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 1-6 of 6 results.

A319075 Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 7, 1, 32, 81, 125, 49, 11, 1, 64, 243, 625, 343, 121, 13, 1, 128, 729, 3125, 2401, 1331, 169, 17, 1, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 1, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23, 1, 1024, 19683, 390625, 823543, 1771561, 371293
Offset: 0

Views

Author

Omar E. Pol, Sep 09 2018

Keywords

Comments

If n = p - 1 where p is prime, then row n lists the numbers with p divisors.
The partial sums of column k give the column k of A319076.

Examples

			The corner of the square array is as follows:
         A000079 A000244 A000351  A000420    A001020    A001022     A001026
A000012        1,      1,      1,       1,         1,         1,          1, ...
A000040        2,      3,      5,       7,        11,        13,         17, ...
A001248        4,      9,     25,      49,       121,       169,        289, ...
A030078        8,     27,    125,     343,      1331,      2197,       4913, ...
A030514       16,     81,    625,    2401,     14641,     28561,      83521, ...
A050997       32,    243,   3125,   16807,    161051,    371293,    1419857, ...
A030516       64,    729,  15625,  117649,   1771561,   4826809,   24137569, ...
A092759      128,   2187,  78125,  823543,  19487171,  62748517,  410338673, ...
A179645      256,   6561, 390625, 5764801, 214358881, 815730721, 6975757441, ...
...

Crossrefs

Rows 0-13: A000012, A000040, A001248, A030078, A030514, A050997, A030516, A092759, A179645, A179665, A030629, A079395, A030631, A138031.
Other rows n: A030635 (n=16), A030637 (n=18), A137486 (n=22), A137492 (n=28), A139571 (n=30), A139572 (n=36), A139573 (n=40), A139574 (n=42), A139575 (n=46), A173533 (n=52), A183062 (n=58), A183085 (n=60), A261700 (n=100).
Columns 1-15: A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Main diagonal gives A093360.
Second diagonal gives A062457.
Third diagonal gives A197987.
Removing the 1's we have A182944/ A182945.
Cf. A006093, A319074, A319076.

Programs

  • PARI
    T(n, k) = prime(k)^n;

Formula

T(n,k) = A000040(k)^n, n >= 0, k >= 1.

A197987 a(n) = prime(n)^(n+1).

Original entry on oeis.org

4, 27, 625, 16807, 1771561, 62748517, 6975757441, 322687697779, 41426511213649, 12200509765705829, 787662783788549761, 243569224216081305397, 37929227194915558802161, 3177070365797955661914307, 566977372488557307219621121, 205442259656281392806087233013
Offset: 1

Views

Author

Bruno Berselli, Oct 20 2011

Keywords

Comments

Subsequence of A000961, A120458.
First five elements are also consecutive members of A133018. - Omar E. Pol, Oct 20 2011
Third diagonal of A319075. - Omar E. Pol, Sep 13 2018

Examples

			The fourth prime number is 7, so a(4) = 7^(4+1) = 7^5 = 16807. - _Omar E. Pol_, Oct 20 2011

Links

Crossrefs

Cf. A000961, A062457, A093360, A120458, A133018, A319075.

Programs

  • Magma
    [NthPrime(n)^(n+1): n in [1..16]];
  • Mathematica
    Table[Prime[n]^(n+1),{n,20}] (* Harvey P. Dale, Dec 16 2012 *)
  • PARI
    for(n=1, 16, print1(prime(n)^(n+1)", "));

Formula

a(n) = A000040(n)^(n+1). - Omar E. Pol, Oct 20 2011

A135485 a(n) = Sum_{i=1..n} prime(i)^(i-1), where prime(i) denotes i-th prime number.

Original entry on oeis.org

1, 4, 29, 372, 15013, 386306, 24523875, 918395614, 79229380895, 14586375356764, 834214662337565, 178751836441797978, 22742242136807984059, 1741006366419098769302, 258407992554086508103671
Offset: 1

Views

Author

Ctibor O. Zizka, Feb 07 2008, Feb 17 2008

Keywords

Comments

The primes in this sequence are 29 = 2^0 + 3^1 + 5^2, 15013 = 2^0 + 3^1 + 5^2 + 7^3 + 11^4, 82630...60939 (a 107-digit number) = 2^0 + 3^1 + 5^2 + ... + 211^46, ...

Links

Crossrefs

Cf. A000040, A087480, A135484.
Partial sums of A093360.

Programs

  • Mathematica
    f[n_] := Sum[Prime[i]^(i - 1), {i, n}]; Array[f, 16] (* Robert G. Wilson v, Feb 12 2008 *)
  • PARI
    a(n) = sum(k=1, n, prime(k)^(k-1)); \\ Michel Marcus, Oct 15 2016

Extensions

Edited and extended by Robert G. Wilson v, Feb 12 2008

A319074 a(n) is the sum of the first n nonnegative powers of the n-th prime.

Original entry on oeis.org

1, 4, 31, 400, 16105, 402234, 25646167, 943531280, 81870575521, 15025258332150, 846949229880161, 182859777940000980, 23127577557875340733, 1759175174860440565844, 262246703278703657363377, 74543635579202247026882160, 21930887362370823132822661921, 2279217547342466764922495586798
Offset: 1

Views

Author

Omar E. Pol, Sep 11 2018

Keywords

Examples

			For n = 4 the 4th prime is 7 and the sum of the first four nonnegative powers of 7 is 7^0 + 7^1 + 7^2 + 7^3 = 1 + 7 + 49 + 343 = 400, so a(4) = 400.

Crossrefs

Main diagonal of A319076.
Cf. A000040, A000203, A006093, A062457, A069459, A093360, A319075.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.

Programs

  • PARI
    a(n) = sum(k=0, n-1, prime(n)^k); \\ Michel Marcus, Sep 13 2018

Formula

a(n) = Sum_{k=0..n-1} A000040(n)^k.
a(n) = Sum_{k=0..n-1} A319075(k,n).
a(n) = (A000040(n)^n - 1)/(A000040(n) - 1).
a(n) = (A062457(n) - 1)/A006093(n).
a(n) = A069459(n)/A006093(n).
a(n) = A000203(A000040(n)^(n-1)).
a(n) = A000203(A093360(n)).

A130606 a(n) = prime(n+1)^n - prime(n)^n where prime(n) is the n-th prime number.

Original entry on oeis.org

1, 16, 218, 12240, 210242, 19310760, 483533066, 61327422240, 12705993314406, 398921053680600, 152509144883055582, 15980538294526150800, 793161021967277155922, 182781628843528905568920, 61073803538208251485772814
Offset: 1

Views

Author

Cino Hilliard, Jun 17 2007

Keywords

Examples

			For n=2, prime(2+1)^2 - prime(2)^2 = 5^2 - 3^2 = 4^2, the second entry.

Programs

  • Maple
    a := proc (n) options operator, arrow; ithprime(n+1)^n-ithprime(n)^n end proc: seq(a(n), n = 1 .. 15); # Emeric Deutsch, Jul 09 2007
  • Mathematica
    n[x_]:=Module[{pn=Prime[x]},(NextPrime[pn])^x-pn^x]; n/@Range[20]  (* Harvey P. Dale, Apr 11 2011 *)
  • PARI
    g1(n) = for(x=1,n,y=prime(x+1)^x-prime(x)^x;print1(y","))

Formula

a(n) = A093360(n+1) - A062457(n). - R. J. Mathar, Nov 25 2008

Extensions

More terms from Emeric Deutsch, Jul 09 2007

A130607 a(n) = prime(n+1)^n + prime(n)^n.

Original entry on oeis.org

5, 34, 468, 17042, 532344, 28964378, 1304210412, 95294548322, 16308298637332, 1240335520281002, 203326098675865244, 29146442306206221362, 2643367226597304414564, 330552343531805913099818, 85200500239848987963203500, 25435446457194919247155743362, 3513844792272393084250431362040
Offset: 1

Views

Author

Cino Hilliard, Jun 17 2007

Keywords

Comments

Prime(n) is the n-th prime number.

Examples

			For n=2, prime(2+1)^2 - prime(2)^2 = 5^2 + 3^2 = 34, the second term.

Crossrefs

Cf. A062457, A093360.

Programs

  • Mathematica
    Table[Prime[n+1]^n + Prime[n]^n, {n, 1, 20}] (* Amiram Eldar, Jun 30 2024 *)
  • PARI
    g2(n) = for(x=1,n,y=prime(x+1)^x+prime(x)^x;print1(y","))

Formula

a(n) = A062457(n) + A093360(n+1). - Amiram Eldar, Jun 30 2024

Extensions

More terms from Amiram Eldar, Jun 30 2024
Showing 1-6 of 6 results.

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