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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A225389 Prime powers (A025475) representable as (p+q)/2, where p and q are distinct prime powers, in two or more ways.

Original entry on oeis.org

781370209, 59015782001929, 109012413691801, 185341023228001, 275502533796361, 315952509152809, 613721000732449, 1579642847367841, 3182047597748881, 5927491050020401, 13602074755852489, 22706626517726761
Offset: 1

Views

Author

Alex Ratushnyak, May 05 2013

Keywords

Comments

Prime powers p such that there are x and y such that p+x, p-x, p+y, p-y are four distinct prime powers.
First 12 terms are squares of prime numbers.
Conjecture: the sequence is infinite.

Crossrefs

Cf. A025475, A225388.

A225674 Triangular numbers of the form p*w, where p is a prime number and w is a prime power (A025475).

Original entry on oeis.org

3, 28, 45, 136, 153, 171, 325, 351, 496, 1431, 3321, 4753, 7381, 8128, 13203, 29161, 31375, 32896, 56953, 65341, 118341, 166753, 195625, 354061, 780625, 1063611, 2390391, 2883601, 3544453, 5649841, 6060421, 6835753, 6924781, 9563751, 11527201, 12708361, 19478161, 24231241
Offset: 1

Views

Author

Alex Ratushnyak, May 12 2013

Keywords

Crossrefs

Cf. A000040, A000217, A025475.

A225791 Numbers n such that the sum of first n prime powers (A025475) is divisible by n.

Original entry on oeis.org

1, 9, 54, 85, 853, 1162, 4209, 11566, 20060, 68048, 76221, 441294, 3007789, 4521955, 39443840, 39755851
Offset: 1

Views

Author

Alex Ratushnyak, May 16 2013

Keywords

Comments

a(17) > 146306913. There are 146306913 prime powers <= 2^63. - Donovan Johnson, May 16 2013

Examples

			The sum of first 9 prime powers is 1 + 4 + 8 + 9 + 16 + 25 + 27 + 32 + 49 = 171. Because 171 is divisible by 9, the latter is in the sequence.

Crossrefs

Cf. A025475, A225792.

Extensions

a(13)-a(16) from Donovan Johnson, May 16 2013

A226102 Prime powers (A025475) representable as triangular(k)+1.

Original entry on oeis.org

1, 4, 16, 121, 529, 4096, 139129, 160554241, 812293020529, 379188080252621270252095321
Offset: 1

Views

Author

Alex Ratushnyak, May 26 2013

Keywords

Comments

Indices of triangular numbers are in A226103.
53072032161200090602953513048447623^2 is also a term. - Giovanni Resta, May 26 2013

Crossrefs

Cf. A025475, A000217, A226103.

Programs

  • C
    #include 
#include 
#include 
#define TOP (1ULL<<32)  // Memory usage: 0.5 Gb
int main() {
  unsigned long long i, j, p, t, r;
  unsigned char *c = (unsigned char *)malloc(TOP/8);
  memset(c, 0, TOP/8);
  for (printf("1, "), i=1; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1)&7))) == 0) {
      for (p = i + (i==1), j = p*p; ; j*=p) {
        t = j - 1;
        r = sqrt(t*2);
        if (r*(r+1)==t*2)  printf("%llu, ", j);
        double k = ((double)j) * ((double)p);
        if (k >= ((double)(1ULL<<62)*4.0)) break;
      }
      if (i>1) for (j=i*i>>1; j>3] |= 1 << (j&7);
    }
  // SORT the output
  return 0;
}

Formula

a(n) = A000217(A226103(n)) + 1.

Extensions

a(10) from Giovanni Resta, May 26 2013

A226103 Numbers k such that triangular(k)+1 is a prime power (A025475).

Original entry on oeis.org

0, 2, 5, 15, 32, 90, 527, 17919, 1274592, 27538630330959
Offset: 1

Views

Author

Alex Ratushnyak, May 26 2013

Keywords

Comments

Generated prime powers are in A226102.
75055187665070250755513356704300447 is also a term. - Giovanni Resta, May 26 2013

Crossrefs

Cf. A025475, A000217, A226102.

Programs

  • C
    #include 
#include 
#include 
#define TOP (1ULL<<32)  // Memory usage: 0.5 Gb
int main() {
  unsigned long long i, j, p, t, r;
  unsigned char *c = (unsigned char *)malloc(TOP/8);
  memset(c, 0, TOP/8);
  for (printf("0, "), i=1; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1)&7))) == 0) {
      for (p = i + (i==1), j = p*p; ; j*=p) {
        t = j - 1;
        r = sqrt(t*2);
        if (r*(r+1)==t*2)  printf("%llu, ", r);
        double k = ((double)j) * ((double)p);
        if (k >= ((double)(1ULL<<62)*4.0)) break;
      }
      if (i>1) for (j=i*i>>1; j>3] |= 1 << (j&7);
    }
  // SORT the output
  return 0;
}

Formula

A000217(a(n)) + 1 = a(n) * (a(n)+1) / 2 + 1 = A226102(n).

Extensions

a(10) from Giovanni Resta, May 26 2013

A239521 Prime powers (A025475) such that the distance to the nearest prime power is an oblong number (A002378).

Original entry on oeis.org

25, 27, 2209, 3481, 3721, 6859, 6889, 18769, 22201, 22801, 157609, 1079521, 1630729, 1635841, 2621161, 2627641, 2825761, 3179089, 5257849, 5276209, 7447441, 7458361, 17032129, 17048641, 34304449, 34351321, 40436881, 40462321, 75290329, 75359761, 137288089, 137334961
Offset: 1

Views

Author

Alex Ratushnyak, Mar 20 2014

Keywords

Crossrefs

Cf. A002378, A025475, A239520 - A239523.

Programs

  • PARI
    lista(nn) = {precp = 1; currp = 4; for (nextp=5, nn, if (is_A025475(nextp), if (currp - precp < nextp - currp, tt = currp - precp, tt = nextp - currp); if (!(tt % 2) && ispolygonal(tt/2, 3), print1(currp, ", ")); precp = currp; currp = nextp;););} \\ Michel Marcus, Apr 11 2014

A239524 Prime powers p (A025475) such that either d1 divides d2, or d2 divides d1, where d1 and d2 are the distances from p to the two nearest prime powers.

Original entry on oeis.org

8, 9, 343, 361
Offset: 1

Views

Author

Alex Ratushnyak, Mar 20 2014

Keywords

Comments

Note the two nearest prime powers can be both above or both below p.
a(5) > 1.6*10^19. - Giovanni Resta, Mar 21 2014

Crossrefs

Cf. A025475.

A239582 Prime powers (A025475) such that the distance to the nearest square is a square.

Original entry on oeis.org

1, 8, 25, 32, 125, 169, 625, 1681, 3721, 12769, 32761, 97969, 177241, 375769, 579121, 707281, 1026169, 1442401, 1692601, 3031081, 3463321, 4464769, 5669161, 6355441, 7189057, 9740641, 13053769, 20367169, 26020201, 53597041, 73633561, 93334921, 98823481, 110523169
Offset: 1

Views

Author

Alex Ratushnyak, Mar 21 2014

Keywords

Crossrefs

Cf. A025475, A239583.

A239583 Prime powers (A025475) such that the distance to the nearest square is a prime power.

Original entry on oeis.org

1, 8, 25, 32, 125, 169, 625, 1681, 3721, 32761, 97969, 177241, 707281, 1442401, 3031081, 3463321, 6355441, 9740641, 26020201, 53597041, 73633561, 86938307, 93334921, 203946961, 268337161, 392079601, 815730721, 1348431841, 1743981121, 3708931801, 5158256041, 6995816881
Offset: 1

Views

Author

Alex Ratushnyak, Mar 21 2014

Keywords

Comments

The first term that does not appear in A239582 is a(22) = 86938307 = 443^3.

Crossrefs

Cf. A025475, A239582.

A239584 Prime powers p^q (A025475) such that either x divides y, or y divides x, where x and y are the distances from p^q to the nearest prime powers above and below p^q.

Original entry on oeis.org

8, 9, 121, 343, 2209
Offset: 1

Views

Author

Alex Ratushnyak, Mar 21 2014

Keywords

Examples

			The nearest prime powers above and below 121 are 81 and 125, because 125-121=4 divides 121-81=40, 121 is the sequence.

Crossrefs

Cf. A025475, A239524, A239879.
Previous Showing 31-40 of 214 results. Next

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