))

;; With Antti Karttunen's IntSeq-library.
(define A245709 (MATCHING-POS 1 1 (lambda (k) (= (A245608 k) (A245708 k)))))

A253789 Fixed points of f(n) = A252753(n-1).

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, 481, 512, 640, 768, 962, 1024, 1280, 1536, 1924, 2048, 2560, 3072, 3848, 4096, 5120, 6144, 7696, 8192, 10240, 12288, 15392, 16384, 20480, 24576

Offset: 1

Antti Karttunen, Jan 13 2015

nonn

Cf. A252753.
Also fixed points of f(n) = A252754(n)+1.
Differs from a subsequence A029747 for the first time at n=25, where a(25) = 481, while A029747 contains no odd terms after 1, 3 and 5.
No other odd numbers can occur than those listed at A253790.

A364541 Numbers k for which A005940(k) <= k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 35, 36, 40, 48, 64, 65, 66, 67, 68, 69, 70, 72, 80, 96, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 140, 144, 160, 192, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 270, 272, 273, 274, 276, 280, 288, 289, 320, 384, 385, 512

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

A029747 is included as a subsequence, because it gives the known fixed points of map n -> A005940(n).

Positions of nonpositive terms in A364499.
Cf. A005940, A364542.
Subsequences: A029747, A364540.

nn = 512; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Select[Range[nn], a[#] <= # &] (* Michael De Vlieger, Jul 28 2023 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
isA364541(n) = (A005940(n)<=n);

A364546 Numbers k such that k is a multiple of A005940(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, 1035, 1280, 1536, 2048, 2070, 2560, 3072, 4096, 4140, 5120, 6144, 8192, 8280, 10240, 12288, 16384, 16560, 20480, 24576, 32768, 33120, 40960, 49152, 65536, 66240, 81920, 98304, 131072, 132480, 163840

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Sequence A005941(A364548(.)) sorted into ascending order.
If k is a term, then also 2*k is present in this sequence, and vice versa.
A029747 is included as a subsequence, because it gives the known fixed points of map n -> A005940(n).

Positions of 1's in A364502.
Subsequence of A364541.
Subsequences: A029747, A364547 (odd terms).
Cf. A005940, A364544, A364548.
Cf. also A364496.

nn = 2^18; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Select[Range[nn], Divisible[#, a[#]] &] (* Michael De Vlieger, Jul 28 2023 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
isA364546(n) = !(n%A005940(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);

A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

Original entry on oeis.org

9, 18, 25, 36, 49, 50, 72, 81, 98, 100, 121, 144, 162, 169, 196, 200, 225, 242, 288, 289, 324, 338, 361, 392, 400, 441, 450, 484, 529, 576, 578, 625, 648, 676, 722, 729, 784, 800, 841, 882, 900, 961, 968, 1058, 1089, 1152, 1156, 1225, 1250, 1296, 1352, 1369

Offset: 1

Benoit Cloitre, Apr 18 2002

Previous name: sum(d|n,6d/(2+mu(d))) is odd, where mu(.) is the Moebius function, A008683.
From Peter Munn, Jul 06 2020: (Start)
Numbers that have an odd number of odd nonsquarefree divisors.
[Proof of equivalence to the name, where m denotes a positive integer:
(1) These properties are equivalent: (a) m has an even number of odd squarefree divisors; (b) m has a nontrivial odd part.
(2) These properties are equivalent: (a) m has an odd number of odd divisors; (b) the odd part of m is square.
(3) m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are both true or both false.
(4) The trivial odd part, 1, is a square, so (1)(b) and (2)(b) cannot both be false, which (from (1), (2)) means (1)(a) and (2)(a) cannot both be false.
(5) From (3), (4), m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are true.
(6) m satisfies the condition in the name if and only if (1)(b) and (2)(b) are true, which (from (1), (2)) is equivalent to (1)(a) and (2)(a) being true, and hence from (5), to m satisfying the condition at the start of this comment.]
(End)
Numbers whose sum of non-unitary divisors (A048146) is odd. - Amiram Eldar, Sep 16 2024

			To determine the odd part of 18, remove all factors of 2, leaving 9. 9 is a nontrivial square, so 18 is in the sequence. - _Peter Munn_, Jul 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Odd part.

A000265, A008683 are used in definitions of this sequence.
Lists of numbers whose odd part satisfies other conditions: A028982 (square), A028983 (nonsquare), A029747 (less than 6), A029750 (less than 8), A036349 (even number of prime factors), A038550 (prime), A070776 U {1} (power of a prime), A072502 (square of a prime), A091067 (has form 4k+3), A091072 (has form 4k+1), A093641 (noncomposite), A105441 (composite), A116451 (greater than 4), A116882 (less than or equal to even part), A116883 (greater than or equal to even part), A122132 (squarefree), A229829 (7-rough), A236206 (11-rough), A260488\{0} (has form 6k+1), A325359 (proper prime power), A335657 (odd number of prime factors), A336101 (prime power).
Cf. A048146, A091476.

Select[Range[1000], (odd = #/2^IntegerExponent[#, 2]) > 1 && IntegerQ @ Sqrt[odd] &] (* Amiram Eldar, Sep 29 2020 *)

upto(n) = { my(res = List()); forstep(i = 3, sqrtint(n), 2, for(j = 0, logint(n\i^2, 2), listput(res, i^2<David A. Corneth, Sep 28 2020

Sum_{n>=1} 1/a(n) = 2 * Sum_{k>=1} 1/(2*k+1)^2 = Pi^2/4 - 2 = A091476 - 2 = 0.467401... - Amiram Eldar, Feb 18 2021

New name from Peter Munn, Jul 06 2020

A364544 Numbers k such that k divides A005940(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 125, 128, 160, 192, 245, 250, 256, 320, 375, 384, 490, 500, 512, 640, 715, 750, 768, 845, 847, 980, 1000, 1024, 1215, 1280, 1430, 1500, 1536, 1690, 1694, 1960, 2000, 2048, 2430, 2560, 2860, 2873, 3000, 3072, 3380, 3388, 3920, 4000, 4096, 4860, 5120

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

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

A029747 is included as a subsequence, because it gives the known fixed points of map n -> A005940(n).

Positions of 1's in A364501.
Subsequence of A364542.
Subsequences: A029747, A364545 (odd terms).
Cf. A005940.
Cf. also A364494, A364546.

nn = 5120; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Select[Range[nn], Divisible[a[#], #] &] (* Michael De Vlieger, Jul 28 2023 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
isA364544(n) = !(A005940(n)%n);

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));

A364565 Numbers k at which point A364557 (the Möbius transform of A005941) is equal to A000010 (Euler phi function).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 85, 96, 128, 160, 170, 192, 256, 320, 340, 384, 512, 640, 680, 768, 1024, 1280, 1360, 1536, 2048, 2560, 2720, 3072, 4096, 5120, 5440, 6144, 8192, 10240, 10880, 12288, 16384, 20480, 21760, 24576, 32768, 40960, 43520, 49152, 65536, 81920, 87040, 98304, 131072

Offset: 1

Antti Karttunen, Jul 29 2023

nonn

Question: Are there any other odd terms apart from 1, 3, 5, 85?

Positions of 0's in A364558.

Cf. A000010, A029747, A364557, A364566.

A364557(n) = if(1==n, 1, 2^(primepi(vecmax(factor(n)[, 1]))+(bigomega(n)-omega(n))-1));
isA364565(n) = (A364557(n)==eulerphi(n));

cp's OEIS Frontend

A364576 Starting from k=1, each subsequent term is the next larger odd k such that A156552(k) < k and the ratio A156552(k)/k is nearer to 1.0 than for any previous k in the sequence.

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A245709 Fixed points of A245705 and A245706.

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A253789 Fixed points of f(n) = A252753(n-1).

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A364541 Numbers k for which A005940(k) <= k.

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A364546 Numbers k such that k is a multiple of A005940(k).

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

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A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

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A364544 Numbers k such that k divides A005940(k).

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

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