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.

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A325129 Heinz numbers of integer partitions into nonsquares (A087153).

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 15, 17, 19, 25, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 51, 55, 57, 59, 61, 65, 67, 71, 73, 75, 79, 81, 83, 85, 87, 89, 93, 95, 99, 101, 103, 107, 109, 111, 113, 117, 121, 123, 125, 127, 129, 131, 135, 137, 139, 141, 143, 145, 149, 153, 155
Offset: 1

Views

Author

Gus Wiseman, Apr 01 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   5: {3}
   9: {2,2}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}
  45: {2,2,3}

Crossrefs

Cf. A001156, A011757, A033461, A056239, A062457, A066328, A087153, A112798.
Cf. A324571, A324587, A324588, A325128, A325130.

Programs

  • Mathematica
    Select[Range[100],!MemberQ[If[#==1,{},FactorInteger[#]],{p_,_}/;IntegerQ[Sqrt[PrimePi[p]]]]&]

A378792 Numbers k such that tau(k) == 1 (mod(2*(tau(prime(k) - k + 1)))), where tau(k) = A000005(k).

Original entry on oeis.org

1, 196, 225, 441, 484, 625, 1089, 1156, 1225, 1296, 1444, 3025, 3249, 3844, 4225, 5929, 6561, 7225, 7396, 7569, 8281, 11236, 12321, 13225, 13924, 15129, 16641, 17689, 20164, 21025, 24025, 25281, 25600, 34225, 34969, 40401, 42025, 47089, 50625, 51076, 55225
Offset: 1

Views

Author

Claude H. R. Dequatre, Dec 07 2024

Keywords

Comments

118 terms < 5*10^5 were found.
All terms are squares because their number of divisors is odd (see formula field in A000005: a(n) is odd iff n is square).

Examples

			1 is a term because tau(1) = 1, tau(2 -1  + 1) = 2 and 1 modulo 4 is 1.
196 is a term because tau(196) = 9, tau(1193 - 196 + 1) = 4 and 9 modulo 8 is 1.
200 is not a term because tau(200) = 12, tau(1223 -200 + 1) = 11 and 12 modulo 22 = 12.

Crossrefs

Cf. A000005, A000040, A011757.
Cf. A378789, A378794.

Programs

  • PARI
    isok(k)=my(d_1=numdiv(k),d_2=numdiv(prime(k)-k+1));d_1%(2*d_2)==1;
for(k=1,1000,if(isok(k),print1(k", ")))

A022463 a(n) = prime(n^2) mod prime(n).

Original entry on oeis.org

0, 1, 3, 4, 9, 8, 6, 7, 5, 19, 10, 13, 25, 32, 17, 29, 50, 8, 25, 43, 17, 64, 68, 28, 78, 1, 60, 15, 50, 104, 80, 39, 106, 28, 97, 57, 17, 150, 51, 5, 7, 78, 24, 149, 76, 153, 59, 100, 53, 94, 27, 220, 164, 218, 240, 126, 188, 58, 20, 174, 40, 178, 47, 309, 167, 114, 244, 13
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Magma
    [NthPrime(n^2) mod NthPrime(n): n in [1..90]]; // Vincenzo Librandi, Dec 09 2014
  • Maple
    seq(ithprime(i^2) mod ithprime(i), i=1..1000); # Robert Israel, May 26 2014
  • Mathematica
    Table[Mod[Prime[n^2], Prime[n]], {n, 100}] (* Vincenzo Librandi, Dec 09 2014 *)
  • PARI
    a(n) = prime(n^2) % prime(n); \\ Michel Marcus, Sep 30 2013

Formula

a(n) = A011757(n) modulo A000040(n). - Michel Marcus, Sep 30 2013

A074328 Numbers m such that prime(m^2+1)-prime(m^2)=2, where prime(j) is the j-th prime.

Original entry on oeis.org

7, 8, 9, 12, 15, 16, 22, 25, 27, 34, 53, 83, 85, 88, 95, 107, 108, 144, 149, 187, 196, 223, 234, 238, 249, 255, 268, 274, 315, 324, 350, 355, 358, 367, 386, 410, 411, 416, 424, 436, 440, 445, 450, 462, 469, 471, 481, 494, 501, 509, 511, 517, 522, 549, 554, 564
Offset: 1

Views

Author

Labos Elemer, Aug 21 2002

Keywords

Comments

Square roots of squares in A029707. - Michel Marcus, Oct 20 2022

Examples

			25 is a term because the 626th and 625th primes are twin primes: 4639 - 4637 = 2.

Links

Crossrefs

Cf. A029707, A001359, A006512, A011757.

Programs

  • Mathematica
    t=Table[0, {250}]; t1=Table[0, {250}]; s=0; k=0; Do[s=Prime[1+n^2]-Prime[n^2]; If[s==2, k=k+1; t[[k]]=n; t1[[k]]=Prime[n^2]; Print[{k, n, Prime[n^2]}]], {n, 1, 2500}] t t1
  • PARI
    isok(m) = my(p=prime(m^2)); nextprime(p+1) - p == 2; \\ Michel Marcus, Oct 20 2022
  • PARI
    list(lim) = {my(k = 1, prv = 2); forprime(p = 3, lim, if(p - prv == 2 && issquare(k), print1(sqrtint(k), ", ")); k++; prv = p);} \\ Amiram Eldar, Mar 20 2025

A109796 a(n) = prime(1^4) + prime(2^4) + ... + prime(n^4).

Original entry on oeis.org

2, 55, 474, 2093, 6730, 17357, 38748, 77621, 143308, 248037, 407558, 641437, 973380, 1432721, 2052922, 2874563, 3944166, 5314265, 7045924, 9206477, 11874460, 15134597, 19083406, 23826383, 29480190, 36172177, 44039724
Offset: 1

Views

Author

Jonathan Vos Post, Aug 15 2005

Keywords

Comments

Analog of prime(1^2) + prime(2^2) + ... + prime(n^2) (A109724). For a(n) to be prime for n > 1 it is necessary but not sufficient that n == 0 (mod 4).

Examples

			a(1) = 2 because prime(1^4) = prime(1) = 2.
a(2) = 55 because prime(1^4) + prime(2^4) = prime(1) + prime(16) = 2 + 53.
a(3) = 474 because prime(1^4) + prime(2^4) + prime(3^4) = prime(1) + prime(16) + prime(81) = 2 + 53 + 419.
a(4) = 2093 because prime(1^4) + prime(2^4) + prime(3^4) + prime(4^4) = 2 + 53 + 419 + prime(256) = 2 + 53 + 419 + 1619.
a(8) = 2 + 53 + 419 + 1619 + 4637 + 10627 + 21391 + 38873 = 77621 (which is prime).
a(12) = 2 + 53 + 419 + 1619 + 4637 + 10627 + 21391 + 38873 + 65687 + 104729 + 159521 + 233879 = 641437 (which is prime).
a(28) = 2 + 53 + 419 + 1619 + 4637 + 10627 + 21391 + 38873 + 65687 + 104729 + 159521 + 233879 + 331943 + 459341 + 620201 + 821641 + 1069603 + 1370099 + 1731659 + 2160553 + 2667983 + 3260137 + 3948809 + 4742977 + 5653807 + 6691987 + 7867547 + 9195889 = 53235613 (which is prime).
It is a coincidence that a(1), a(2) and a(3) are all palindromes.

Links

Crossrefs

First differences are A109791.
Cf. A000040, A000290, A000583, A011757, A109724, A109770.

Programs

Formula

a(n) = Sum_{i=1..n} A000040(A000583(i)).

A145290 Numbers n such that prime[(n + 1)^2] - prime[n^2] is a perfect square.

Original entry on oeis.org

2, 30, 32, 180, 269, 820, 1282, 1403, 2102, 2152, 2344, 2616, 4459, 6329, 6604, 7325, 7893, 9988, 10270, 10441, 10662, 11700, 11740, 11846, 12279, 18754, 18927, 23781, 27096, 27294, 27531, 31005, 33616, 36374, 38266, 43557, 43680, 44263, 45945
Offset: 1

Views

Author

Zak Seidov, Oct 06 2008

Keywords

Comments

n such that A011757(n + 1) - A011757(n) is a perfect square

Examples

			n = 2, prime[(n + 1)^2] - prime[n^2]=23 - 7 = 16 = 4^2; n = 30, prime[(n + 1)^2] - prime[n^2]=7573 - 6997 = 576 = 24^2

Crossrefs

A011757 prime(n^2)

Programs

  • Mathematica
    Do[If[IntegerQ[Sqrt[Prime[(i + 1)^2] - Prime[i^2]]], Print[i]], {i, 1, 20000}]

Extensions

a(28)-a(39) from Donovan Johnson, Sep 16 2009

A217623 a(n) = prime(prime(n^2)).

Original entry on oeis.org

3, 17, 83, 241, 509, 877, 1433, 2063, 2897, 3911, 4943, 6353, 8011, 9661, 11909, 13693, 16141, 18787, 21727, 24781, 28307, 32261, 35801, 40093, 44621, 49139, 54251, 59417, 64853, 70621, 77047, 83617, 90203, 97039, 103991, 112097, 120223, 128683, 136813, 145903
Offset: 1

Views

Author

Vincenzo Librandi, Oct 13 2012

Keywords

Links

Crossrefs

Cf. A000040, A011757.
Subsequence of A006450.

Programs

  • Magma
    [NthPrime(NthPrime(n^2)): n in [1..50]];
  • Maple
    a:= n-> (ithprime@@2)(n^2):
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 17 2021
  • Mathematica
    Table[Prime[Prime[n^2]], {n, 100}]
  • PARI
    a(n) = prime(prime(n^2)); \\ Michel Marcus, Mar 17 2021
  • Python
    from sympy import prime
def a(n): return prime(prime(n**2))
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Mar 17 2021

A248354 Least positive integer m such that m + n divides prime(m^2) + prime(n^2).

Original entry on oeis.org

1, 1, 2, 1, 3, 8, 2, 6, 6, 45, 9, 4, 15, 2, 13, 17, 4, 12, 9, 8, 11, 6, 101, 20, 2, 15, 7, 50, 4, 183, 48, 15, 9, 5, 4, 4, 157, 1, 123, 4, 13, 112, 76, 4, 7, 13, 44, 2, 16, 28, 83, 202, 114, 50, 85, 31, 14, 62, 19, 25
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 05 2014

Keywords

Comments

Conjecture: a(n) exists for any n > 0. Moreover, a(n) <= n*(n-1)/2 for all n > 1.
See also the comments in A248052.

Examples

			a(3) = 2 since 2 + 3 = 5 divides prime(2^2) + prime(3^2) = 7 + 23 = 30.

Links

Crossrefs

Cf. A000040, A000290, A011757, A247824, A247975, A248052.

Programs

  • Mathematica
    Do[m = 1; Label[aa]; If[Mod[Prime[m^2] + Prime[n^2], m + n] == 0, Print[n, " ", m]; Goto[bb]]; m = m + 1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]
  • PARI
    a(n)=my(N=prime(n^2),m); while((prime(m++^2)+N)%(m+n), ); m \\ Charles R Greathouse IV, Oct 05 2014

A261619 a(n) = floor(prime(n^2) / prime(n)).

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 16, 18, 18, 21, 22, 24, 27, 30, 30, 31, 35, 36, 38, 42, 43, 45, 47, 47, 50, 53, 56, 59, 61, 59, 62, 63, 67, 66, 70, 72, 73, 76, 78, 80, 83, 83, 86, 89, 92, 92, 91, 94, 97, 100, 101, 105, 105, 107, 109, 111, 115, 117, 119
Offset: 1

Views

Author

Altug Alkan, Sep 09 2015

Keywords

Comments

Inspired by A213926.
The reason of "/" operation between prime(n^2) and prime(n) is n^2 / n = n.
Sequence is not monotone: 61 = a(30) > a(31) = 59. In the first thousand terms there are 83 less than the preceding term; in the first ten thousand, 865. - Charles R Greathouse IV, Sep 12 2015

Examples

			For n=2, a(n) = floor(prime(n^2) / prime(n)) =  floor(7/3) = 2.

Links

Crossrefs

Cf. A000040, A011757, A213926.

Programs

  • Magma
    [NthPrime(n^2) div NthPrime(n): n in [1..70]]; // Vincenzo Librandi, May 24 2019
  • Mathematica
    Table[Floor[Prime[n^2] / Prime[n]], {n, 1, 100}] (* Vincenzo Librandi, May 24 2019 *)
  • PARI
    a(n) = floor(prime(n^2) / prime(n));
vector(70, n, a(n))
  • PARI
    first(n)=my(v=List(),p,k); forprime(q=2,, if(issquare(k++), p=nextprime(p+1); listput(v, q\p); if(#v==n, return(Vec(v))))) \\ Charles R Greathouse IV, Sep 12 2015
  • Sage
    [floor(nth_prime(n^2)/nth_prime(n)) for n in (1..70)] # G. C. Greubel, May 24 2019

Formula

a(n) = floor(A011757(n) / A000040(n)).
a(n) ~ n/(2 log^2 n). - Charles R Greathouse IV, Sep 12 2015

A343512 Numbers k such that Sum_{i=1..k} prime(i^3) is prime.

Original entry on oeis.org

1, 6, 28, 72, 90, 92, 96, 112, 118, 148, 160, 162, 184, 222, 282, 312, 314, 316, 330, 336, 390, 396, 418, 440, 444, 448, 472, 488, 524, 534, 552, 598, 604, 614, 638, 748, 758, 798, 824, 848, 906, 916, 970, 992, 1008, 1010, 1012, 1016, 1056, 1078, 1084, 1094, 1098
Offset: 1

Views

Author

Chai Wah Wu, Apr 17 2021

Keywords

Comments

Numbers n such that A109789(n) is prime. For n > 1, a(n) is even.

Examples

			72 is a term since Sum_{i=1..72} prime(i^3) = 94154923 is prime.

Links

Crossrefs

Cf. A055875, A011757, A109724, A109770, A109789.
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