A301764 -id:A301764 - cp's OEIS frontend

A301421 Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 6 data.

1, 6, 46, 371, 3026, 24707, 201748, 1647429, 13452565, 109850886, 897019828, 7324880157, 59813470848, 488424550081, 3988374821616, 32568251770049, 265945672309613, 2171657880797162, 17733313387923690, 144806604435722311, 1182461068019218530, 9655734852907204771

Offset: 1

Gregory Gerard Wojnar, Mar 20 2018

nonn

Re-express the Girard-Waring formulae to yield the mean powers in terms of the mean symmetric polynomials in the data values. Then for a family of 6 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/6)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) See extended comment in A301417.

Gregory Gerard Wojnar, Table of n, a(n) for n = 1..62 [a(21) corrected by _Georg Fischer_, Aug 18 2021]
G. G. Wojnar, D. S. Wojnar, and L. Q. Brin, Universal peculiar linear mean relationships in all polynomials, arXiv:1706.08381 [math.GM], 2017. See Table GW.n=6 p. 24.

Cf. A301764, A024537, A195350, A301417, A301420, A301424.

lista(6, nn) \\ use pari script file in A301417; Michel Marcus, Apr 21 2018

G.f.: (-x*(x+1)^5+1)/(x^7+5*x^6+9*x^5+5*x^4-5*x^3-9*x^2-7*x+1); this denominator equals (1-x)*(2-(1+x)^6) (conjectured).

a(21) corrected by Georg Fischer, Aug 18 2021

A302601 Numbers that are powers of a prime number whose prime index is also a prime power (not including 1).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 17, 19, 23, 25, 27, 31, 41, 49, 53, 59, 67, 81, 83, 97, 103, 109, 121, 125, 127, 131, 157, 179, 191, 211, 227, 241, 243, 277, 283, 289, 311, 331, 343, 353, 361, 367, 401, 419, 431, 461, 509, 529, 547, 563, 587, 599, 617, 625, 661, 691, 709

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			49 is in the sequence because 49 = prime(4)^2 = prime(prime(1)^2)^2.
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset multisystems.
001: {}
003: {{1}}
005: {{2}}
007: {{1,1}}
009: {{1},{1}}
011: {{3}}
017: {{4}}
019: {{1,1,1}}
023: {{2,2}}
025: {{2},{2}}
027: {{1},{1},{1}}
031: {{5}}
041: {{6}}
049: {{1,1},{1,1}}
053: {{1,1,1,1}}
059: {{7}}
067: {{8}}
081: {{1},{1},{1},{1}}
083: {{9}}
097: {{3,3}}
103: {{2,2,2}}
109: {{10}}
121: {{3},{3}}
125: {{2},{2},{2}}
127: {{11}}
131: {{1,1,1,1,1}}

Cf. A000961, A001222, A003963, A005117, A007425, A007716, A056239, A275024, A281113, A295931, A301764, A302242, A302243, A302498.

Select[Range[1000],#===1||MatchQ[FactorInteger[#],{{?(PrimePowerQ[PrimePi[#]]&),}}]&]

isok(n) = (n==1) || ((isprimepower(n, &p)) && isprimepower(primepi(p))); \\ Michel Marcus, Apr 10 2018

A302602 Numbers that are powers of a prime number whose prime index is either 1 or also a prime number.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 11, 16, 17, 25, 27, 31, 32, 41, 59, 64, 67, 81, 83, 109, 121, 125, 127, 128, 157, 179, 191, 211, 241, 243, 256, 277, 283, 289, 331, 353, 367, 401, 431, 461, 509, 512, 547, 563, 587, 599, 617, 625, 709, 729, 739, 773, 797, 859, 877, 919

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			25 is in the sequence because 25 = prime(3)^2 and 3 is a prime number.
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
08: {{},{},{}}
09: {{1},{1}}
11: {{3}}
16: {{},{},{},{}}
17: {{4}}
25: {{2},{2}}
27: {{1},{1},{1}}
31: {{5}}
32: {{},{},{},{},{}}
41: {{6}}
59: {{7}}
64: {{},{},{},{},{},{}}
67: {{8}}
81: {{1},{1},{1},{1}}
83: {{9}}

Cf. A000961, A001222, A003963, A005117, A007425, A007716, A056239, A275024, A281113, A295931, A301764, A302242, A302243.

Select[Range[1000],#===1||MatchQ[FactorInteger[#],{{?(#===2||PrimeQ[PrimePi[#]]&),}}]&]

isok(n) = (n==1) || ((isprimepower(n, &p)) && ((p==2) || isprime(primepi(p)))); \\ Michel Marcus, Apr 10 2018

cp's OEIS Frontend

A301421 Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 6 data.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A302601 Numbers that are powers of a prime number whose prime index is also a prime power (not including 1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A302602 Numbers that are powers of a prime number whose prime index is either 1 or also a prime number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI