keyword:new

A376328

G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x)))^5.

+0
0

1, 5, 40, 380, 3970, 44051, 509575, 6077435, 74194780, 922644310, 11646083631, 148827827450, 1921724362880, 25034267112600, 328614891689845, 4342322118727300, 57715241768897445, 771087466276360970, 10349495416322497575, 139486475071720234920, 1886980259513934080860, 25613816043115261657425

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..21.

Index entries for reversions of series

FORMULA

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).

G.f.: (1/x) * Series_Reversion( x / (1+x+x^2)^5 ).

G.f.: B(x)^5, where B(x) is the g.f. of A365189.

PROG

(PARI) a(n, s=1, t=5) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^2)^5)/x)

CROSSREFS

Cf. A143927, A365128, A376320.

Cf. A365189.

KEYWORD

nonn,new

AUTHOR

Seiichi Manyama, Sep 20 2024

STATUS

approved

A376320

G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x)))^4.

+0
0

1, 4, 26, 200, 1691, 15180, 142038, 1370076, 13526645, 136024876, 1388394234, 14346699052, 149790104030, 1577765967600, 16745718467070, 178912981116840, 1922688816819276, 20769064846817136, 225384498769815750, 2455985319885345820, 26862562977746930145, 294807644917408047060

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..21.

Index entries for reversions of series

FORMULA

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).

G.f.: (1/x) * Series_Reversion( x / (1+x+x^2)^4 ).

G.f.: B(x)^4, where B(x) is the g.f. of A365183.

PROG

(PARI) a(n, s=1, t=4) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^2)^4)/x)

CROSSREFS

Cf. A143927, A365128, A376328.

Cf. A365183.

KEYWORD

nonn,new

AUTHOR

Seiichi Manyama, Sep 20 2024

STATUS

approved

A376326

Expansion of (1/x) * Series_Reversion( x * (1-x-x^2)^4 ).

+0
0

1, 4, 30, 272, 2737, 29380, 329614, 3818540, 45329440, 548511612, 6740687924, 83898110660, 1055441468145, 13398494365088, 171422870731600, 2208161418665872, 28614197357895055, 372754395074051500, 4878709294080115494, 64123505084010848580, 846018700129069313495

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..20.

Index entries for reversions of series

FORMULA

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(4*n+k+3,k) * binomial(5*n-k+3,n-2*k).

G.f.: B(x)^4, where B(x) is the g.f. of A365188.

PROG

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x-x^2)^4)/x)

(PARI) a(n, s=2, t=4, u=0) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

CROSSREFS

Cf. A001002, A368961, A368963.

Cf. A365188.

KEYWORD

nonn,new

AUTHOR

Seiichi Manyama, Sep 20 2024

STATUS

approved

A376330

Decimal expansion of 1 bit in J/K.

+0
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

-23,1

LINKS

Table of n, a(n) for n=-23..63.

Index entries for transcendental numbers

FORMULA

Exactly equal to log(2) * 1.380649 * 10^-23 under the 2019 SI redefinition.

EXAMPLE

9.569929616929079315015921112591100205368711850043520563464627344270444107788370394 * 10^-24 J/K.

PROG

(PARI) 1.380649e-23*log(2) \\ Charles R Greathouse IV, Sep 20 2024

CROSSREFS

Cf. A070063.

KEYWORD

cons,nonn,new

AUTHOR

Charles R Greathouse IV, Sep 20 2024

STATUS

approved

A376291

Numbers k such that k^k is not a cube and can be expressed as (a^3 + b^3)/2 for positive integers a, b.

+0
0

76, 112, 172, 364, 427, 532

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Sequence is equal to all terms in A267415 that are not in A376279.

If a or b are allowed to be 0, then 2, 4, 128, 256, 686, 1372, 2000, 4000, 4394, ... are also terms.

LINKS

Table of n, a(n) for n=1..6.

EXAMPLE

76^76 is not a cube and is equal to (523974089123227128080087214816032969930445946880^3 + 314384453473936276848052328889619781958267568128^3)/2.

112^112 is not a cube and is equal to (39739105680019344543609706294181022974041418385894471812303329593638887882752^3 + 13246368560006448181203235431393674324680472795298157270767776531212962627584^3)/2.

172^172 is not a cube and is equal to (186302856478727791003189416646781404088735216286873615701287403506960347135763572314025783484358072432973760558663310530664464384^3 + 26614693782675398714741345235254486298390745183839087957326771929565763876537653187717969069194010347567680079809044361523494912^3)/2

CROSSREFS

Cf. A267415, A376279.

KEYWORD

nonn,more,new

AUTHOR

Chai Wah Wu, Sep 19 2024

STATUS

approved

A375786

a(n) is the minimum volume of an integer-sided cuboid having the same surface as a cube with edge length n.

+0
0

1, 8, 13, 36, 37, 104, 73, 188, 121, 252, 181, 428, 253, 540, 337, 764, 433, 828, 541, 1196, 661, 1448, 793, 1476, 937, 2024, 1093, 2160, 1261, 2592, 1441, 2628, 1633, 3464, 1837, 3884, 2053, 3708, 2281, 4796, 2521, 5148, 2773, 5616, 3037, 5436, 3313, 6660, 3601

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Conjecture: From the integer-sided cuboids with same surface 6*n^2 always the one with the smallest edge length has the minimum volume. If there are several integer-sided cuboids having the smallest edge length, then the one with the smallest second smallest edge length has the minimum volume (checked up to a(1000)).

The maximum volume is always A000578(n) = n^3.

LINKS

Felix Huber, Table of n, a(n) for n = 1..10000

Felix Huber, Maple programs

Eric Weisstein's World of Mathematics, Cuboid

EXAMPLE

a(6) = 104: because from the five integer-sided cuboids (2, 2, 26), (2, 5, 14), (2, 6, 12), (3, 6, 10), (6, 6, 6) having the same surface as a cube with edge length 6 (see example in A375785) has (2, 2, 26) with 2*2*26 = 104 the smallest volume.

MAPLE

See Huber link.

CROSSREFS

Cf. A000578, A369951, A375580, A375785.

KEYWORD

nonn,new

AUTHOR

Felix Huber, Sep 17 2024

STATUS

approved

A375785

a(n) is the number of distinct integer-sided cuboids having the same surface as a cube with edge length n.

+0
0

1, 1, 3, 3, 5, 5, 5, 7, 9, 9, 9, 13, 9, 9, 19, 15, 13, 19, 13, 23, 19, 19, 17, 29, 25, 19, 27, 23, 21, 41, 21, 31, 35, 29, 33, 45, 25, 29, 35, 51, 29, 41, 29, 45, 61, 39, 33, 61, 33, 57, 51, 45, 37, 63, 61, 51, 51, 49, 41, 97, 41, 49, 61, 63, 61, 81, 45, 67, 67

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

a(n) is the number of unordered solutions (x, y, z) to x*y + y*z + x*z = 3*n^2 in positive integers x and y.

Conjecture: All terms are odd.

LINKS

Felix Huber, Table of n, a(n) for n = 1..10000

Felix Huber, Maple programs

Eric Weisstein's World of Mathematics, Cuboid

EXAMPLE

a(6) = 5 because exactly the 5 integer-sided cuboids (2, 2, 26), (2, 5, 14), (2, 6, 12), (3, 6, 10), (6, 6, 6) have the same surface as a cube with edge length 6: 2*(2*2 + 2*26 + 2*26) = 2*(2*5 + 5*14 + 2*14) = 2*(2*6 + 6*12 + 2*12) = 2*(3*6 + 6*10 + 3*10) = 2*(6*6 + 6*6 + 6*6) = 6*6^2.

MAPLE

See Huber link.

CROSSREFS

Cf. A000578, A003167, A066955, A369951, A375580, A375786, A376074.

KEYWORD

nonn,new

AUTHOR

Felix Huber, Sep 17 2024

STATUS

approved

A375744

Numbers that are the sum of 4 but no fewer nonzero squares and admit a representation with 4 distinct squares.

+0
0

39, 63, 71, 79, 87, 95, 111, 119, 127, 135, 143, 151, 156, 159, 167, 175, 183, 191, 199, 207, 215, 223, 231, 239, 247, 252, 255, 263, 271, 279, 284, 287, 295, 303, 311, 316, 319, 327, 335, 343, 348, 351, 359, 367, 375, 380, 383, 391, 399, 407, 415, 423, 431

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Intersection of A004215 and A004433.

LINKS

Table of n, a(n) for n=1..53.

EXAMPLE

39 is a term, since it requires 4 squares to be represented and admits the representation 39 = 5^2 + 3^2 + 2^2 + 1^2.

30 is not a term, although it can be represented as a sum of 4 different squares 30 = 4^2+ 3^2 + 2^2 + 1^2 also admits a representation as a sum of 3 squares: 30 = 5^2 + 2^2 + 1^2.

7 is not a term, since although it requires 4 squares to be represented as follows 7 = 2^2 + 1^2 + 1^2 + 1^2, it is noted that 1 is used on more than one occasion.

CROSSREFS

Cf. A004215, A004433.

KEYWORD

nonn,new

AUTHOR

Gonzalo Martínez, Aug 26 2024

STATUS

approved

A375473

a(n) is the area of the largest rectangle with integer sides that can be inscribed under the parabola y = -x^2 + n and on or above the x-axis.

+0
0

0, 0, 2, 4, 6, 8, 10, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Given the function defined by f(x) = -x^2 + n, the area of each rectangle inscribed under the parabola associated with f and on the x-axis is modeled by the function g(x) = 2x*(-x^2 + n), where 2x is the base of the rectangle and ( -x^2 +n) is its height. The value of x that maximizes the area is x = sqrt(n/3). However, this value is not always an integer, so x should be chosen as the nearest integer to sqrt(n/3), which corresponds to floor(1/2 + sqrt(n/3 - 1/12)).

LINKS

Table of n, a(n) for n=0..58.

FORMULA

a(n) = 2*floor(1/2 + sqrt(n/3 - 1/12))*(-(floor(1/2 + sqrt(n/3 - 1/12)))^2 + n).

CROSSREFS

Cf. A361795, A000194.

KEYWORD

nonn,new

AUTHOR

Gonzalo Martínez, Aug 17 2024

STATUS

approved

A375358

a(n) is the greatest difference between m and k, with m, k both prime such that k + m = p + q, where (p, q) is the n-th twin prime pair.

+0
0

2, 2, 14, 26, 46, 74, 106, 134, 194, 206, 266, 286, 346, 374, 382, 442, 454, 506, 550, 614, 686, 818, 854, 914, 1034, 1118, 1186, 1226, 1274, 1294, 1606, 1630, 1618, 1702, 1754, 2018, 2042, 2078, 2102, 2174, 2290, 2434, 2546, 2534, 2582, 2626, 2846, 2890, 2950

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

If p and q are twin primes and x is their average, then among all pairs of primes (k, m) such that |x - k| = |x - m|, it is observed that p and q are at the smallest distance from x, which is 1. Our interest lies in finding the pair (m, k) such that the distance to x is maximum and then determining |k - m|.

LINKS

Table of n, a(n) for n=1..49.

EXAMPLE

Since the 3rd pair of twin primes is (11, 13), whose sum is 24, and the other pairs of primes that sum to 24 are (5, 19) and (7, 17), the greatest difference is 19 - 5 = 14. Therefore, a(3) = 14.

PROG

(Python)

from itertools import islice

from sympy import isprime, nextprime, primerange

def agen(): # generator of terms

p, q = 2, 3

while True:

if q - p == 2:

s = p + q

yield max(m-k for k in primerange(2, s//2+1) if isprime(m:=s-k))

p, q = q, nextprime(q)

print(list(islice(agen(), 80))) # Michael S. Branicky, Aug 13 2024

CROSSREFS

Cf. A001359, A020482, A014574.

KEYWORD

nonn,new

AUTHOR

Gonzalo Martínez, Aug 13 2024

STATUS

approved