A262512 -id:A262512 - cp's OEIS frontend

A262686 a(n) = largest number k such that k - d(k) = n, or 0 if no such number exists, where d(n) = the number of divisors of n (A000005).

2, 4, 6, 5, 8, 7, 12, 0, 0, 11, 14, 16, 18, 0, 20, 17, 24, 21, 22, 0, 0, 23, 30, 27, 0, 0, 32, 36, 0, 33, 34, 35, 40, 0, 42, 39, 0, 0, 48, 45, 0, 43, 46, 0, 50, 47, 54, 51, 60, 0, 0, 55, 0, 57, 58, 0, 0, 64, 66, 61, 72, 65, 70, 0, 0, 69, 0, 0, 0, 75, 80, 73, 84, 77, 0, 0, 81, 79, 90, 0, 88, 85, 86, 87, 96, 0, 92, 91, 0, 93, 94, 100, 98, 99, 102, 97, 108, 105, 0, 101

Offset: 0

Antti Karttunen, Sep 28 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10080

Cf. A000005, A002183, A049820, A060990, A261100, A262512.

Cf. also A082284 (the smallest such number), A262511 (positions where these are equal and nonzero).

Table[k = 2 n + 3; While[Nor[k - DivisorSigma[0, k] == n, k == 0], k--]; k, {n, 0, 99}] (* Michael De Vlieger, Sep 29 2015 *)

(definec (A262686 n) (if (zero? n) 2 (let ((u (+ n (A002183 (+ 2 (A261100 n)))))) (let loop ((k u)) (cond ((= (A049820 k) n) k) ((< k n) 0) (else (loop (- k 1))))))))

A262511 Numbers k for which there is exactly one solution to x - d(x) = k, where d(k) is the number of divisors of k (A000005). Positions of ones in A060990.

Original entry on oeis.org

2, 3, 4, 5, 9, 10, 12, 14, 15, 16, 18, 21, 23, 26, 30, 31, 32, 41, 42, 44, 45, 47, 53, 54, 59, 60, 61, 71, 72, 73, 76, 77, 80, 82, 83, 84, 86, 89, 90, 92, 93, 94, 95, 97, 99, 101, 104, 105, 106, 110, 115, 119, 121, 122, 127, 135, 139, 146, 148, 149, 151, 154, 158, 161, 169, 171, 173, 176, 177, 183, 186, 188, 189, 190, 191, 192, 194, 195, 199, 200, 202

Offset: 1

Antti Karttunen, Sep 25 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000005, A049820, A060990.
Cf. A262512 (gives the corresponding x).
Cf. A262510 (a subsequence).
Subsequence of A236562.

allocatemem(123456789);
uplim = 14414400 + 504; \\ = A002182(49) + A002183(49).
v060990 = vector(uplim);
for(n=3, uplim, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
uplim2 = 14414400;
n=0; k=1; while(n <= uplim2, if(1==A060990(n), write("b262511_big.txt", k, " ", n); k++); n++;);

;; With Antti Karttunen's IntSeq-library.
(define A262511 (ZERO-POS 1 1 (COMPOSE -1+ A060990)))

Other identities. For all n >= 1:

a(n) = A049820(A262512(n)).

A262522 a(n)=0 if n is in A259934, otherwise the largest term in A045765 from which one can reach n by iterating A049820 zero or more times.

Original entry on oeis.org

0, 8, 0, 7, 8, 7, 0, 7, 8, 79, 20, 79, 0, 13, 20, 79, 24, 79, 0, 19, 20, 79, 0, 79, 24, 25, 40, 79, 28, 79, 0, 79, 40, 33, 0, 79, 36, 37, 140, 79, 40, 43, 0, 43, 50, 79, 0, 79, 140, 49, 50, 79, 52, 79, 0, 55, 56, 79, 0, 79, 140, 79, 0, 63, 64, 79, 66, 67, 68, 79, 0, 79, 140, 79, 74, 75, 123, 79, 0, 79, 88, 123, 98, 123, 140, 85, 98, 123, 88, 103, 0, 123, 98, 103, 0, 123

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

If n is itself in A045765, we iterate 0 times, and thus a(n) = n.

			For n=1, its transitive closure (as defined by edge-relation A049820(child) = parent) is the union of {1} itself together with all its descendants: {1, 3, 4, 5, 7, 8}. We see that there are no other nodes in a subtree whose root is 1, because A049820(3) = 3 - d(3) = 1, A049820(4) = 1, A049820(5) = 3, A049820(7) = 5, A049820(8) = 4 and of these only 7 and 8 are terms of A045765. The largest term (which by necessity is always a term of A045765) is here 8, thus a(1) = 8. Note however that it is not always the largest leaf from which starts the longest path leading back to n. (In this case it is 7 instead of 8, see the example in A262695).
For n=9, its transitive closure is {9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79}. The largest term is 79, thus a(9) = 79.

Antti Karttunen, Table of n, a(n) for n = 0..17724

Cf. A000005, A045765, A049820, A060990, A082284, A259934, A262511, A262512, A262686, A262693.

Cf. A262679, A262695, A262696, A262697.

If A262693(n) = 1 [when n is in A259934],
  then a(n) = 0,
otherwise, if A060990(n) = 0 [when n is one of the leaves, A045765],
  then a(n) = n,
otherwise:
  a(n) = Max_{k = A082284(n) .. A262686(n)} [A049820(k) = n] * a(k).
(In the last clause [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = n, and 0 otherwise).
Other identities. For all n >= 1:
a(A262511(n)) = a(A262512(n)) = a(A082284(A262511(n))).

A262513 Numbers where A049820 takes a unique value; numbers n for which A060990(A049820(n)) = 1.

Original entry on oeis.org

5, 6, 7, 8, 11, 14, 17, 18, 20, 22, 23, 24, 27, 32, 34, 35, 40, 43, 46, 47, 50, 51, 57, 58, 61, 65, 72, 73, 77, 79, 81, 84, 86, 87, 88, 92, 93, 94, 96, 97, 98, 99, 101, 102, 103, 105, 107, 114, 116, 119, 120, 123, 125, 130, 135, 137, 143, 151, 154, 155, 158, 160, 163, 164, 173, 175, 177, 179, 184, 187, 191, 193, 194, 197, 198, 200, 203, 204, 206, 209, 210, 212

Offset: 1

Antti Karttunen, Sep 25 2015

nonn

Sequence A262512 sorted into ascending order.

Numbers n such that there is no other number k for which A049820(k) = A049820(n).

Antti Karttunen, Table of n, a(n) for n = 1..45700

Cf. A060990, A049820, A262511, A262512.

Cf. A262509 (a subsequence).

lim = 212; s = Table[n - DivisorSigma[0, n], {n, 2 lim + 3}]; t = Length@ Position[s, #] & /@ Range[0, lim]; Position[t[[# + 1]] & /@ Take[s, lim], 1] // Flatten (* Michael De Vlieger, Sep 29 2015, after Wesley Ivan Hurt at A049820 *)

cp's OEIS Frontend

A262686 a(n) = largest number k such that k - d(k) = n, or 0 if no such number exists, where d(n) = the number of divisors of n (A000005).

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A262511 Numbers k for which there is exactly one solution to x - d(x) = k, where d(k) is the number of divisors of k (A000005). Positions of ones in A060990.

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A262522 a(n)=0 if n is in A259934, otherwise the largest term in A045765 from which one can reach n by iterating A049820 zero or more times.

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A262513 Numbers where A049820 takes a unique value; numbers n for which A060990(A049820(n)) = 1.

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