A263098 -id:A263098 - cp's OEIS frontend

A263093 Numbers whose squares are in A045765.

5, 6, 7, 8, 10, 14, 16, 18, 20, 22, 26, 27, 28, 34, 35, 37, 46, 47, 50, 54, 56, 58, 59, 60, 62, 67, 73, 78, 82, 85, 89, 90, 94, 95, 98, 100, 103, 104, 106, 110, 114, 116, 118, 122, 124, 125, 126, 127, 128, 130, 135, 140, 141, 142, 148, 150, 155, 158, 161, 164, 170, 172, 174, 177, 178, 182, 184, 188, 190, 199, 202, 205, 207

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

Numbers n such that there is no such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

Numbers n for which A060990(n^2) = A263087(n) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..12443
A. Karttunen, Ratio a(n)/A263092(n) drawn with the help of OEIS Plot2-utility

Cf. A000005, A000196, A045765, A049820, A060990, A263098.
Complement: A263092.
Positions of zeros in A263087 and positions of ones in A263088.
Cf. A263095 (the squares of these numbers).

\\ Compute A263093 and A263095 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 is good for at least up to A002182(67) = 1102701600 as A002183(67) = 1440.
n = 1; k = 0; while((n^2)<1102701600, if((0 == A060990(n*n)), k++; write("b263093.txt", k, " ", n); write("b263095.txt", k, " ", (n*n)); ); n++; if(!(n%8192),print1(n,",k=", k, ", ")); );

;; With Antti Karttunen's IntSeq-library.
(define A263093 (MATCHING-POS 1 1 (lambda (n) (zero? (A060990 (* n n))))))
(define A263093 (ZERO-POS 1 0 A263087))

a(n) = A000196(A263095(n)).

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))

A263095 Squares in A045765; numbers n^2 such that there is no 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

25, 36, 49, 64, 100, 196, 256, 324, 400, 484, 676, 729, 784, 1156, 1225, 1369, 2116, 2209, 2500, 2916, 3136, 3364, 3481, 3600, 3844, 4489, 5329, 6084, 6724, 7225, 7921, 8100, 8836, 9025, 9604, 10000, 10609, 10816, 11236, 12100, 12996, 13456, 13924, 14884, 15376, 15625, 15876, 16129, 16384, 16900, 18225, 19600, 19881, 20164, 21904, 22500, 24025, 24964, 25921, 26896

Offset: 1

Antti Karttunen, Oct 10 2015

nonn

Some of the terms are shared with A262687, but none with A262514.

Antti Karttunen, Table of n, a(n) for n = 1..12443

Cf. A000005, A049820, A010052, A060990, A262514, A262687, A263098.
Cf. A263093 (gives the square roots).
Intersection of A000290 and A045765.
Cf. also A263091.

lim = 40000; Take[Select[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], IntegerQ@ Sqrt@ # &], 60] (* Michael De Vlieger, Oct 13 2015 *)

PARI
```
\\ See code in A263093.
```

;; With Antti Karttunen's IntSeq-library.
(define A263095 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010052 n)) (zero? (A060990 n))))))

A263094 Squares in A236562; numbers n^2 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, 4, 9, 16, 81, 121, 144, 169, 225, 289, 361, 441, 529, 576, 625, 841, 900, 961, 1024, 1089, 1296, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2304, 2401, 2601, 2704, 2809, 3025, 3249, 3721, 3969, 4096, 4225, 4356, 4624, 4761, 4900, 5041, 5184, 5476, 5625, 5776, 5929, 6241, 6400, 6561, 6889, 7056, 7396, 7569, 7744, 8281, 8464, 8649, 9216, 9409, 9801, 10201, 10404, 11025

Offset: 0

Antti Karttunen, Oct 11 2015

nonn

Starting offset is zero, because a(0)=0 is a special case in this sequence.

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

Cf. A000005, A049820, A060990.
Intersection of A000290 and A236562.
Cf. A263092 (gives the square roots of these terms).
Cf. A263095 (complement among squares).
Cf. A262514 (a subsequence).
Cf. also A263090, A263098.

Take[Select[Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, 20000}], IntegerQ@ Sqrt@ # &], 68] (* Michael De Vlieger, Oct 13 2015 *)

PARI
```
\\ See code in A263092.
```

(define (A263094 n) (A000290 (A263092 n)))

a(n) = A000290(A263092(n)).

A263096 Square roots of highly composite numbers, floored down: a(n) = A000196(A002182(n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 6, 7, 10, 13, 15, 18, 26, 28, 35, 40, 50, 70, 86, 100, 122, 141, 158, 166, 212, 224, 235, 288, 332, 407, 470, 526, 576, 706, 744, 815, 848, 1039, 1200, 1470, 1697, 1898, 2079, 2546, 2684, 2940, 3287, 3796, 4158, 4649, 5694, 6062, 6575, 7826, 8573, 10500, 11068, 12125, 13556, 15653, 17147, 19172, 23480, 26426, 27113, 33206, 37373, 45772, 46961, 48248, 52853, 59092, 68233, 74746, 83568, 102350

Offset: 1

Antti Karttunen, Oct 10 2015

nonn

a(n) = number of strictly positive squares <= A002182(n).

Antti Karttunen, Table of n, a(n) for n = 1..1000 (based on the b-file of A002182 provided by _T. D. Noe_.)

Cf. A000196, A002182, A263097, A263098.

(define (A263096 n) (A000196 (A002182 n)))

a(n) = A000196(A002182(n)).

A263097 First differences of A263096.

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 0, 1, 3, 3, 2, 3, 8, 2, 7, 5, 10, 20, 16, 14, 22, 19, 17, 8, 46, 12, 11, 53, 44, 75, 63, 56, 50, 130, 38, 71, 33, 191, 161, 270, 227, 201, 181, 467, 138, 256, 347, 509, 362, 491, 1045, 368, 513, 1251, 747, 1927, 568, 1057, 1431, 2097, 1494, 2025, 4308, 2946, 687, 6093, 4167, 8399, 1189, 1287, 4605, 6239, 9141, 6513, 8822, 18782, 15834, 26561, 22392, 37564, 16401, 32375, 17317, 12602

Offset: 1

Antti Karttunen, Oct 10 2015

nonn

a(n) = number of perfect squares in range [A002182(n)+1 .. A002182(n+1)].

			A002182 begins as 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ...
In range [2,2] there are no squares, so a(1) = 0.
In range [3,4] there is one square, so a(2) = 1.
In range [5,6] there are no squares, so a(3) = 0.
In range [7,12] there is one square, so a(4) = 1.
In range [13,24] there is one square, so a(5) = 1.
In range [25,36] there are two squares, so a(6) = 2.
In range [37,48] there are no squares, so a(7) = 0.
In range [49,60] there is one square, so a(8) = 1.
In range [61,120] there are three squares (64, 81, 100), thus a(9) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..999 (based on the b-file of A002182 provided by _T. D. Noe_.)

Cf. A000196, A002182, A263096, A263098.

(define (A263097 n) (- (A263096 (+ n 1)) (A263096 n)))

a(n) = A263096(n+1) - A263096(n).

cp's OEIS Frontend

A263093 Numbers whose squares are in A045765.

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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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A263095 Squares in A045765; numbers n^2 such that there is no such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

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A263094 Squares in A236562; numbers n^2 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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A263096 Square roots of highly composite numbers, floored down: a(n) = A000196(A002182(n)).

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A263097 First differences of A263096.

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