Bernard Schott - cp's OEIS frontend

A363968 Least number of 1's needed to represent n using only additions +, subtractions -, multiplications *, divisions /, concatenations # and parentheses ().

2, 1, 2, 3, 4, 5, 5, 6, 5, 4, 3, 2, 3, 4, 5, 6, 6, 7, 6, 5, 4, 3, 4, 5, 5, 6, 6, 7, 7, 6, 5, 4, 5, 5, 6, 7, 6, 6, 7, 7, 6, 5, 5, 6, 6, 7, 7, 8, 7, 8, 7, 6, 7, 7, 7, 6, 6, 7, 8, 8, 7, 6, 6, 6, 7, 8, 7, 8, 8, 8, 8, 7, 8, 8, 8, 9, 9, 8, 8, 8, 7, 6, 7, 8, 7, 8, 8, 8, 7, 7, 6, 5, 6, 7, 8, 9, 8, 8, 7, 6, 5

Offset: 0

Bernard Schott, Jun 30 2023

Fractions are not allowed as intermediate results.
The unique difference with A362471 is that concatenation is here allowed; in fact, in A362471, concatenation is only allowed for getting repunits as 111 = 1#1#1 but not for getting other integers.
Also, for example, the concatenation of 5 and -3 is not possible, so it should not be interpreted as 5-3 = 2.
The first differences with A362471 in the data appear at n = 16, 19, 20, 21, 29, ... see Example section.

			For n = 16, 16 = 1 # ((1+1)*(1+1+1)), so a(16) = 6 while A362471(16) = 7.
For n = 19, 19 = 1 # (11-1-1), so a(19) = 5 while A362471(19) = 6.
For n = 20, 20 = (1+1) # (1-1), so a(20) = 4 while A362471(20) = 5.
For n = 31, 31 = (1+1+1) # (1), so a(31) = 4 while A362471(31) = 7.
For n = 43, 43 = (1+1)*((1+1) # (1)) + 1, so a(43) = 6 while A362471(43) = 7.

Michael S. Branicky, Table of n, a(n) for n = 0..10000.
Index to sequences related to the complexity of n.

Cf. A002275, A005245, A091333, A133344, A362471, A362626.

|a(n+1) - a(n)| <= 1; improved by Pontus von Brömssen, Jun 30 2023
a(n) <= A362471(n).
a(n) <= Sum_{k=1..m} a(dk), where d1d2..dm are the decimal digits of n. - Michael S. Branicky, Jun 30 2023

a(72) and beyond from Michael S. Branicky, Jun 30 2023

A363906 Decimal expansion of Sum_{n>=1} (arcsin(1/n) - sin(1/n)).

Original entry on oeis.org

7, 9, 9, 5, 8, 8, 6, 2, 3, 5, 5, 3, 3, 7, 6, 9, 9, 0, 1, 1, 3, 9, 9, 1, 1, 1, 3, 5, 2, 7, 2, 3, 9, 8, 2, 5, 0, 4, 0, 1, 7, 2, 2, 8, 4, 1, 9, 0, 7, 7, 7, 9, 6, 8, 3, 6, 4, 1, 1, 6, 5, 9, 2, 8, 4, 3, 6, 7, 7, 3, 0, 4, 0, 6, 7, 7, 5, 5, 7, 2, 1, 7, 9, 1, 8, 1, 7

Offset: 0

Bernard Schott, Jun 27 2023

Series Sum_{n>=1} arcsin(1/n) and Sum_{n>=1} sin(1/n) -> oo but with v(n) = (arcsin(1/n) - sin(1/n)), as v(n) ~ 1 / (3*n^3) when n -> oo, the series Sum_{n>=1} v(n) is convergent.

			0.79958862355337699...

Cf. A096444, A248946, A342680, A362662.

NSum[ArcSin[1/n]-Sin[1/n], {n, Infinity}, WorkingPrecision -> 95, NSumTerms -> 82] // RealDigits[#, 10, 87] &//First (* Stefano Spezia, Jun 27 2023 *)

sumpos(n=1, asin(1/n) - sin(1/n)) \\ Michel Marcus, Jun 27 2023

Equals Sum_{k>=1} (binomial(2*k,k)/((2*k+1)*2^(2*k)) - (-1)^k/(2*k+1)!) * zeta(2*k+1). - Vaclav Kotesovec, Jun 27 2023

More terms from Stefano Spezia, Jun 27 2023

A362670 Integer inradii for which there exists an isosceles triangle with integer sides (a, a, c) where a < c.

Original entry on oeis.org

3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 87, 88, 90, 92, 93, 96, 99, 100, 102, 104, 105, 108, 111, 112, 114, 116, 117, 120, 123, 124, 126, 128, 129, 132, 135

Offset: 1

Bernard Schott, May 05 2023

nonn

The inradius for isosceles triangle (a, a, c) is r = (c/2)*sqrt((2*a-c)/(2*a+c)).
If m is a term, so is k*m with k > 1.
As r = 3 and r = 4 are terms, A008585 and A008586 are respective subsequences; the only terms < 100 that are not multiples of 3 or 4 are 35 and 70, the next one is r = 154 = 2*7*11 for triple (765, 765, 1386).
By the triangle inequality, a+1 <= c <= 2*a-1.
Differs from A059267. Examples: 154 is not in A059267 but in this sequence at radius r=154 with side lengths c=1386 and a=765. 442 is not in A059267 but in this sequences with r=442, c=6630, a=3435. - R. J. Mathar, Jun 26 2023

			The smallest inradius r = 3 corresponds to isosceles triangle (10, 10, 12).
The second inradius r = 4 corresponds to isosceles triangle (15, 15, 24).
r = 15 is the first inradius for which there exist two such isosceles triangles: (50, 50, 60) and (68, 68, 120).
r = 35 is the smallest inradius that is not multiple of 3 or of 4, this inradius corresponds to isosceles triangle (222, 222, 420).

R. J. Mathar, Solution strategy and Maple program
Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Isosceles Triangle.

Cf. A008585, A008586, A070204, A120062, A120570.

Cf. A362669 (similar but with (a,b,b)).

A362669 Integer inradii for which there exists an isosceles triangle with integer sides (a, b, b) where a < b.

Original entry on oeis.org

10, 20, 21, 24, 30, 36, 40, 42, 48, 50, 55, 60, 63, 70, 72, 78, 80, 84, 90, 96, 100, 105, 108, 110, 112, 120, 126, 130, 136, 140, 144, 147, 150, 156, 160, 165, 168, 170, 171, 180, 189, 190, 192, 195, 200, 210, 216, 220, 224, 230, 231, 234, 240, 250, 252, 253, 260, 264, 270, 272, 273, 275

Offset: 1

Bernard Schott, Apr 29 2023

nonn

The inradius for isosceles triangle (a, b, b) is r = (a/2)*sqrt((2*b-a)/(2*b+a)).

If m is a term, so is k*m with k > 1; hence, A008592 \ {0} is a subsequence.

			The smallest inradius, r = 10, corresponds to isosceles triangle (30, 39, 39).
The third inradius, r = 21, corresponds to isosceles triangle (56, 100, 100).
r = 60 is the first inradius for which there exist two such isosceles triangles: (168, 259, 259) and (180, 234, 234).

Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Isosceles Triangle.

Cf. A008592, A070204, A120062, A120570, A362670 (similar but with (a,a,c)).

Select[Range[300], Length @ Reduce[#^2 == a^2*(2*b - a)/(4*(2*b + a)) && 0 < a < b, {a, b}, Integers] > 0 &] (* Amiram Eldar, May 05 2023 *)

A362662 Decimal expansion of Sum_{n>=1} (tan(1/n) - sin(1/n)).

Original entry on oeis.org

8, 2, 2, 0, 8, 2, 2, 0, 0, 8, 0, 3, 5, 8, 8, 2, 0, 2, 9, 3, 5, 8, 7, 0, 1, 1, 8, 7, 1, 5, 9, 9, 3, 5, 2, 0, 7, 3, 0, 4, 4, 6, 0, 4, 3, 8, 1, 1, 6, 5, 3, 2, 6, 3, 9, 0, 8, 3, 6, 8, 5, 9, 3, 9, 3, 4, 3, 7, 1, 0, 5, 3, 4, 5, 3, 5, 4, 3, 6, 8, 1, 3, 2, 4, 6, 0, 0, 4, 7, 1, 3, 4, 7, 4, 3, 2, 2

Offset: 0

Bernard Schott, Apr 29 2023

Series Sum_{n>=1} sin(1/n) and Sum_{n>=1} tan(1/n) -> oo but with u(n) = (tan(1/n) - sin(1/n)), as u(n) ~ 1 / (2*n^3) when n -> oo, the series Sum_{n>=1} u(n) is convergent.

			Equals 0.822082200803588202935870118715993520730...

J. Guégand and M.-A. Maingueneau, Exercices d'Analyse, Exercice 1 - 41.2, p. 47, Classes Préparatoires aux Grandes Ecoles, Ellipses, 1988.

Cf. A096444, A248946, A248948, A342680.

evalf(sum(tan(1/n) - sin(1/n), n=1..infinity), 120);

sumpos(n=1, tan(1/n) - sin(1/n)) \\ Michel Marcus, Apr 29 2023

A362148 Numbers that are neither cubefree nor cubefull.

Original entry on oeis.org

24, 40, 48, 54, 56, 72, 80, 88, 96, 104, 108, 112, 120, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 224, 232, 240, 248, 250, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 344, 351, 352, 360, 368, 375, 376, 378, 384, 392, 400

Offset: 1

Bernard Schott, Apr 09 2023

In fact, every cubefull number > 1 is noncubefree, but the converse is false.
This sequence = A046099 \ A036966 and lists these counterexamples.
Numbers k such that for some primes p and q, k is divisible by p^3*q but not by q^3. - Robert Israel, Apr 28 2023
The asymptotic density of this sequence is 1 - 1/zeta(3) = 0.168092... - Charles R Greathouse IV, Apr 28 2023
From Amiram Eldar, Sep 17 2023: (Start)
Numbers k such that A360539(k) > 1 and A360540(k) > 1.
Equivalently, numbers that have in their prime factorization at least one exponent that is smaller than 3 and at least one exponent that is larger than 2. (End)

			24 = 2^3 * 3 is noncubefree as it is divisible by the cube 2^3, but it is not cubefull because 3 divides 24 but 3^3 does not divide 24, hence 24 is a term.
648 = 2^4 * 3^3 is noncubefree as it is divisible by the cube 3^3, but it is also cubefull because primes 2 and 3 divide 648, and 2^3 and 3^3 divide also 648, so 648 is not a term.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Intersection of A046099 (not cubefree) and A362147 (not cubefull)

Cf. A004709 (cubefree), A036966 (cubefull), A360539, A360540.

filter:= proc(n) local F;
F:= ifactors(n)[2][..,2];
  min(F) < 3 and max(F) >= 3
end proc:
select(filter, [$1..400]); # Robert Israel, Apr 28 2023

Select[Range[500], Min[(e = FactorInteger[#][[;; , 2]])] < 3 && Max[e] > 2 &] (* Amiram Eldar, Apr 09 2023 *)

isok(k) = (k>1) && (vecmax(factor(k)[, 2])>2) && (vecmin(factor(k)[, 2])<=2); \\ Michel Marcus, Apr 19 2023

Equals A362147 \ A004709.

Sum_{n>=1} 1/a(n) = 1 + zeta(s) - zeta(s)/zeta(3*s) - Product_{p prime}(1 + 1/(p^(2*s)*(p^s-1))), s > 1. - Amiram Eldar, Sep 17 2023

A362533 Decimal expansion of lim_{n->oo} ( Sum_{k=2..n} 1/(k * log(k) * log log(k)) - log log log(n) ).

Original entry on oeis.org

2, 6, 9, 5, 7, 4

Offset: 1

Bernard Schott, Apr 24 2023

If u(n) = Sum_{k=2..n} ( 1/(k*log(k)*log log(k)) - log log log(n) ), then (u(n)) is convergent, while the series v(n) = Sum_{k=2..n} 1/(k*log(k)*log log log(k)) diverges (see link). This is an extension of A001620 and A361972.

Note that ( log log log(x) )' = 1 / ( x * log(x) * log log(x) ).

			2.69574...

Patrice Lassère, Divergence douce de Sum_{k>1} 1/( k*log(k)*log log(k) ) par le TAF, Les-Mathematiques.net.

Cf. A001620, A361972.

Limit_{n->oo} 1/( 2*log(2)*log log(2) ) + 1/( 3*log(3)*log log(3) ) + ... + 1/( n*log(n)*log log(n) ) - log log log(n).

A362147 Numbers that are not cubefull.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84

Offset: 1

Bernard Schott, Apr 09 2023

nonn

Integers m for which there is a prime p that divides m, but p^3 does not divide m.

Complement of A036966.

			2|24 and 2^3|24, but 3|24 and 3^3 does not divide 24, so 24 is a term.

Cf. A004709 (cubefree), A046099 (not cubefree), A036966 (cubefull), A362148 (non-cubefree that are not cubefull).

Select[Range[2, 100], Min[FactorInteger[#][[;; , 2]]] < 3 &] (* Amiram Eldar, Apr 09 2023 *)

isok(k) = (k!=1) && (vecmin(factor(k)[, 2])<=2); \\ Michel Marcus, Apr 12 2023

from math import gcd
from sympy import integer_nthroot, factorint
def A362147(n):
    def f(x):
        c = n
        for w in range(1,integer_nthroot(x,5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
                    if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c += integer_nthroot(z//y**4,3)[0]
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Nov 22 2024

A361972 Decimal expansion of lim_{n->oo} ( Sum_{k=2..n} 1/(k*log(k)) - log(log(n)) ).

Original entry on oeis.org

7, 9, 4, 6, 7, 8, 6, 4, 5, 4, 5, 2, 8, 9, 9, 4, 0, 2, 2, 0, 3, 8, 9, 7, 9, 6, 2, 0, 6, 5, 1, 4, 9, 5, 1, 4, 0, 6, 4, 9, 9, 9, 5, 9, 0, 8, 8, 2, 8, 0, 4, 9, 6, 8, 9, 0, 1, 5, 1, 2, 0, 9, 5, 0, 1, 4, 8, 1, 7, 8, 5, 8, 9, 6, 0, 6, 8, 7, 5, 6, 6, 6, 9, 6, 6, 1, 4, 7, 7, 7, 6, 2, 7, 3, 3

Offset: 0

Bernard Schott, Apr 08 2023

Let u(n) = Sum_{k=2..n} 1/(k*log(k)) - log(log(n)), then (u(n)) is strictly decreasing and lower bounded by -log(log(2)) = A074785, so (u(n)) is convergent, while the series v(n) = Sum_{k=2..n} 1/(k*log(k)) diverges (see Mathematics Stack Exchange link).

Compare with w(n) = Sum_{k=1..n} 1/k - log(n) that converges (A001620), while the harmonic series H(n) = Sum_{k=1..n} 1/k diverges.

			0.79467864545289940220389796...

J. Guégand and M.-A. Maingueneau, Exercices d'Analyse, Exercice 1.18 p. 23, 1988, Classes Préparatoires aux Grandes Ecoles, Ellipses.

Mathematics Stack Exchange, Infinite series Sum_{n=2..oo} 1/(n*log(n)).

Cf. A001620, A074785, A241005.

limit(sum(1/(k*log(k)), k=2..n) - log(log(n)), n = infinity);

Limit_{n->oo} 1/(2*log(2)) + 1/(3*log(3)) + ... + 1/(n*log(n)) - log(log(n)).

Equals A241005 - log(log(2)) = A241005 + A074785. - Amiram Eldar, Apr 08 2023

A361914 Primes that are repunits with three or more digits for exactly one base b >= 2.

Original entry on oeis.org

7, 13, 43, 73, 127, 157, 211, 241, 307, 421, 463, 601, 757, 1093, 1123, 1483, 1723, 2551, 2801, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 19531, 20023, 20593, 21757, 22621, 22651, 23563

Offset: 1

Bernard Schott, Mar 29 2023

Brazilian primes that have exactly one Brazilian representation as a repunit.
As these primes p satisfy beta(p) = tau(p) / 2 (= 1), where beta = A220136 and tau = A000005, this sequence is a subsequence of A326380.
Equals A085104 \ {31, 8191}, since according to the Goormaghtigh conjecture (link), 31 and 8191 which are both Mersenne numbers, are the only primes which are Brazilian in two different bases.
The three following sequences realize a partition of the set of primes: A220627 (primes not Brazilian), this sequence (primes 1-Brazilian) and {31,8191} (primes 2-Brazilian).

			7 = 111_2 is a term.
13 = 111_3 is a term.
19 = 11_18 is not a term.
31 = 11111_5 = 111_5 is not a term.
127 = 1111111_2 is a term.
8191 = 1111111111111_2 = 111_90 is not a term.

Wikipedia, Goormaghtigh conjecture.
Index entries for sequences related to Brazilian numbers.

Cf. A000005, A085104, A220136, A220627, A326380.
Equals A326380 \ {A326385 Union A326387}.
Subsequence of A288783.

q[n_] := Count[Range[2, n - 2], ?(Length[d = IntegerDigits[n, #]] > 2 && Length[Union[d]] == 1 &)] == 1;; Select[Prime[Range[3000]], q] (* _Amiram Eldar, Mar 30 2023 *)

cp's OEIS Frontend

User: Bernard Schott

A363968 Least number of 1's needed to represent n using only additions +, subtractions -, multiplications *, divisions /, concatenations # and parentheses ().

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A363906 Decimal expansion of Sum_{n>=1} (arcsin(1/n) - sin(1/n)).

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A362670 Integer inradii for which there exists an isosceles triangle with integer sides (a, a, c) where a < c.

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A362669 Integer inradii for which there exists an isosceles triangle with integer sides (a, b, b) where a < b.

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Mathematica

A362662 Decimal expansion of Sum_{n>=1} (tan(1/n) - sin(1/n)).

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Maple

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A362148 Numbers that are neither cubefree nor cubefull.

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A362533 Decimal expansion of lim_{n->oo} ( Sum_{k=2..n} 1/(k * log(k) * log log(k)) - log log log(n) ).

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A362147 Numbers that are not cubefull.

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