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.

Previous Showing 51-60 of 62 results. Next

A195007 Number of primes in the range (n*sqrt(n-1), (n+1)*sqrt(n)].

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 3, 2, 0, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 0, 3, 2, 1, 2, 1, 3, 2, 1, 1, 3, 0, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 1, 2, 1, 1, 2, 3, 2, 2, 4, 1, 2, 2, 2, 3, 1, 2, 1, 2, 3, 2, 2, 1, 1
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Sep 07 2011

Keywords

Examples

			a(1)=1`because 1*sqrt(1-1)<(prime 2)<=(1+1)*sqrt(1),
a(2)=1 because 2*sqrt(2-1)<(prime 3)<=(2+1)*sqrt(2).

Crossrefs

Cf. A006005, A192454.

Programs

  • Mathematica
    Table[PrimePi[(n+1)*Sqrt[n]] - PrimePi[n*Sqrt[n-1]], {n, 100}] (* T. D. Noe, Sep 14 2011 *)

A280084 1 together with the Pythagorean primes.

Original entry on oeis.org

1, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 25 2016

Keywords

Comments

Positive noncomposite numbers of the form 4k + 1.
Positive noncomposite numbers in A020668.
Essentially the same as A002313 and A002144. - R. J. Mathar, Jan 04 2017

Crossrefs

Cf. A002144, A006005, A008578, A020668.

Programs

  • Magma
    [1] cat [a: n in [0..200] | IsPrime(a) where a is 4*n + 1 ];
  • PARI
    is(n)=if(isprime(n), n%4==1, n==1) \\ Charles R Greathouse IV, Oct 10 2018

Formula

A002144 UNION {1}. - R. J. Mathar, Jan 04 2017

A352609 Inverse Euler transform of odd primes.

Original entry on oeis.org

3, -1, 0, 2, -4, 6, -8, 7, 0, -14, 40, -73, 88, -70, -4, 194, -524, 926, -1234, 1078, 140, -3038, 7900, -14065, 18858, -16368, -2710, 48861, -127826, 228788, -306494, 263377, 55144, -829014, 2160120, -3865459, 5172792, -4405043, -1118324, 14530041, -37606792, 67159094
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 22 2022

Keywords

Crossrefs

Cf. A006005, A030010, A030011, A065091, A353065.

Formula

Product_{n>=1} 1 / (1 - x^n)^a(n) = 1 + Sum_{n>=1} prime(n+1) * x^n.

A353950 Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + Sum_{n>=1} prime(n+1)*x^n.

Original entry on oeis.org

3, -4, -8, -26, -52, -126, -320, -1214, -2016, -7068, -16064, -48142, -122552, -390574, -903176, -3549556, -7597004, -22902332, -61172890, -198872948, -486889660, -1555059566, -4093173788, -12448334478, -33815484714, -105268420776, -279683446078, -894795490384, -2366564864546
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2022

Keywords

Crossrefs

Cf. A006005, A065091, A147557, A305882, A353605, A353606, A353951.

Programs

  • Mathematica
    A[m_, n_] := A[m, n] = Which[m == 1, Prime[n + 1], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m - 1, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 29]

A384231 Index of the largest odd noncomposite divisor in the list of divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 3, 1, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 2, 4, 2, 4, 2, 1, 3, 3, 3, 3, 2, 3, 3, 4, 2, 5, 2, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 3, 2, 5, 2, 3, 3, 1, 3, 5, 2, 4, 3, 4, 2, 3, 2, 3, 3, 4, 3, 5, 2, 4, 2, 3, 2, 6, 3, 3, 3, 5, 2, 4, 3, 4, 3, 3, 3, 3, 2, 3, 4, 4
Offset: 1

Views

Author

Omar E. Pol, May 29 2025

Keywords

Comments

a(n) = 1 if and only if n is a power of 2.

Examples

			For n = 30 the divisors of 30 are [1, 2, 3, 5, 6, 10, 15, 30] and the largest odd noncomposite divisor is 5 and 5 is its 4th divisor, so a(30) = 4.

Crossrefs

Companion of A383401.
Right border of A384234.
Cf. A006005 (odd noncomposite numbers).
Cf. A000079, A027750, A384232, A384233.

Programs

  • Mathematica
    a[n_] := Module[{m = n/2^IntegerExponent[n, 2]}, If[m == 1, 1, Position[Divisors[n], FactorInteger[m][[-1, 1]]][[1, 1]]]]; Array[a, 100] (* Amiram Eldar, May 29 2025 *)

A384234 Irregular triangle read by rows: T(n,k) is the index of the k-th odd noncomposite divisor in the list of divisors of n, with n >=1, k >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 4, 1, 2, 1, 1, 2, 3, 1, 3, 1, 2, 3, 1, 3, 1, 2, 1, 3, 1, 2, 3, 1, 4, 1, 2, 1, 3, 5, 1, 2, 1, 4, 1, 2, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3
Offset: 1

Views

Author

Omar E. Pol, May 29 2025

Keywords

Comments

Row n lists the indices of the odd noncomposite divisors in the list of divisors of n.
Row n is [1] if and only if n is a power of 2 (A000079).

Examples

			Triangle begins (n = 1..21):
  1;
  1;
  1, 2;
  1;
  1, 2;
  1, 3;
  1, 2;
  1;
  1, 2;
  1, 3;
  1, 2;
  1, 3;
  1, 2;
  1, 3;
  1, 2, 3;
  1;
  1, 2;
  1, 3;
  1, 2;
  1, 4;
  1, 2, 3;
  ...
For n = 30 the divisors of 30 are [1, 2, 3, 5, 6, 10, 15, 30] and the odd noncomposite divisors are [1, 3, 5] and the indices of them in the list of divisors are [1, 3, 4] respectively, so the 30th row of the triangle is [1, 3, 4].

Crossrefs

Companion of A383962.
Column 1 gives A000012.
Right border gives A384231.
Cf. A006005 (odd noncomposite numbers).
Cf. A000079, A027750, A384232, A384233.

Programs

  • Mathematica
    row[n_] := Module[{m = n/2^IntegerExponent[n, 2]}, Join[{1}, If[m == 1, {}, Position[Divisors[n], #] & /@ FactorInteger[m][[;; , 1]] // Flatten]]]; Array[row, 50] // Flatten (* Amiram Eldar, May 29 2025 *)

A130892 a(n) = n if n is not an odd prime number. Otherwise, a(n) = k*ceiling(n/10), where k is the smallest integer such that n < 10^k.

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 6, 1, 8, 9, 10, 4, 12, 4, 14, 15, 16, 4, 18, 4, 20, 21, 22, 6, 24, 25, 26, 27, 28, 6, 30, 8, 32, 33, 34, 35, 36, 8, 38, 39, 40, 10, 42, 10, 44, 45, 46, 10, 48, 49, 50, 51, 52, 12, 54, 55, 56, 57, 58, 12, 60, 14, 62, 63, 64, 65, 66, 14, 68
Offset: 1

Views

Author

Mohammad K. Azarian, Aug 21 2007

Keywords

Crossrefs

Cf. A131516, A130758, A000027, A001477, A006005.

Programs

  • Mathematica
    Join[{0,1,2},Table[If[PrimeQ[n],IntegerLength[n]Ceiling[n/10],n],{n,3,99}]] (* Harvey P. Dale, Sep 14 2013 *)

Extensions

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A194188 Triangle read by rows in which row n lists the numbers k >= 0 such that n-k and n+k are both odd noncomposite numbers.

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 0, 2, 1, 5, 0, 4, 6, 3, 5, 2, 4, 8, 3, 7, 9, 0, 6, 8, 1, 5, 7, 11, 0, 6, 10, 3, 9, 2, 4, 8, 14, 3, 13, 15, 0, 6, 12, 14, 1, 5, 11, 13, 0, 12, 18, 3, 9, 17, 2, 8, 10, 16, 20, 9, 15, 19, 21, 0, 6, 18, 20, 5, 7, 13, 17, 19, 23, 6, 12, 18, 22, 3, 15, 21, 4, 10, 14, 16, 20, 26, 9, 15, 25, 0, 12, 18, 24, 1, 7, 11, 13, 17, 23, 29
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Aug 18 2011

Keywords

Examples

			Triangle begins;
0;
1;
0,2;
1,3;
0,2;
1,5;
0,4,6;
3,5;
2,4,8;
3,7,9;
0,6,8;
1,5,7,11;

Crossrefs

Cf. A006005(the odd noncomposite numbers), A027750

Extensions

Corrected by R. J. Mathar Aug 27 2011

A309676 Number of compositions (ordered partitions) of n into odd primes (including 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 86, 138, 222, 357, 573, 921, 1481, 2381, 3828, 6153, 9890, 15898, 25556, 41082, 66039, 106156, 170644, 274307, 440945, 708815, 1139412, 1831589, 2944253, 4732847, 7607989, 12229743, 19659153, 31601828, 50799517, 81659549
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 12 2019

Keywords

Links

Crossrefs

Cf. A002124, A006005, A008578, A023360, A080545, A280917, A298603, A298604.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, a(n-1)+
      add(`if`(isprime(j), a(n-j), 0), j=3..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 12 2019
  • Mathematica
    nmax = 42; CoefficientList[Series[1/(1 - x - Sum[x^Prime[k], {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[PrimeOmega[k] < 2 && OddQ[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

Formula

G.f.: 1/(1 - x - Sum_{k>=2} x^prime(k)).

A353155 Logarithmic transform of odd primes.

Original entry on oeis.org

3, -4, 16, -94, 740, -7322, 87096, -1209242, 19190176, -342623408, 6797028096, -148325493672, 3531032617412, -91064679012376, 2529198638215228, -75262590212948118, 2388933783463085676, -80567150574145456164, 2876970976034496438802, -108441134639989639371264
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 27 2022

Keywords

Crossrefs

Cf. A006005, A007447, A065091, A343622, A353079.

Programs

  • Maple
    a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*
      binomial(n, j)*t(n-j)*a(j), j=1..n-1)/n))(i->ithprime(i+1))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Apr 27 2022
  • Mathematica
    nmax = 20; CoefficientList[Series[Log[1 + Sum[Prime[k + 1] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]! // Rest
a[n_] := a[n] = Prime[n + 1] - (1/n) Sum[Binomial[n, k] Prime[n - k + 1] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 1, 20}]

Formula

E.g.f.: log( 1 + Sum_{k>=1} prime(k+1) * x^k / k! ).
a(n) = prime(n+1) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * prime(n-k+1) * k * a(k).
Previous Showing 51-60 of 62 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.