A263081 -id:A263081 - cp's OEIS frontend

A045765 k - d(k) never takes these values, where d(k) = A000005(k).

7, 8, 13, 19, 20, 24, 25, 28, 33, 36, 37, 40, 43, 49, 50, 52, 55, 56, 63, 64, 66, 67, 68, 74, 75, 79, 85, 88, 98, 100, 103, 108, 109, 112, 113, 116, 117, 123, 124, 126, 131, 132, 133, 134, 136, 140, 143, 145, 150, 153, 156, 159, 160, 163, 164, 167, 168

Offset: 1

David W. Wilson

nonn

Complement of A236562. - Jaroslav Krizek, Feb 09 2014
Positions of zeros in A060990, leaf-nodes in the tree generated by edge-relation A049820(child) = parent. - Antti Karttunen, Oct 06 2015
Since A000005(x) <= 1 + x/2, k is in the sequence if there are no x <= 2*(k+1) with k = x - d(x). - Robert Israel, Oct 12 2015
This can be improved as: k is in the sequence if there are no x <= k + A002183(2+A261100(k)) with k = x - d(x). Cf. also A070319, A262686. - Antti Karttunen, Oct 12 2015
Luca (2005) proved that this seqeunce is infinite. - Amiram Eldar, Jul 26 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000
Florian Luca, On numbers not of the form n-omega(n), Acta Mathematica Hungarica, Vol. 106, No. 1 (2005), pp. 117-135.

Top row of A262898.
Cf. A000005, A002183, A049820, A060990, A070319, A236562, A236565, A261100, A262511, A262686, A262901, A262902, A262903, A262909, A263081.
Cf. A263091 (primes in this sequence), A263095 (squares).
Cf. A259934 (gives the infinite trunk of the same tree, conjectured to be unique).

N:= 1000: # to get all terms <= N
sort(convert({$1..N} minus {seq(x - numtheory:-tau(x), x=1..2*(1+N))},list)); # Robert Israel, Oct 12 2015

lim = 10000; Take[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], 57] (* Michael De Vlieger, Oct 13 2015 *)

allocatemem((2^31)+(2^30));
uplim = 36756720 + 640; \\ = A002182(53) + A002183(53).
v060990 = vector(uplim);
for(n=3, uplim, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
uplim2 = 36756720;
n=0; k=1; while(n <= uplim2, if(0==A060990(n), write("b045765_big.txt", k, " ", n); k++); n++;);
\\ Antti Karttunen, Oct 09 2015

(define A045765 (ZERO-POS 1 1 A060990))
;; Using also IntSeq-library of Antti Karttunen, Oct 06 2015

A262896 If n is in A262892, a(n) = A259934(n), otherwise the largest term in A045765 from which A259934(n) can be reached by iterating A049820, without visiting any other (larger) term of A259934.

Original entry on oeis.org

8, 2, 79, 12, 18, 40, 30, 140, 42, 52, 54, 66, 68, 123, 98, 90, 94, 116, 106, 126, 164, 121, 369, 133, 156, 168, 180, 184, 280, 229, 190, 194, 210, 218, 252, 246, 236, 242, 272, 254, 312, 324, 300, 364, 298, 302, 372, 356, 334, 342, 346, 354, 439, 366, 374, 390, 672, 414, 410, 438, 426, 460, 442, 452, 470, 466, 564, 496, 494, 524, 627, 530, 546, 558, 562, 566, 574, 592, 859, 660, 606, 642, 708, 650

Offset: 0

Antti Karttunen, Oct 06 2015

nonn

a(n) is the largest leaf-node among the finite subtrees branching from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent, and A259934(n) itself if it is one of the nonbranching nodes (A262897).

Note that without (so far undetected) regularity in A262509, there is no a priori upper bound for the value of a(n), and for some n this might not even be finite, if it happens that contrary to its conjectured nature, A259934 is not the unique infinite component, but just the lexicographically earliest instance of multiple infinite branches of the tree. In that case we might consider this sequence to be well-defined only up to the least such node branching to multiple infinite components, or alternatively, we might mark the nonfinite values at those points with -1.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Max Alekseyev & Antti Karttunen, Standalone C++-program for computing this sequence

Cf. A000005, A049820, A082284, A259934, A262509, A262522, A262679, A262686, A262890, A262892, A262897, A262904, A263081.

(define (A262896 n) (let ((t (A259934 n))) (let loop ((m t) (k (A262686 t))) (cond ((<= k t) m) ((= t (A049820 k)) (loop (max m (A262522 k)) (- k 1))) (else (loop m (- k 1)))))))

a(n) = max(A259934(n), Max_{k = A082284(A259934(n)) .. A262686(A259934(n))} [A049820(k) = A259934(n)] * A262522(k)).
(Here [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = A259934(n), and 0 otherwise).
Other identities. For all n >= 0:
A262904(a(n)) = n. [A262904 works as a left inverse for this sequence.]
A259934(n) = A262679(a(n)).
For all n >= 1:
a(A262892(n)) = A259934(A262892(n)) = A262897(n).

A262909 a(n) = greatest k such that A155043(k+A262509(n)) < A155043(A262509(n)).

Original entry on oeis.org

5197, 5193, 5177, 5115, 5113, 4419, 4417, 4259, 4245, 4243, 4239, 4059, 4047, 3991, 3941, 3633, 3593, 3449, 3445, 3437, 3423, 3421, 2897, 2789, 2517, 2261, 2079, 2077, 2067, 2063, 1527, 1379, 1135, 1127, 1117, 1103, 1083, 575, 23457, 23451, 21689, 21671, 20241, 19003, 18977, 18649, 18063, 18019, 14853, 14159, 13659, 12707, 11681, 10993, 10991, 10297, 10281, 9151, 9149, 9145, 9111, 8897, 8535, 8147, 6835, 6813, 5539, 5537

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

a(n) = largest k such that A155043(k+A262509(n)) < A262508(n).
There might occur also negative terms, but no zeros.
For all terms a(n) > 0, a(n)+A262509(n) = A263081(n) is by necessity one of the leaves (A045765) in the tree generated by edge-relation A049820(child) = parent. See also comments in A262908.

Cf. A000005, A049820, A045765, A262508, A262509, A262908, A263078, A263081.

a(n) = A263078(A262509(n)).
a(n) = A263081(n) - A262509(n).
Other identities. For all n >= 1:
a(n) >= A262908(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).

cp's OEIS Frontend

A045765 k - d(k) never takes these values, where d(k) = A000005(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

A262896 If n is in A262892, a(n) = A259934(n), otherwise the largest term in A045765 from which A259934(n) can be reached by iterating A049820, without visiting any other (larger) term of A259934.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A262909 a(n) = greatest k such that A155043(k+A262509(n)) < A155043(A262509(n)).

Views

Author

Keywords

Comments

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula