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 1-5 of 5 results.

A176465 Palindromic primes p(k) = palprime(k) such that their sum of digits ("sod") equals sum of digits of their palprime index k.

Original entry on oeis.org

13331, 1022201, 1311131, 3001003, 3002003, 100707001, 102272201, 103212301, 103323301, 103333301, 104111401, 105202501, 105313501, 105323501, 106060601, 111181111, 111191111, 112494211, 121080121, 140505041, 160020061, 160161061
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 18 2010

Keywords

Comments

p(k) = palprime(k) (see A002385) with sod(p(k)) = sod(k)
List of (p(k),k):
(13331,29) (1022201,116) (1311131,173) (3001003,304) (3002003,305)
(100707001,790) (102272201,818) (103212301,832) (103323301,835) (103333301,836)
(104111401,850) (105202501,862) (105313501,865) (105323501,866) (106060601,875)
(111181111,961) (111191111,962) (112494211,979) (121080121,1096) (140505041,1379)
(160020061,1672) (160161061,1678) (160171061,1679) (181111181,1958) (300151003,2209)
(310131013,2344) (313222313,2387) (320444023,2488) (321242123,2495) (341040143,2765)
(341222143,2767) (342020243,2774) (342202243,2776) (342212243,2777) (342313243,2779)
(343050343,2788) (700090007,3488) (730111037,3884) (910212019,4858)

Examples

			p(1) = 13331 = palprime(29), sod(p(1)) = 1+3+3+3+1 = 11 = sod(29), first term
p(8) = 103212301 = palprime(832), sod(p(8)) = 1+0+3+2+1+2+3+1 = 13 = 8+3+2 = sod(832), 8th term
p(?) = 156300010003651 = palprime(99643), sod(p(?)) = 31 = sod(99733)
Note successive p(i) and p(i+1) which are also consecutive palindromic primes (i = 4, 9, 13, 16, 22, 33)

References

  • A. H. Beiler: Recreations in the Theory of Numbers: The Queen of Mathematical Entertains. Dover Publications, New York, 1964
  • M. Gardner: Mathematischer Zirkus , Ullstein Berlin-Frankfurt/Main-Wien, 1988
  • K. G. Kroeber: Ein Esel lese nie. Mathematik der Palindrome, Rowohlt Tb., Hamburg, 2003

Crossrefs

A000213, A002385, A033548, A155214, A176111

A176790 Honaker primes of the form k^2 + 1.

Original entry on oeis.org

3137, 4357, 13457, 80657, 115601, 184901, 309137, 341057, 1008017, 1073297, 4227137, 5541317, 11806097, 16974401, 18576101, 22848401, 24443137, 24542117, 27625537, 28132417, 30913601, 39112517, 42432197, 46049797, 46321637, 52417601, 71132357, 84713617, 92736901
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 26 2010

Keywords

Comments

The intersection of A033548 with A002522 or with A002496.
The list of associated n is: 56, 66, 116, 284, 340, 430, 556, 584, 1004, 1036, 2056, ...
The associated indices in A002496 are: 14, 15, 21, 48, 53, 61, 73, 76, 113, 115, 215, 243, 341, 395, 414, ...

Examples

			a(1) = 3137 = 56^2 + 1 = A033548(24).
a(2) = 4357 = 66^2 + 1 = A033548(31).

References

  • M. Aigner, Diskrete Mathematik, Vieweg u. Teubner, 6. Aufl., 2006.
  • E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, Berlin, 1985.
  • H. Scheid, Zahlentheorie, Spektrum Akademischer Verlag, 4. Aufl., 2006.

Links

Crossrefs

Cf. A000040, A002496, A033548, A176012, A176111, A176465.

Programs

  • Mathematica
    fHQ[n_]:=Plus@@IntegerDigits@n==Plus@@IntegerDigits@PrimePi@n;Select[Range[10000]^2+1, PrimeQ[#] && fHQ[#] &]  (* K. D. Bajpai, Apr 06 2021 *)
  • PARI
    for(n =1, 50000, my(k=n^2+1); if(isprime(k) && vecsum(digits(k))==vecsum(digits(primepi(k))), print1(k, ", "))); \\ K. D. Bajpai, Apr 06 2021

Extensions

Comments tightened by R. J. Mathar, Jun 07 2010
a(21)-a(29) from K. D. Bajpai, Apr 06 2021

A178237 Smallest prime p of the form prime(n)+k^2 such that sum of digits(p) = prime(n).

Original entry on oeis.org

2, 3, 5, 7, 47, 157, 593, 919, 599, 66593, 46687, 396937, 467897, 467899, 6969647, 16499897, 367488959, 598095997, 2977884967, 4977866987, 2797986889, 58888728979, 58987779959, 679585896989, 4989996468997
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 23 2010

Keywords

Comments

It is still an open problem if there exist infinitely many primes of form k^2 + d (d integer, no negative square).
For n<=4, k=0 suffices: e.g. prime(1)+0^2=2 = sum of digits(prime(1)), so a(n)=prime(n).

Examples

			a(13) = 467897 because its digitsum is 41 which is the 13th prime, it is of the form prime(13)+k^2 with k=684, and it is the least such prime.

Crossrefs

Cf. A000040, A172035, A176111, A176790

Programs

  • PARI
    sod(n) = {digs = digits(n, 10); return (sum(j=1, #digs, digs[j]));}
a(n) = {k = 0; p = prime(n); while (! (isprime(q=p+k^2) && (sod(q) == p)), k++); return (q);} \\ Michel Marcus, Jul 26 2013

Extensions

a(5) corrected and sequence extended by D. S. McNeil, May 25 2010

A176760 Numbers k such that k^2 and k^4 have the same sum of digits.

Original entry on oeis.org

0, 1, 3, 10, 17, 19, 27, 30, 57, 93, 100, 170, 190, 219, 267, 270, 300, 314, 327, 359, 387, 417, 423, 424, 570, 685, 693, 807, 828, 891, 917, 930, 963, 1000, 1207, 1223, 1317, 1333, 1673, 1693, 1700, 1827, 1864, 1900, 1917, 2141, 2190, 2202, 2213, 2364, 2367
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 25 2010

Keywords

Comments

Let sod(n) := digital sum of n (A007953); here we have sod(n^2) = sod(n^4).
Trivial cases:
(I) Powers of 10, as sod((10^k)^2) = sod((10^k)^4) = 1.
(II) If N is a term of sequence, then so is 10 * N.

Examples

			sod(3^2) = sod(9) = 9 = sod(81) = sod(3^4), so 3 is a term.
sod(17^2) = sod(289) = 19 = sod(83521) = sod(17^4), so 17 is a term.

References

  • Hans Schubart, Einfuehrung in die klassische und moderne Zahlentheorie, Vieweg, Braunschweig, 1974.

Crossrefs

Cf. A000290, A000583, A058369, A176012, A176111, A176465.

Programs

  • Mathematica
    Select[Range[0,2000],Total[IntegerDigits[#^2]]==Total[IntegerDigits[#^4]]&]  (* Harvey P. Dale, Jan 19 2011 *)

Extensions

Edited by D. S. McNeil, Nov 21 2010
a(43)-a(51) from Jason Yuen, Oct 13 2024

A178371 The smallest prime p of the form j^3 + prime(n), such that the sum-of-digits of p equals prime(n).

Original entry on oeis.org

2, 3, 5, 7, 227, 229, 13841, 1747, 729023, 474581, 46687, 1259749, 37933097, 6434899, 14886983, 485587709, 2985984059, 2526569989, 56888939803, 60976889927, 60976889929, 879768685447, 8296386686867, 22597978779737
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 26 2010

Keywords

Examples

			n=1: 0^3 + prime(1) = 0+2 = 2.
n=2: 0^3 + prime(2) = 0+3 = 3.
n=3: 0^3 + prime(3) = 0+5 = 5. Next candidate with j>0 would be 6^3 + 7 = 223.
n=4: 0^3 + prime(4) = 0+7 = 7.
n=5: 6^3 + 11 = 227 = prime(49).
n=6: 6^3 + 13 = 229 = prime(50).
n=7: 24^3 + 17 = 13841 = prime(1636).
n=8: 12^3 + 19 = 1747 = prime(272).
n=9: 90^3 + 23 = 729023 = prime(58716).
n=10: 78^3 + 29 = 474581 = prime(39587).
n=11: 36^3 + 31 = 46687 = prime(4825).
n=12: 108^3 + 37 = 1259749 = prime(97168).
n=13: 336^3 + 41 = 37933097 = prime(2315164).
n=14: 186^3 + 43 = 6434899 = prime(440614).
n=15: 246^3 + 47 = 14886983 = prime(963902).
n=16: 786^3 + 53 = 485587709 = prime(25635800).
n=17: 1440^3 + 59 = 2985984059 = prime(143807568).
n=18: 1362^3 + 61 = 2526569989 = prime(122671100).

Crossrefs

Cf. A000040, A000578, A176111, A176790, A178237

Extensions

Redefined the variables in the definition - R. J. Mathar, Jun 07 2010
a(19)-a(24) from Donovan Johnson, Aug 09 2010
Showing 1-5 of 5 results.

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