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 15 results. Next

A316430 Heinz numbers of integer partitions whose length is equal to the GCD of all the parts.

Original entry on oeis.org

1, 2, 9, 21, 39, 57, 87, 91, 111, 125, 129, 159, 183, 203, 213, 237, 247, 267, 301, 303, 321, 325, 339, 377, 393, 417, 427, 453, 489, 519, 543, 551, 553, 559, 575, 579, 597, 669, 687, 689, 707, 717, 753, 789, 791, 813, 817, 843, 845, 879, 923, 925, 933, 951, 973
Offset: 1

Views

Author

Gus Wiseman, Jul 02 2018

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
2 is the only even term in the sequence. 3k is in the sequence if and only if k is in A031215. 5k is in the sequence if and only if k = pq with p and q in A031336.

Examples

			Sequence of integer partitions whose length is equal to their GCD begins: (), (1), (2,2), (4,2), (6,2), (8,2), (10,2), (6,4), (12,2), (3,3,3), (14,2), (16,2), (18,2), (10,4), (20,2), (22,2), (8,6), (24,2), (14,4), (26,2), (28,2), (6,3,3).

Crossrefs

Subsequence of A004280.
Cf. A056239, A067538, A074761, A143773, A289508, A289509, A296150, A316413, A316431, A316432, A316433, A031215, A031336.

Programs

  • Mathematica
    Select[Range[200],PrimeOmega[#]==GCD@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]]&]
  • PARI
    is(n,f=factor(n))=gcd(apply(primepi,f[,1]))==vecsum(f[,2]) \\ Charles R Greathouse IV, Jul 25 2024

Formula

a(n) << n log^2 n, can this be improved? - Charles R Greathouse IV, Jul 25 2024

A246929 a(n) = prime(11*n).

Original entry on oeis.org

31, 79, 137, 193, 257, 317, 389, 457, 523, 601, 661, 743, 823, 887, 977, 1049, 1117, 1213, 1289, 1373, 1453, 1531, 1607, 1693, 1777, 1871, 1951, 2029, 2113, 2213, 2293, 2377, 2447, 2551, 2659, 2713, 2797, 2887, 2971, 3079, 3187, 3271, 3359, 3461, 3539
Offset: 1

Views

Author

Vincenzo Librandi, Sep 08 2014

Keywords

Links

Crossrefs

Cf. sequences of the type prime(k*n): A000040 (k=1), A031215 (k=2), A031336 - A031343 (k=3..10), this sequence (k=11), A246930 (k=12), A126588 (k=13), A246931 (k=14), A246932 (k=15), A246933 (k=16), A129480 (k=17), A031921 (k=100), A031922 (k=1000).

Programs

  • Magma
    [NthPrime(11*n): n in [1..50]];
  • Mathematica
    Prime[11 Range[50]]
  • PARI
    a(n)=prime(11*n) \\ Edward Jiang, Sep 08 2014
  • Sage
    [nth_prime(11*n) for n in (1..50)] # Bruno Berselli, Sep 08 2014

A031343 a(n) = prime(10*n).

Original entry on oeis.org

29, 71, 113, 173, 229, 281, 349, 409, 463, 541, 601, 659, 733, 809, 863, 941, 1013, 1069, 1151, 1223, 1291, 1373, 1451, 1511, 1583, 1657, 1733, 1811, 1889, 1987, 2053, 2129, 2213, 2287, 2357, 2423, 2531, 2617, 2687, 2741, 2819, 2903, 2999
Offset: 1

Views

Author

Jeff Burch

Keywords

Links

Crossrefs

Cf. similar sequences listed in A031336.

Programs

A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, May 06 2018

Keywords

Examples

			a(36) = 8 because 36 = 2^2*3^2 = prime(1)^2*prime(2)^2 and 1*2*2*2 = 8.

Links

Crossrefs

Cf. A000026, A000040, A000720, A001414, A002110, A003963, A005361, A056239, A156061, A304037.

Programs

  • Mathematica
    a[n_] := Times @@ (PrimePi[#[[1]]] #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 80}]
  • PARI
    a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = primepi(f[k,1])*f[k,2]; f[k, 2] = 1); factorback(f); \\ Michel Marcus, May 06 2018

Formula

a(n) = A005361(n)*A156061(n).
a(p^k) = A000720(p)*k where p is a prime.
a(A002110(m)^k) = k^m*m!.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A001248(k)) = a(A031215(k)) = A005843(k).
a(A030078(k)) = a(A031336(k)) = A008585(k)
a(A061742(k)) = A000165(k).
a(A115964(k)) = A032031(k).
a(A002110(k)) = A000142(k).
a(A080696(k)) = A002110(k).

A091734 Permutation of primes generated by array shown below.

Original entry on oeis.org

2, 3, 11, 17, 7, 5, 31, 19, 13, 41, 29, 23, 59, 43, 37, 67, 53, 47, 83, 71, 61, 97, 79, 73, 109, 101, 89, 127, 107, 103, 149, 131, 113, 157, 139, 137, 179, 163, 151, 191, 173, 167
Offset: 1

Views

Author

Giovanni Teofilatto, Mar 06 2004

Keywords

Comments

2 11 17 31 41 59 67 83 97... (A091735)
3 7 19 29 43 53 71 79 101... (A091738)
5 13 23 37 47 61 73 89 103...(A031336)

Extensions

A-number corrected by R. J. Mathar, Apr 22 2010

A126588 a(n) = prime(13*n).

Original entry on oeis.org

41, 101, 167, 239, 313, 397, 467, 569, 643, 733, 823, 911, 1009, 1091, 1187, 1283, 1381, 1481, 1567, 1657, 1753, 1871, 1979, 2069, 2153, 2273, 2371, 2459, 2591, 2687, 2767, 2861, 2971, 3089, 3217, 3323, 3433, 3533, 3623, 3727, 3847, 3931, 4051, 4157
Offset: 1

Views

Author

Cino Hilliard, Jan 05 2007

Keywords

Comments

Old name was "Every 13th prime number".

Examples

			The 13th prime is 41, the first entry in the table.

Links

Crossrefs

Cf. similar sequences listed in A031336.

Programs

A129480 a(n) = Prime(17*n).

Original entry on oeis.org

59, 139, 233, 337, 439, 557, 653, 769, 883, 1013, 1117, 1249, 1381, 1493, 1613, 1747, 1879, 2017, 2141, 2287, 2399, 2551, 2689, 2801, 2953, 3089, 3253, 3373, 3529, 3643, 3793, 3923, 4073, 4219, 4357, 4513, 4651, 4799, 4957, 5087, 5237, 5413, 5527, 5683
Offset: 1

Views

Author

Cino Hilliard, May 29 2007

Keywords

Examples

			The 17th prime is 59.

Links

Crossrefs

Cf. similar sequences listed in A031336.
Cf. A129484.

Programs

  • Magma
    [NthPrime(17*n): n in [1..100]]; // G. C. Greubel, Feb 12 2024
  • Maple
    seq(ithprime(17*i),i=1..100); # Robert Israel, Sep 08 2014
  • Mathematica
    Prime[17*Range[1, 100]] (* G. C. Greubel, Feb 12 2024 *)
  • PARI
    cicada(n) = forstep(x=17,n,17,print1(prime2(x)","))
  • PARI
    a(n)=prime(17*n) \\ Edward Jiang, Sep 08 2014
  • Sage
    [nth_prime(17*n) for n in (1..50)] # Bruno Berselli, May 07 2014

A022460 a(n) = prime(3*n) mod prime(n).

Original entry on oeis.org

1, 1, 3, 2, 3, 9, 5, 13, 11, 26, 13, 3, 3, 9, 9, 11, 56, 7, 1, 68, 15, 1, 15, 3, 88, 94, 7, 5, 13, 11, 106, 110, 112, 1, 124, 140, 136, 130, 142, 140, 140, 158, 154, 164, 170, 190, 178, 158, 172, 176, 184, 194, 214, 200, 206, 208, 212, 220, 220, 226, 244
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Magma
    [NthPrime(3*n) mod NthPrime(n): n in [1..90]]; // Vincenzo Librandi, Dec 09 2014
  • Mathematica
    Table[Mod[Prime[3 n], Prime[n]], {n, 100}] (* Vincenzo Librandi, Dec 09 2014 *)
  • PARI
    a(n) = prime(3*n) % prime(n); \\ Michel Marcus, Sep 30 2013

Formula

a(n) = A031336(n) modulo A000040(n). - Michel Marcus, Sep 30 2013

A031338 a(n) = prime(5*n).

Original entry on oeis.org

11, 29, 47, 71, 97, 113, 149, 173, 197, 229, 257, 281, 313, 349, 379, 409, 439, 463, 499, 541, 571, 601, 631, 659, 691, 733, 761, 809, 829, 863, 907, 941, 977, 1013, 1039, 1069, 1103, 1151, 1187, 1223, 1259, 1291, 1319, 1373, 1427, 1451, 1483, 1511, 1553, 1583
Offset: 1

Views

Author

Jeff Burch

Keywords

Links

Crossrefs

Cf. A000040, similar sequences listed in A031336.

Programs

  • Magma
    [NthPrime(5*n): n in [1..50]]; // Vincenzo Librandi, Apr 10 2011
  • Maple
    seq(ithprime(5*n), n=1..52);
  • PARI
    a(n)=prime(5*n) \\ Edward Jiang, Sep 08 2014
  • Sage
    [nth_prime(5*n) for n in (1..50)] # Bruno Berselli, May 07 2014

Formula

a(n) = A000040(5n).

A031339 a(n) = prime(6*n).

Original entry on oeis.org

13, 37, 61, 89, 113, 151, 181, 223, 251, 281, 317, 359, 397, 433, 463, 503, 557, 593, 619, 659, 701, 743, 787, 827, 863, 911, 953, 997, 1033, 1069, 1109, 1163, 1213, 1249, 1291, 1321, 1399, 1439, 1481, 1511, 1559, 1601, 1627, 1693, 1733
Offset: 1

Views

Author

Jeff Burch

Keywords

Links

Crossrefs

Cf. similar sequences listed in A031336.

Programs

Showing 1-10 of 15 results. Next

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

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