cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 42 results. Next

A279643 Exponential transform of the tenth powers A008454.

Original entry on oeis.org

1, 1, 1025, 62122, 4436645, 635999636, 70891240117, 9361749284896, 1491531423411913, 235989968039151760, 40944833826904310921, 7754112338325635303264, 1525672210381158739381165, 318496972975593582426074560, 70389888724665631249754800189
Offset: 0

Views

Author

Alois P. Heinz, Dec 16 2016

Keywords

Links

Crossrefs

Column k=10 of A279636.
Cf. A008454.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^10*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);

Formula

E.g.f.: exp(exp(x)*(x^10+45*x^9+750*x^8+5880*x^7+22827*x^6+42525*x^5 +34105*x^4 +9330*x^3 +511*x^2+x)).

A253710 Second partial sums of tenth powers (A008454).

Original entry on oeis.org

1, 1026, 61100, 1169750, 12044025, 83384476, 437200176, 1864757700, 6779099625, 21693441550, 62545208076, 165314338826, 405941961425, 935824239000, 2042356907200, 4248401203176, 8470439399601, 16262944822650, 30186516503500, 54350088184350, 95193540843401, 162596916293876, 271426802958000, 443660070587500
Offset: 1

Views

Author

Luciano Ancora, Jan 10 2015

Keywords

Comments

The formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) are the m-th Faulhaber's formulas.

Links

Programs

  • Mathematica
    a253710[n_] := Block[{f}, f[1] = 1; f[2] = 1026; f[x_] := 2*f[x - 1] - f[x - 2] + x^10; Array[f, n]]; a253710[21] (* Michael De Vlieger, Jan 11 2015 *)
CoefficientList[Series[(1 + 1013 x + 47840 x^2 + 455192 x^3 + 1310354 x^4 + 1310354 x^5 + 455192 x^6 + 47840 x^7 + 1013 x^8 + x^9) / (1 - x)^13, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 19 2015 *)
Nest[Accumulate,Range[30]^10,2] (* Harvey P. Dale, May 10 2019 *)

Formula

a(n) = n*(n+1)^2*(n+2)*(n^2 + 2*n - 2)*(2*n^6 + 12*n^5 + 16*n^4 - 16*n^3 - 17*n^2 + 30*n - 5)/264.
a(n) = 2*a(n-1)-a(n-2)+n^10.
G.f.: x*(1 + 1013*x + 47840*x^2 + 455192*x^3 + 1310354*x^4 + 1310354*x^5 + 455192*x^6 + 47840*x^7 + 1013*x^8 + x^9)/(1-x)^13. - Vincenzo Librandi, Jan 19 2015

A239504 Number of digits in the decimal expansion of n^10 (A008454).

Original entry on oeis.org

1, 1, 4, 5, 7, 7, 8, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19
Offset: 0

Views

Author

Lorianne Kwak, Mar 20 2014

Keywords

Examples

			a(1) = 1, because 1^10 = 1.
a(2) = 4, because 2^10 = 1024.

Programs

  • Mathematica
    Join[{1},IntegerLength[Range[80]^10]] (* Harvey P. Dale, Jun 29 2021 *)
  • PARI
    a(n) = #Str(n^10); \\ Michel Marcus, Mar 21 2014

Formula

a(n) = floor(10*log_10(n))+1 for n>0, a(0) = 1.
a(n) = A055642(n^10).

A001016 Eighth powers: a(n) = n^8.

Original entry on oeis.org

0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, 4294967296, 6975757441, 11019960576, 16983563041, 25600000000, 37822859361, 54875873536, 78310985281, 110075314176
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Besides the first term, this sequence lists the denominators in Pi^8/9450 = 1 + 1/256 + 1/6561 + 1/65536 + 1/390625 + 1/1679616 + ... - Mohammad K. Azarian, Nov 01 2011, edited by M. F. Hasler, Jul 03 2025
For n > 0, a(n) is the largest number k such that k + n^4 divides k^2 + n^4. - Derek Orr, Oct 01 2014
Fourth powers of squares and squares of 4th powers. Squares composed with themselves twice. - Wesley Ivan Hurt, Apr 01 2016

References

  • Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000290 (squares), A000583 (fourth powers), A001014 - A001017 (6th - 9th powers), A008454 (10th powers), A010801 (13th powers).
Cf. A000542 (partial sums), A022524 (first differences), A013666 (zeta(8)).
Cf. A003380 - A003390 (sums of 2, ..., 12 eighth powers).

Programs

Formula

Multiplicative with a(p^e) = p^(8e). - David W. Wilson, Aug 01 2001
Totally multiplicative sequence with a(p) = p^8 for primes p. - Jaroslav Krizek, Nov 01 2009
G.f.: -x*(1+x)*(x^6+246*x^5+4047*x^4+11572*x^3+4047*x^2+246*x+1)/(x-1)^9. - R. J. Mathar, Jan 07 2011
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) + 40320. - Ant King, Sep 24 2013
From Wesley Ivan Hurt, Apr 01 2016: (Start)
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n > 8.
a(n) = A000290(n)^4 = A000290(A000290(A000290(n))).
a(n) = A000583(n)^2. (End)
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(8) = Pi^8/9450 (A013666).
Sum_{n>=1} (-1)^(n+1)/a(n) = 127*zeta(8)/128 = 127*Pi^8/1209600. (End)
E.g.f.: exp(x)*x*(1 + 127*x + 966*x^2 + 1701*x^3 + 1050*x^4 + 266*x^5 + 28*x^6 + x^7). - Stefano Spezia, Jul 29 2022

Extensions

More terms from James Sellers, Sep 19 2000

A008455 11th powers: a(n) = n^11.

Original entry on oeis.org

0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, 1792160394037, 4049565169664, 8649755859375, 17592186044416, 34271896307633, 64268410079232, 116490258898219, 204800000000000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000583, A000584, A001014, A001017, A008454, A013669.
Cf. A004813 - A004823 (sums of 2, ..., 12 positive eleventh powers).

Programs

Formula

a(n) = A000584(n)*A001014(n).
Multiplicative with a(p^e) = p^(11*e). - David W. Wilson, Aug 01 2001
Totally multiplicative with a(p) = p^11 for primes p. - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(11) (A013669).
Sum_{n>=1} (-1)^(n+1)/a(n) = 1023*zeta(11)/1024. (End)

A004812 Numbers that are the sum of 12 positive 10th powers.

Original entry on oeis.org

12, 1035, 2058, 3081, 4104, 5127, 6150, 7173, 8196, 9219, 10242, 11265, 12288, 59060, 60083, 61106, 62129, 63152, 64175, 65198, 66221, 67244, 68267, 69290, 70313, 118108, 119131, 120154, 121177, 122200, 123223, 124246, 125269, 126292, 127315, 128338, 177156, 178179, 179202
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 02 2020: (Start)
40172177 is in the sequence as 40172177 = 1^10 + 1^10 + 1^10 + 1^10 + 2^10 + 2^10 + 3^10 + 4^10 + 5^10 + 5^10 + 5^10 + 5^10.
90873751 is in the sequence as 90873751 = 1^10 + 1^10 + 1^10 + 2^10 + 2^10 + 2^10 + 3^10 + 4^10 + 5^10 + 5^10 + 5^10 + 6^10.
122264704 is in the sequence as 122264704 = 1^10 + 3^10 + 4^10 + 4^10 + 4^10 + 5^10 + 5^10 + 5^10 + 5^10 + 5^10 + 5^10 + 6^10. (End)

Links

Crossrefs

Cf. A008454 (tenth powers), A003335 - A004823 (same for 3rd - 11th powers).

Programs

  • PARI
    A004812_upto(N, n=12, p=10)={my(P=[x^p|x<-[1..sqrtnint(N-n+1, p)]], S=P); while(n--, S=Set(concat([[x+y|y<-S, x+y<=N]|x<-P]))); S} \\ M. F. Hasler, Jul 03 2025

A030629 Numbers with 11 divisors.

Original entry on oeis.org

1024, 59049, 9765625, 282475249, 25937424601, 137858491849, 2015993900449, 6131066257801, 41426511213649, 420707233300201, 819628286980801, 4808584372417849, 13422659310152401
Offset: 1

Views

Author

Jeff Burch

Keywords

Comments

Let p be a prime. Then the n-th number with p divisors is equal to prime(n)^(p-1). - Omar E. Pol, May 06 2008

Links

Crossrefs

Cf. A000005, A000040, A001248, A008454, A009087, A030514, A030516.
Cf. A013668, A013678.

Programs

Formula

a(n) = A000040(n)^10, i.e. tenth power of n-th prime. - Henry Bottomley, Aug 20 2001
From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(10)/zeta(20) = 16368226875/(174611*Pi^10) = A013668/A013678.
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(10) = 93555/Pi^10 = 1/A013668. (End)

A004809 Numbers that are the sum of 9 positive 10th powers.

Original entry on oeis.org

9, 1032, 2055, 3078, 4101, 5124, 6147, 7170, 8193, 9216, 59057, 60080, 61103, 62126, 63149, 64172, 65195, 66218, 67241, 118105, 119128, 120151, 121174, 122197, 123220, 124243, 125266, 177153, 178176, 179199, 180222, 181245, 182268, 183291, 236201, 237224, 238247, 239270
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 03 2020: (Start)
251633402 is in the sequence as 251633402 = 1^10 + 2^10 + 2^10 + 2^10 + 5^10 + 6^10 + 6^10 + 6^10 + 6^10.
383052503 is in the sequence as 383052503 = 1^10 + 1^10 + 4^10 + 5^10 + 5^10 + 5^10 + 5^10 + 6^10 + 7^10.
626642399 is in the sequence as 626642399 = 1^10 + 1^10 + 3^10 + 3^10 + 3^10 + 4^10 + 6^10 + 7^10 + 7^10. (End)

Links

Crossrefs

Cf. A008454 (tenth powers).

Programs

  • Mathematica
    k = 9; p = 10; amax = 10^6; bmax = amax^(1/p) // Ceiling; Clear[b]; b[0] = 1; Select[Table[Total[Array[b, k]^p], {b[1], b[0], bmax}, Evaluate[ Sequence @@ Table[{b[j], b[j-1], bmax}, {j, 1, k}]]] // Flatten // Union, # <= amax&] (* Jean-François Alcover, Jul 19 2017 *)

A004810 Numbers that are the sum of 10 positive 10th powers.

Original entry on oeis.org

10, 1033, 2056, 3079, 4102, 5125, 6148, 7171, 8194, 9217, 10240, 59058, 60081, 61104, 62127, 63150, 64173, 65196, 66219, 67242, 68265, 118106, 119129, 120152, 121175, 122198, 123221, 124244, 125267, 126290, 177154, 178177, 179200, 180223, 181246, 182269, 183292, 184315
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 03 2020: (Start)
72332028 is in the sequence as 72332028 = 1^10 + 1^10 + 1^10 + 2^10 + 2^10 + 2^10 + 4^10 + 4^10 + 5^10 + 6^10.
243962883 is in the sequence as 243962883 = 1^10 + 1^10 + 1^10 + 2^10 + 4^10 + 4^10 + 6^10 + 6^10 + 6^10 + 6^10.
312998872 is in the sequence as 312998872 = 1^10 + 2^10 + 3^10 + 3^10 + 3^10 + 4^10 + 5^10 + 5^10 + 5^10 + 7^10. (End)

Links

Crossrefs

Cf. A008454.
Column k=10 of A336725.

Programs

  • Mathematica
    M = 1413602992; m = M^(1/10) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++, For[h = g, h <= m, h++, For[i = h, i <= m, i++,
For[j = i, j <= m, j++,
s = a^10 + b^10 + c^10 + d^10 + e^10 + f^10 + g^10 + h^10 + i^10 + j^10;
If[s <= M, Sow[s]]]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

A004807 Numbers that are the sum of 7 positive 10th powers.

Original entry on oeis.org

7, 1030, 2053, 3076, 4099, 5122, 6145, 7168, 59055, 60078, 61101, 62124, 63147, 64170, 65193, 118103, 119126, 120149, 121172, 122195, 123218, 177151, 178174, 179197, 180220, 181243, 236199, 237222, 238245, 239268, 295247, 296270, 297293, 354295, 355318, 413343, 1048582
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 03 2020: (Start)
1129904068 is in the sequence as 1129904068 = 2^10 + 2^10 + 2^10 + 7^10 + 7^10 + 7^10 + 7^10.
1357385795 is in the sequence as 1357385795 = 2^10 + 2^10 + 3^10 + 3^10 + 4^10 + 7^10 + 8^10.
3831882028 is in the sequence as 3831882028 = 1^10 + 3^10 + 4^10 + 4^10 + 6^10 + 7^10 + 9^10. (End)

Links

Crossrefs

Cf. A008454 (tenth powers).

Programs

  • Mathematica
    k = 7; p = 10; amax = 10^6; bmax = amax^(1/p) // Ceiling; Clear[b]; b[0] = 1; Select[Table[Total[Array[b, k]^p], {b[1], b[0], bmax}, Evaluate[ Sequence @@ Table[{b[j], b[j-1], bmax}, {j, 1, k}]]] //Flatten // Union, # <= amax&] (* Jean-François Alcover, Jul 19 2017 *)
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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