A155043 -id:A155043 - cp's OEIS frontend

A262676 Number of nonzero even numbers encountered when iterating A049820 starting from n: a(0) = 0 and for n >= 1, a(n) = (1-A000035(n)) + a(A049820(n)).

0, 0, 1, 0, 1, 0, 2, 0, 2, 2, 3, 2, 3, 2, 4, 2, 3, 2, 4, 2, 5, 2, 5, 2, 4, 5, 6, 2, 6, 2, 6, 2, 7, 2, 7, 2, 3, 2, 8, 2, 8, 2, 8, 2, 9, 2, 9, 2, 9, 9, 10, 2, 10, 2, 10, 2, 10, 2, 11, 2, 10, 2, 12, 2, 3, 2, 12, 2, 13, 2, 13, 2, 11, 2, 14, 2, 14, 2, 14, 2, 14, 14, 15, 14, 12, 14, 16, 14, 15, 14, 15, 14, 17, 14, 16, 14, 13, 14, 18, 14, 15, 14, 17

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

Number of even numbers encountered before zero is reached when starting from k = n and repeatedly applying the map that replaces k by k - d(k), where d(k) is the number of divisors of k (A000005). This count includes n itself if it is even, but excludes the zero.

Antti Karttunen, Table of n, a(n) for n = 0..10080
A. Karttunen, Ratio a(n)/A262677(n) drawn with the help of OEIS Plot2-script

Cf. A000005, A000035, A049820, A155043, A262677, A262680.

a(0) = 0; for n >= 1, a(n) = (1-A000035(n)) + a(A049820(n)).
Other identities. For all n >= 0:
A155043(n) = a(n) + A262677(n).

A262681 Odd bisection of A262680.

Original entry on oeis.org

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, 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, 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, 3, 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, 3, 3, 1

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

Number of perfect squares (A000290) encountered before zero is reached when starting from k = 2n+1 and repeatedly applying the map that replaces k by k - d(k), where d(k) is the number of divisors of k (A000005). This count includes n itself if it is a square, but excludes the zero.

Antti Karttunen, Table of n, a(n) for n = 0..10082

Cf. A000005, A000290, A010052, A049820, A155043, A262677, A262680, A262682.

(define (A262681 n) (A262680 (+ n n 1)))

a(n) = A262680((2*n)+1).

A262682 Even bisection of A262680.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

Number of perfect squares (A000290) encountered before zero is reached when starting from k = 2n and repeatedly applying the map that replaces k by k - d(k), where d(k) is the number of divisors of k (A000005). This count includes n itself if it is a square, but excludes the zero.

Antti Karttunen, Table of n, a(n) for n = 0..10082

Cf. A000005, A000290, A010052, A049820, A155043, A262677, A262680, A262681.

Scheme
```
(define (A262682 n) (A262680 (+ n n)))
```

a(n) = A262680(2*n).

A264988 The left edge of A263267.

Original entry on oeis.org

0, 1, 3, 5, 7, 13, 17, 19, 23, 27, 29, 31, 35, 37, 41, 43, 51, 53, 57, 59, 61, 65, 67, 71, 73, 77, 79, 143, 149, 151, 155, 157, 161, 163, 173, 177, 179, 181, 185, 191, 193, 199, 203, 209, 211, 215, 219, 223, 231, 233, 237, 239, 241, 249, 251, 263, 267, 269, 271, 277, 285, 291, 293, 299, 303, 315, 317, 321, 327, 333, 335, 337, 341, 347, 349, 357, 359, 369, 517, 531, 535, 523, 527

Offset: 0

Antti Karttunen, Nov 29 2015

nonn

The first point where the sequence is nonmonotonic is the dip from a(80) = 535 to a(81) = 523.

Antti Karttunen, Table of n, a(n) for n = 0..10001

Cf. A263260, A262508, A262509.
The left edge of irregular table A263267.
Cf. A263269 (the other edge).
Differs from A261089 for the first time at n=69, where a(69) = 333, while A261089(69) = 331.

(define (A264988 n) (if (zero? n) n (A263267 (A263260 (- n 1)))))

a(0) = 0; for n >= 1, a(n) = A263267(A263260(n-1)).
Other identities. For all n >= 0:
A155043(a(n)) = n.
a(A262508(n)) = A262509(n) = A263269(A262508(n)). [In case A262508 and A262509 are infinite sequences.]

A262907 a(n) = number of steps needed to reach a fixed point when starting with k = n, and repeatedly replacing k with k - A262904(k).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 4, 6, 5, 7, 6, 8, 7, 6, 8, 7, 9, 8, 8, 9, 8, 10, 8, 11, 9, 12, 10, 13, 12, 14, 9, 15, 13, 16, 13, 17, 15, 13, 16, 14, 16, 15, 14, 16, 14, 17, 18, 18, 16, 19, 17, 20, 18, 21, 17, 22, 15, 23, 18, 24, 22, 25, 20, 26, 21, 27, 19, 28, 23, 16, 16, 19, 28, 23, 23, 21, 27, 22, 27, 20, 20, 24, 20, 17, 23, 29, 29, 24, 22, 28, 24, 28, 30, 21

Offset: 0

Antti Karttunen, Oct 07 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65538

Cf. A262904, A262906, A262907.

Cf. also A155043, A262905.

If A262904(n) = 0 then a(n) = 0, otherwise a(n) = 1 + a(A262906(n)).

A262908 a(n) = largest k such that A049820(k + A262509(n)) <= A262509(n).

Original entry on oeis.org

53, 49, 69, 55, 53, 31, 47, 39, 25, 35, 31, 39, 37, 51, 33, 43, 33, 69, 65, 57, 43, 41, 57, 49, 33, 33, 43, 41, 37, 33, 37, 39, 35, 27, 41, 27, 43, 75, 177, 171, 173, 155, 45, 133, 107, 121, 111, 139, 78, 119, 123, 47, 65, 79, 77, 97, 81, 151, 149, 145, 111, 197, 375, 71, 59, 81, 259, 257

Offset: 1

Antti Karttunen, Oct 08 2015

nonn

For all nonzero terms a(n), A263083(n) = a(n) + A262509(n) and A155043(A263083(n)) < A155043(A262509(n)) because at each A262509(n) the "distance to zero", A155043 obtains a unique value A262508(n), thus no A049820-iteration trajectory starting from any k larger than A262509(n) and using a greater or equal number of steps to reach zero may bypass A262509(n) [i.e., without going through A262509(n)], because then A262508(n) would not be unique anymore. See also comments in A262909.

Cf. A049820, A155043, A262508, A262509, A262909, A263083.

(define (A262908 n) (let ((t (A262509 n))) (let loop ((k (A002183 (+ 2 (A261100 t))))) (cond ((<= (A049820 (+ t k)) t) k) (else (loop (- k 1)))))))

Other identities. For all n >= 1:

a(n) <= A262909(n).

A263083 a(n) = largest k such that A049820(k) <= A262509(n).

Original entry on oeis.org

119196, 119196, 119232, 119280, 119280, 119952, 119970, 120120, 120120, 120132, 120132, 120320, 120330, 120400, 120432, 120750, 120780, 120960, 120960, 120960, 120960, 120960, 121500, 121600, 121856, 122112, 122304, 122304, 122310, 122310, 122850, 123000, 123240, 123240, 123264, 123264, 123300, 123840, 24660720, 24660720, 24662484, 24662484, 24663804, 24665130, 24665130, 24665472, 24666048

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

When a(n) > A262509(n), then a(n) is the "farthest immediate bypasser" of A262509(n) [the n-th "constriction point" in the tree generated by edge-relation A049820(child) = parent], bypassing it in the single A049820-step. In contrast, A263081(n) gives the farthest node (by necessity a leaf-node) which bypasses A262509(n) in multiple A049820-steps.

Sequence b(n) = A155043(A262509(n)) - A155043(a(n)) = A262508(n) - A155043(a(n)) gives the following terms: 395, 396, 354, 363, 364, 399, 390, 419, 422, 420, 421, 442, 430, 437, 460, 456, 498, 511, 512, 513, 515, 516, 506, 509, 533, 543, 564, 565, 557, 558, 591, 608, 612, 613, 614, 617, 617, 655, 3240, 3241, 3236, 3239, 3291, 3346, 3350, 3373, 3451, 3455, 2, 3598, 3637, 3605, 3674, 3688, 3689, 3748, 3749, 3792, 3793, 3794, 3800, 3803, 3858, 3843, 3902, 3947, 3985, 3986, ... which tells how many steps shorter trajectory there is to zero (using A049820) for those bypassers than for the constriction points themselves.

Antti Karttunen, Table of n, a(n) for n = 1..68

Cf. A049820, A155043, A262509, A262908, A263081.

a(n) = A262509(n)+A262908(n).

A263269 The right edge of irregular table A263267.

Original entry on oeis.org

0, 2, 6, 12, 18, 22, 30, 34, 42, 46, 54, 58, 66, 70, 80, 88, 94, 102, 112, 116, 126, 124, 130, 138, 150, 148, 160, 158, 164, 184, 190, 194, 210, 214, 222, 234, 252, 246, 250, 258, 266, 272, 296, 312, 306, 320, 328, 340, 352, 364, 372, 354, 358, 368, 384, 392, 408, 402, 414, 418, 426, 434, 448, 460, 462, 470, 474, 486, 496, 510, 522, 530, 546, 558, 562, 566, 574, 582, 592, 598, 606, 630

Offset: 0

Antti Karttunen, Nov 29 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10001

Cf. A000005, A049820, A155043, A262508, A262509, A262686, A263260, A263267.

Cf. A264988 (the other edge).

a(n) = A263267(A263260(n)-1).
Other identities. For all n >= 0:
A155043(a(n)) = n.
a(A262508(n)) = A262509(n) = A264988(A262508(n)). [In case A262508 and A262509 are infinite sequences.]

A322987 Number of iterations of A049820(x) = x - A000005(x) needed to reach a square, when starting from x = n.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 3, 1, 0, 3, 1, 3, 2, 4, 2, 0, 3, 4, 4, 5, 4, 5, 5, 1, 0, 6, 6, 6, 7, 6, 8, 7, 8, 7, 9, 0, 10, 8, 10, 8, 11, 8, 12, 9, 11, 9, 12, 9, 0, 10, 13, 10, 14, 10, 14, 10, 15, 11, 16, 10, 17, 12, 16, 0, 18, 12, 19, 13, 19, 13, 20, 11, 21, 14, 20, 14, 22, 14, 23, 14, 0, 15, 1, 12, 1, 16, 2, 15, 3, 15, 3, 17, 4, 16, 4, 13, 5, 18, 5, 0

Offset: 0

Antti Karttunen, Jan 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..110880

Cf. A000290, A010052, A049820, A155043, A259934, A262514, A262680, A322996, A323073.

A322987(n) = if(issquare(n),0,1+A322987(n-numdiv(n)));

A322987(n) = { for(j=0,oo,if(issquare(n),return(j)); n -= numdiv(n)); };

If A010052(n) == 1 [when n is in A000290], then a(n) = 0, otherwise a(n) = 1+a(A049820(n)).
a(n) <= A155043(n).
For n >= 83, a(2*n) = A322996(2*n)-1. [Note that 2*83 = 166 > 144 = A262514(2).]

A263089 Least monotonic left inverse for A261089; a(n) = largest k for which A261089(k) <= n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 17, 17, 17, 17, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 26

Offset: 0

Antti Karttunen, Nov 24 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..12332

Cf. A155043, A261089, A263259.

cp's OEIS Frontend

A262676 Number of nonzero even numbers encountered when iterating A049820 starting from n: a(0) = 0 and for n >= 1, a(n) = (1-A000035(n)) + a(A049820(n)).

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A262681 Odd bisection of A262680.

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A262682 Even bisection of A262680.

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A264988 The left edge of A263267.

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A262907 a(n) = number of steps needed to reach a fixed point when starting with k = n, and repeatedly replacing k with k - A262904(k).

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A262908 a(n) = largest k such that A049820(k + A262509(n)) <= A262509(n).

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A263083 a(n) = largest k such that A049820(k) <= A262509(n).

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A263269 The right edge of irregular table A263267.

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A322987 Number of iterations of A049820(x) = x - A000005(x) needed to reach a square, when starting from x = n.

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