A261089 -id:A261089 - cp's OEIS frontend

A263077 a(n) = greatest k where A155043(k) < A155043(n).

0, 0, 2, 2, 6, 2, 12, 6, 6, 6, 12, 6, 18, 12, 18, 18, 22, 12, 30, 18, 30, 18, 34, 22, 22, 22, 42, 22, 48, 22, 60, 30, 60, 30, 72, 48, 84, 34, 84, 34, 96, 34, 108, 42, 96, 42, 108, 42, 48, 48, 120, 48, 132, 48, 132, 48, 140, 60, 140, 48, 140, 72, 140, 140, 140, 72, 140, 84, 140, 84, 140, 60, 140, 96, 140, 96, 150, 96, 156, 96, 108, 108, 120, 72, 120, 120, 132, 108, 140, 108, 140, 132, 140, 120, 140, 84

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..124340

Cf. A155043, A261089, A262503, A263078, A263079, A263080, A263082.

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Table[k = 3 n;
While[a@ k >= a@ n, k--]; k, {n, 96}] (* Michael De Vlieger, Oct 13 2015 *)

allocatemem((2^31)+(2^30));
uplim1 = 36756720 + 640; \\ = A002182(53) + A002183(53).
uplim2 = 36756720; \\ = A002182(53).
uplim3 = 32432400; \\ = A002182(52). Really just some Ad Hoc value smaller than above.
v155043 = vector(uplim1);
vother = vector(uplim3); \\ Contains A262503 and A263082 in succession.
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim1, v155043[i] = 1 + v155043[i-numdiv(i)]; if(!(i%1048576),print1(i,", ")));
A155043 = n -> if(!n,n,v155043[n]);
maxlen = 0; for(i=1, uplim2, len = v155043[i]; vother[len] = i; maxlen = max(maxlen,len); if(!(i%1048576),print1(i,", "))); \\ First it will be A262503.
print("uplim2=", uplim2, " uplim3=", uplim3, " maxlen=", maxlen);
\\ Then we convert it to A263082:
m = 0; for(i=1, maxlen, m = max(m, vother[i]); vother[i] = m; if(!(i%1048576),print1(i,", ")));
A263082 = n -> if(!n,n,vother[n]);
A263077 = n -> A263082(A155043(n)-1);
\\ Finally we can compute A263077:
for(i=1, uplim3, write("b263077.txt", i, " ", A263077(i)); );

a(n) = A263082(A155043(n)-1).

A263078 a(n) = greatest k for which A155043(n+k) < A155043(n); a(n) = A263077(n)-n.

Original entry on oeis.org

-1, -2, -1, -2, 1, -4, 5, -2, -3, -4, 1, -6, 5, -2, 3, 2, 5, -6, 11, -2, 9, -4, 11, -2, -3, -4, 15, -6, 19, -8, 29, -2, 27, -4, 37, 12, 47, -4, 45, -6, 55, -8, 65, -2, 51, -4, 61, -6, -1, -2, 69, -4, 79, -6, 77, -8, 83, 2, 81, -12, 79, 10, 77, 76, 75, 6, 73, 16, 71, 14, 69, -12, 67, 22, 65, 20, 73, 18, 77, 16, 27, 26, 37, -12, 35, 34, 45, 20, 51, 18, 49, 40, 47, 26, 45, -12, 43, 42, 41, 40, 39, 30

Offset: 1

Antti Karttunen, Oct 09 2015

sign

			For n=1 we have A049820(1) = 0, thus A155043(1) = 1, and 0 is the only (and thus the largest) number from which zero can be reached with less steps (namely in zero steps, A155043(0) = 0), thus a(1) = 0 - 1 = -1.
For n=7, we have A155043(7) = 4 [as A049820(7) = 5, A049820(5) = 3, A049820(3) = 1, A049820(1) = 0], but there exists x=12 for which we have A049820(12) = 6, A049820(6) = 2, A049820(2) = 0, and this is the largest x such that A155043(x) < A155043(7), thus a(7) = 12 - 7 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..124340

Cf. A155043, A261089, A262503, A262909, A263077.

Cf. A263079 (indices of the negative terms), A263080 (of the positive terms).

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Table[k = 3 n;
While[a@ k >= a@ n, k--]; k - n, {n, 102}] (* Michael De Vlieger, Oct 13 2015 *)

A263078 = n -> A263077(n) - n;
for(n=1,124340,write("b263078.txt",n," ",A263078(n)));
\\ Other code as in A263077

a(n) = A263077(n)-n.

A263079 Numbers n for which there does not exist any x > n such that A155043(x) < A155043(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 38, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 60, 72, 84, 96, 104, 108, 120, 128, 132, 136, 140, 142, 144, 150, 152, 156, 160, 162, 168, 170, 180, 182, 184, 186, 188, 190, 192, 194, 198, 200, 204, 208, 210, 216, 220, 225, 228, 240, 248, 252, 260, 264, 276, 280, 288, 296, 300, 308, 312, 320, 328, 340, 352, 360

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

Numbers n for which A263077(n) < n.
Numbers n for which A263078(n) is negative.
Numbers n at which point A155043(n) is the greatest lower bound for the rest of its terms from A155043(n) onward.

			1 is present because A049820(1) = 0, thus A155043(1) = 1, while all the larger numbers require at least the same number of steps to reach zero.

Antti Karttunen, Table of n, a(n) for n = 1..9165

Cf. A155043, A261089, A262503, A263077, A263078.

Complement: A263080.

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Position[Table[k = 3 n; While[a@ k >= a@ n, k--]; k - n, {n, 200}], Integer?Negative] // Flatten (* _Michael De Vlieger, Oct 13 2015 *)

n=0; i=0; while(n < 124340, n++; if((A263077(n) < n), i++; write("b263079.txt",i," ",n)));
\\ Other code as in A263077.

;; With Antti Karttunen's IntSeq-library.
(define A263079 (MATCHING-POS 1 1 (COMPOSE negative? A263078)))

A263259 a(n) = number of integers k <= n for which A155043(k) = A155043(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 3, 4, 2, 5, 1, 3, 2, 3, 1, 4, 1, 4, 2, 5, 1, 2, 3, 4, 1, 5, 1, 6, 1, 3, 2, 4, 1, 2, 1, 2, 2, 3, 1, 4, 1, 2, 2, 3, 2, 4, 3, 4, 1, 5, 1, 6, 2, 7, 1, 3, 1, 8, 1, 2, 2, 3, 1, 3, 1, 3, 2, 4, 1, 4, 1, 3, 2, 4, 1, 5, 1, 6, 3, 4, 2, 4, 3, 4, 3, 5, 2, 6, 3, 4, 4, 5, 5, 5, 2, 4, 3, 6, 2, 5, 3, 7, 3, 5, 4, 8, 3, 6, 4, 7, 2, 7, 3, 8, 4, 4, 2, 7

Offset: 0

Antti Karttunen, Nov 24 2015

nonn

a(n) = one-based index of n in row A155043(n) of table A263265.

Antti Karttunen, Table of n, a(n) for n = 0..12333

Cf. A000005, A155043, A261089, A262503, A262507, A263089, A263265, A263279.

Other identities. For all n >= 0:
a(A261089(n)) = 1.
a(A262503(n)) = A262507(n).
A263265(A155043(n), a(n)) = n.

A262502 a(n) = least k such that A261104(k) = n; positions of records in A261104.

Original entry on oeis.org

0, 1, 3, 7, 11, 17, 23, 31, 40, 50, 62, 74, 86, 98, 110, 126, 142, 158, 174, 192, 210, 228, 248, 268, 288, 308, 328, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 738, 768, 798, 828, 860, 892, 924, 956, 988, 1020, 1052, 1084, 1116, 1148, 1180, 1212, 1244, 1280, 1316, 1352, 1388, 1424, 1460, 1496, 1532, 1568, 1604, 1640, 1676, 1716

Offset: 0

Antti Karttunen, Sep 24 2015

nonn

a(n+2) should give a safe upper bound for A262503(n), and actually seems to significantly overshoot it when n grows.

Antti Karttunen, Table of n, a(n) for n = 0..1641

Cf. A261104, A261089, A262503, A262507.

Cf. A262504 (first differences).

Other identities. For all n >= 0:

A261104(a(n)) = n.

A263080 Numbers n for which there exists x > n such that A155043(x) < A155043(n); numbers n for which A263078(n) is positive.

Original entry on oeis.org

5, 7, 11, 13, 15, 16, 17, 19, 21, 23, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

			5 is present, because if we start iterating A049820 from it as: A049820(5) = 3, A049820(3) = 1, A049820(1) = 0, we get a path of length 3 to reach zero, thus A155043(5) = 3. On the other hand, if we start from 6, the path is one step shorter: A049820(6) = 2, A049820(2) = 0 [i.e., A155043(6) = 2], thus there exists a number larger than 5 which results a shorter path to zero.

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

Cf. A155043, A261089, A262503, A263077, A263078.

Complement: A263079.

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Position[Table[k = 3 n; While[a@ k >= a@ n, k--]; k - n, {n, 121}], Integer?Positive] // Flatten (* _Michael De Vlieger, Oct 13 2015 *)

n=0; i=0; while(i < 10000, n++; if((A263077(n) > n), i++; write("b263080.txt",i," ",n)));
\\ Other code as in A263077.

;; With Antti Karttunen's IntSeq-library.
(define A263080 (MATCHING-POS 1 1 (COMPOSE positive? A263078)))

A266114 Least siblings in A263267-tree: numbers n for which there doesn't exist any k < n such that k - d(k) = n - d(n), where d(n) = A000005(n), the number of divisors of n.

Original entry on oeis.org

1, 3, 5, 6, 7, 8, 9, 11, 13, 14, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 56, 57, 58, 59, 61, 62, 65, 67, 68, 71, 72, 73, 74, 77, 79, 81, 82, 83, 84, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 109, 113, 114, 116, 118, 119, 120, 121, 123, 125, 127, 128

Offset: 1

Antti Karttunen, Dec 21 2015

nonn

Sequence A082284 sorted into ascending order, with zeros removed.

At least initially, most of the odd squares (A016754) seem to be in A266114, while most of the even squares (A016742) seem to be in A266115. The first exceptions to this are 63^2 = 3969 = A266115(1296), and 20^2 = 400 = A266114(269).

			3 is present, as 3 - A000005(3) = 1, but there are no any number k less than 3 for which k - A000005(k) = 1. (Although there is a larger sibling 4, for which 4 - A000005(4) = 1 also). Thus 3 is a smallest children of 1 in a tree A263267 defined by edge-relation child - A000005(child) = parent.

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

Cf. A000005, A082284, A263267.
Cf. A266112 (characteristic function).
Cf. A266113 (least monotonic left inverse).
Cf. A266115 (complement).
Cf. A065091, A261089, A264988, A262509 (subsequences).
Cf. also A016742, A016754.

Other identities. For all n >= 1:

A266113(a(n)) = n.

A262506 a(n) = A262503(n) - A259934(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 14, 22, 26, 30, 30, 38, 38, 6, 2, 8, 3, 5, 9, 6, 4, 6, 14, 2, 2, 2, 10, 6, 14, 18, 18, 28, 38, 0, 4, 0, 14, 14, 18, 8, 18, 18, 26, 22, 22, 26, 0, 0, 2, 10, 6, 26, 0, 4, 0, 6, 16, 26, 34, 42, 54, 66, 74, 78, 70, 72, 82, 64, 60, 60, 62, 74, 90, 2, 0, 2, 0, 2, 10, 0, 4, 14, 18, 22, 18, 22, 40, 58, 30, 2, 8, 22, 38, 42, 0, 0, 0, 4, 0, 0, 0, 4, 0, 2

Offset: 0

Antti Karttunen, Sep 25 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..64800
A. Karttunen, Ratio A261103(n)/a(n) drawn with the help of OEIS Plot2-script

Cf. A259934, A261089, A261103, A262503, A262505.

(define (A262506 n) (- (A262503 n) (A259934 n)))

a(n) = A262503(n) - A259934(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.]

cp's OEIS Frontend

A263077 a(n) = greatest k where A155043(k) < A155043(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A263078 a(n) = greatest k for which A155043(n+k) < A155043(n); a(n) = A263077(n)-n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A263079 Numbers n for which there does not exist any x > n such that A155043(x) < A155043(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

A263259 a(n) = number of integers k <= n for which A155043(k) = A155043(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A262502 a(n) = least k such that A261104(k) = n; positions of records in A261104.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A263080 Numbers n for which there exists x > n such that A155043(x) < A155043(n); numbers n for which A263078(n) is positive.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

A266114 Least siblings in A263267-tree: numbers n for which there doesn't exist any k < n such that k - d(k) = n - d(n), where d(n) = A000005(n), the number of divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A262506 a(n) = A262503(n) - A259934(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A264988 The left edge of A263267.