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 1-9 of 9 results.

A277852 "Late birds" (values a(n) < a(k) for all k > n) in A075771 = quotient + remainder of Euclidean division of n^2 by prime(n).

Original entry on oeis.org

1, 2, 4, 5, 9, 15, 16, 24, 59, 61, 81, 100, 124, 129, 152, 169, 196, 249, 305, 400, 425, 475, 520, 556, 565, 676, 771, 795, 904, 939
Offset: 1

Views

Author

M. F. Hasler, Nov 25 2016

Keywords

Comments

The lower bound A075771(n) >= n^2/prime(n) ensures that a given number can't occur beyond a certain index in that sequence.
See A277853 for the corresponding indices m.

Crossrefs

Cf. A075771, A277851, A277853, A277854.

Programs

  • PARI
    A277852(N,L=N^2/prime(N),A=A075771,S=List())={forstep(n=N,1,-1,A(n)

Formula

a(n) = A075771(A277853(n)), i.e., equals A075771 o A277853.

A277853 Indices m of "late birds", i.e., values a(m) < a(k) for all k > m, in A075771 = quotient + remainder in Euclidean division of n^2 by prime(n).

Original entry on oeis.org

1, 2, 4, 5, 21, 27, 44, 104, 173, 365, 369, 500, 590, 735, 840, 987, 1564, 1797, 2415, 3368, 3545, 4025, 4466, 5018, 5477, 6686, 7239, 8025, 8182, 9369
Offset: 1

Views

Author

M. F. Hasler, Nov 25 2016

Keywords

Comments

See A277852 (= A075771 o A277853) for the corresponding values of the "late birds".

Crossrefs

Cf. A075771, A277851, A277852, A277854.

Programs

  • PARI
    A277853(N,L=N^2/prime(N),A=A075771,S=List())={forstep(n=N,1,-1,A(n)

A277851 Numbers not occurring in A075771 (= q(n) + r(n), with n^2 = prime(n)*q(n) + r(n), 0 <= r(n) < prime(n)).

Original entry on oeis.org

3, 6, 7, 8, 11, 13, 14, 18, 19, 20, 21, 22, 23, 25, 26, 29, 30, 32, 34, 35, 38, 39, 40, 43, 44, 45, 47, 51, 53, 54, 56, 58, 62, 67, 68, 69, 70, 71, 72, 74, 75, 77, 78, 80, 82, 83, 87, 89, 90, 91, 92, 94, 97, 98, 99, 102, 103, 104, 105, 106, 107, 108, 110, 111, 112, 115, 117, 118, 119, 120, 122, 123, 125, 126
Offset: 1

Views

Author

M. F. Hasler, Nov 25 2016

Keywords

Comments

The lower bound A075771(n) >= n^2/prime(n) ensures that a given number can't occur beyond a certain index in that sequence.

Crossrefs

Cf. A075771, A277852, A277853, A277854.

Programs

  • Mathematica
    mx = 6300; Take[ Complement[ Range@ mx, Array[Plus @@ IntegerDigits[#^2, Prime[#]] &, mx]], mx/90] (* Robert G. Wilson v, Nov 25 2016 *)
  • PARI
    A277851_vec(N,L=N^2\prime(N),A=A075771)=setminus([1..L],Set(vector(N,n,A(n))))

A277854 Frequent terms, i.e., values such that no smaller value appears more often, in A075771 = quotient + remainder of Euclidean division of n^2 by prime(n).

Original entry on oeis.org

1, 2, 4, 5, 9, 15, 16, 48, 64, 86, 100, 144, 169, 3364, 3969, 4096, 195364
Offset: 1

Views

Author

M. F. Hasler, Nov 25 2016

Keywords

Comments

Equivalently, record values (in the weak sense of >=) in the sequence of frequencies of values of A075771. (The lower bound A075771(n) >= n^2/prime(n) ensures that no number below this limit can occur beyond the index n in that sequence.)
It appears that this sequence contains mainly squares, but there are exceptions such as 2, 5, 15, 48, 86, and some squares (25 = 5^2, 36 = 6^2, 49 = 7^2, 81 = 9^2, 121 = 11^2) do not occur. Is there an explanation for this and/or the fact that exceptions are close to missing squares: 48 ~ 49 = 7^2, 86 ~ 81 = 9^2 ? Can one prove or disprove that
- from some point on, only squares will occur?
- all sufficiently large squares (or: even squares?) will occur?
- from a(12) = 144 (or some later point) on, a(n) will occur in A075771 strictly more often than the preceding value?
a(18) > 10^6. - Robert G. Wilson v, Nov 25 2016

Examples

			Values that occur in A075771 not less often than any smaller value are 1, 2, 4 (which appear once), 5, 9, 15 (which appear twice), 16, 48, 64, 86, 100 (which appear three times), 144 (which appears five times), 169 (which appears seven times), ...

Crossrefs

Cf. A075771, A277851, A277852, A277853.

Extensions

a(14)-a(17) from Robert G. Wilson v, Nov 25 2016

A135101 Digital sum (base the n-th prime) of n^n.

Original entry on oeis.org

1, 2, 3, 10, 15, 24, 55, 46, 71, 116, 101, 180, 213, 196, 205, 276, 307, 444, 337, 610, 621, 646, 687, 808, 985, 876, 921, 996, 1049, 1184, 1417, 1576, 1665, 1576, 2127, 1836, 2377, 1660, 2201, 2088, 2731, 2844, 2847, 2944, 3317, 3232, 3503, 3294, 3165
Offset: 1

Views

Author

Hieronymus Fischer, Dec 24 2007

Keywords

Examples

			a(2) = ds_prime(2)(2^2) = ds_3(4) = 1+1 = 2;
a(10) = ds_prime(5)(5^5) = ds_11(3125) = 2+3+9+1 = 15.

Links

Crossrefs

Cf. A000040, A000045, A007953, A054899, A075771, A131451, A133620, A133900, A134599.

Programs

  • Mathematica
    Table[Total[IntegerDigits[n^n, Prime[n]]], {n, 50}] (* G. C. Greubel, Sep 23 2016 *)
  • PARI
    a(n) = vecsum(digits(n^n, prime(n))); \\ Michel Marcus, Sep 24 2016

Formula

a(n) = ds_prime(n)(n^n), where ds_prime(n) = digital sum base the n-th prime.
a(n) = n^n - (prime(n)-1)*Sum{k>0} ( floor(n^n/prime(n)^k) ).

A135102 Digital sum (base the n-th prime) of Fibonacci(n).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 3, 12, 27, 29, 36, 33, 41, 58, 51, 31, 64, 89, 45, 74, 83, 39, 168, 145, 193, 170, 129, 149, 104, 211, 289, 274, 175, 257, 252, 125, 161, 318, 347, 447, 316, 317, 285, 450, 107, 253, 648, 363, 301, 498, 409, 773, 522, 429, 515, 782, 649, 641
Offset: 1

Views

Author

Hieronymus Fischer, Dec 24 2007

Keywords

Examples

			a(2) = ds_prime(2)(Fib(2)) = ds_3(1) = 1;
a(10) = ds_prime(10)(55) = ds_29(55) = 1+26 = 27.

Links

Crossrefs

Cf. A000040, A000045, A007953, A054899, A075771, A131451, A133620, A133900, A134599.

Programs

  • Mathematica
    Table[Total[IntegerDigits[Fibonacci[n],Prime[n]]],{n,60}] (* Harvey P. Dale, May 05 2013 *)
  • PARI
    a(n) = vecsum(digits(fibonacci(n), prime(n))); \\ Michel Marcus, Sep 24 2016

Formula

a(n) = ds_prime(n)(Fib(n)), where ds_prime(n) = digital sum base the n-th prime.
a(n) = Fibonacci(n) - (prime(n)-1)*Sum{k>0} ( floor(Fibonacci(n)/prime(n)^k) ).

A135103 Digital sum (base the n-th prime) of n^3.

Original entry on oeis.org

1, 4, 3, 4, 5, 12, 7, 26, 25, 20, 41, 36, 37, 56, 63, 40, 41, 72, 61, 90, 117, 118, 113, 96, 73, 76, 99, 116, 89, 120, 181, 138, 169, 112, 251, 156, 109, 116, 57, 188, 35, 108, 87, 128, 181, 118, 83, 258, 129, 284, 179, 188, 317, 214, 231, 338, 273, 442, 311, 400, 253
Offset: 1

Views

Author

Hieronymus Fischer, Dec 24 2007

Keywords

Examples

			a(2)=ds_prime(2)(2^3)=ds_3(8)=2+2=4; a(6)=ds_prime(6)(6^3)=ds_13(216)=1+3+8=12.

Links

Crossrefs

Cf. A000040, A007953, A054899, A075771, A131451, A133620, A133900, A134599.

Programs

  • Mathematica
    Table[Total[IntegerDigits[n^3,Prime[n]]],{n,70}] (* Harvey P. Dale, Oct 21 2011 *)

Formula

a(n)=ds_prime(n)(n^3), where ds_prime(n)=digital sum base the n-th prime.
a(n)=n^3-(prime(n)-1)*sum{k>0, floor(n^3/prime(n)^k)}.

A135104 Digital sum (base the n-th prime) of n^4.

Original entry on oeis.org

1, 4, 5, 10, 15, 24, 17, 28, 27, 60, 31, 36, 81, 70, 71, 68, 117, 96, 37, 120, 81, 100, 139, 192, 97, 176, 123, 174, 205, 128, 193, 126, 137, 220, 201, 216, 133, 196, 397, 296, 189, 396, 321, 256, 305, 280, 331, 396, 445, 292, 313, 256, 481, 556, 417, 326, 553, 256
Offset: 1

Views

Author

Hieronymus Fischer, Dec 24 2007

Keywords

Examples

			a(2) = ds_prime(2)(2^4) = ds_3(16) = 1+2+1 = 4;
a(6) = ds_prime(6)(6^4) = ds_13(1296) = 7+8+9 = 24.

Links

Crossrefs

Cf. A000040, A007953, A054899, A075771, A131451, A133620, A133900, A134599.

Programs

  • Mathematica
    Table[Total[IntegerDigits[n^4, Prime[n]]], {n, 50}] (* G. C. Greubel, Sep 23 2016 *)
  • PARI
    a(n) = vecsum(digits(n^4, prime(n))); \\ Michel Marcus, Sep 24 2016

Formula

a(n) = ds_prime(n)(n^4), where ds_prime(n) = digital sum base the n-th prime.
a(n) = n^4 - (prime(n)-1)*Sum{k>0} ( floor(n^4/prime(n)^k) ).

A135105 Digital sum (base the n-th prime) of n^5.

Original entry on oeis.org

1, 4, 11, 16, 15, 12, 23, 44, 45, 40, 41, 72, 93, 98, 99, 48, 133, 108, 109, 160, 117, 172, 81, 120, 217, 176, 159, 102, 221, 144, 187, 262, 169, 304, 375, 276, 241, 158, 211, 316, 273, 72, 313, 320, 397, 406, 227, 582, 335, 236, 187, 460, 293, 274, 663, 178, 433, 538
Offset: 1

Views

Author

Hieronymus Fischer, Dec 24 2007

Keywords

Examples

			a(2) = ds_prime(2)(2^5) = ds_3(32) = 1+0+1+2 = 4;
a(6) = ds_prime(5)(5^5) = ds_11(3125) = 2+9+3+1 = 15.

Links

Crossrefs

Cf. A000040, A007953, A054899, A075771, A131451, A133620, A133900, A134599.

Programs

  • Mathematica
    Table[Total[IntegerDigits[n^5,Prime[n]]],{n,60}] (* Harvey P. Dale, Jul 19 2013  *)
  • PARI
    a(n) = vecsum(digits(n^5, prime(n))); \\ Michel Marcus, Sep 24 2016

Formula

a(n) = ds_prime(n)(n^5), where ds_prime(n) = digital sum base the n-th prime.
a(n) = n^5 - (prime(n)-1)*Sum_{k>0} ( floor(n^5/prime(n)^k) ).
Showing 1-9 of 9 results.

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