A262515 -id:A262515 - cp's OEIS frontend

A262514 Squares present in A259934.

0, 121, 144, 113569, 123904, 398161, 399424, 1708249, 1710864, 24591681, 24681024, 53890281, 53934336, 184063489, 184742464, 8338063969, 8339342400, 470959650225, 470972003076, 25876358308161, 25876429524544, 168967735540081, 168967969517824, 303965800198225

Offset: 0

Antti Karttunen, Sep 26 2015

nonn

Every other term is even and every other odd. From each odd square starts a new run of odd terms in A259934 (see A262516 for the lengths of these and A262517 for the odd terms themselves), which lasts as long as the next even square is encountered, at which point the parity of terms is switched back to even.
The terms were grepped from the data file of A259934 provided by Max Alekseyev, which contained terms up to A259934(397420670) = 9999999974.
The indexing starts from zero, because a(0) = 0 is a special case in this sequence.
a(31) > 10^16. - Hiroaki Yamanouchi, Oct 07 2015

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..30

Cf. A262515 (square roots of these terms), A262516, A262517, A259934.

a(n) = A259934(A263276(n)).

a(17)-a(30) from Hiroaki Yamanouchi, Oct 07 2015

A263092 Numbers whose squares are in A236562; numbers n such that there is at least one such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 11, 12, 13, 15, 17, 19, 21, 23, 24, 25, 29, 30, 31, 32, 33, 36, 38, 39, 40, 41, 42, 43, 44, 45, 48, 49, 51, 52, 53, 55, 57, 61, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 83, 84, 86, 87, 88, 91, 92, 93, 96, 97, 99, 101, 102, 105, 107, 108, 109

Offset: 0

Antti Karttunen, Oct 11 2015

nonn

Starting offset is zero, because a(0)=0 is a special case in this sequence.
Numbers n for which A060990(n^2) = A263087(n) > 0.
Numbers n for which A049820(x) = n^2 has a solution.

Antti Karttunen, Table of n, a(n) for n = 0..20763

Complement: A263093.
Cf. A000005, A000196, A049820, A060990, A236562, A263087, A263098.
Cf. A263094 (the squares of these numbers).
Cf. A262515 (a subsequence).

\\ Compute A263092 and A263094 at the same time:
A060990(n) = { my(k = n + 1440, s=0); while(k > n, if(((k-numdiv(k)) == n),s++); k--;); s}; \\ Hard limit 1440 good for at least up to A002182(67) = 1102701600 as A002183(67) = 1440.
n = 0; k = 0; while((n^2)<1102701600, if((A060990(n*n) > 0), write("b263092.txt", k, " ", n); write("b263094.txt", k, " ", (n*n)); k++; ); n++; if(!(n%8192),print1(n,",k=", k, ", ")); );

;; With Antti Karttunen's IntSeq-library.
(define A263092 (MATCHING-POS 0 0 (lambda (n) (not (zero? (A060990 (* n n)))))))
(define A263092 (NONZERO-POS 0 0 A263087))

A263088 a(n) = A262697(n^2).

Original entry on oeis.org

0, 6, 2, 38, 2, 1, 1, 1, 1, 22, 1, 0, 0, 2, 1, 3, 1, 9, 1, 39, 1, 47, 1, 51, 4, 114, 1, 1, 1, 529, 2, 6, 2, 3, 1, 1, 22, 1, 11, 3, 2, 4, 7, 93, 7, 967, 1, 1, 3, 4, 1, 3, 2, 4, 1, 3, 1, 3, 1, 1, 1, 2, 1, 139, 2, 265, 2, 1, 6, 464, 12, 4, 22, 1, 2, 1503, 2, 6, 1, 5, 2, 2, 1, 2, 5, 1, 2, 4, 2, 1, 1, 6, 3, 386, 1, 1, 3, 800, 1, 2, 1, 7, 5, 1, 1, 3353, 1, 2, 21, 3, 1, 17, 3, 3, 1, 4, 1, 5, 1, 3, 9, 2

Offset: 0

Antti Karttunen, Oct 12 2015

nonn

a(n)=0 if n^2 is in A259934, otherwise number of nodes in that finite subtree whose root is n^2 and edge-relation is defined by A049820(child) = parent. This count includes also leaves and n^2 itself.

Antti Karttunen, Table of n, a(n) for n = 0..222

Cf. A000290, A049820, A259934, A262697.
Cf. also A261088, A263087.
Cf. A262515 (positions of zeros), A263093 (positions of ones).

(define (A263088 n) (A262697 (A000290 n)))

a(n) = A262697(A000290(n)) = A262697(n^2).
Other identities. For all n >= 0:
If A263087(n) = 0, a(n) = 1.

A262688 a(n) = square root of the index where A262680 reaches n for the first time.

Original entry on oeis.org

0, 1, 2, 13, 18, 39, 354, 355, 652, 1309, 1310, 2227, 4976

Offset: 0

Antti Karttunen, Oct 03 2015

Natural numbers whose squares set new records for A262680.

Even and odd terms alternate.

Cf. A000196, A262680, A262687, A262515.

PARI
```
\\ See A262687.
```

(define (A262688 n) (A000196 (A262687 n)))

a(n) = A000196(A262687(n)).

cp's OEIS Frontend

A262514 Squares present in A259934.

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A263092 Numbers whose squares are in A236562; numbers n such that there is at least one such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

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A263088 a(n) = A262697(n^2).

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A262688 a(n) = square root of the index where A262680 reaches n for the first time.

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