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 1-3 of 3 results.

A176111 Honaker primes of the form p = 2*k-1 with sum-of-digits(p) = sum-of-digits(k).

Original entry on oeis.org

4357, 6301, 6553, 7741, 8011, 12277, 13339, 14437, 14923, 16273, 18307, 24733, 26731, 27091, 34471, 34543, 35227, 36217, 36307, 36433, 36523, 37783, 41491, 41851, 41941, 42373, 43543, 45181, 47017, 49411, 52543, 53407, 54217, 55207, 57943, 58321, 58411, 64513
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 08 2010

Keywords

Comments

The set of Honaker primes A033548 is intersected with the set {37, 73, 109, 127, 163, 181, 271, 307, 397, 433, 523, 541, 577, 613,...} of primes p = 2k-1, where A007953(p) = A007953(k) for the digit sums.
The requirement on the digit sum defining the Honaker primes plus the additional requirement on the digits sum of k means both digit sums are of the form 9*m+1, m>=1.
The sequence contains prime(n) for n = 595, 820, 847, 982, 1009, 1099, 1468, 15856, 1693, 1747,...
The fourth to sixth member of the sequence are three consecutive Honaker primes.
As a curiosity we have that for p=120709 = prime(11359) = A033548(469), k=60355 even the index in the Honaker primes has the same sum, 19.

Examples

			p = 2719 = prime(397) has digit sum 19, but k = 1360 has digit sum 10, which yields no term.
p = 6301 = prime(820) with k = 3151, digit sum 10, is the 2nd term.
p = 10711 = prime(1306) with digit sum 10, but k = 5356 has digit sum 10: no contribution to the sequence.
p = 57943 = prime(5869) with k = 28972 have common digit sum 28 and p is in the sequence.

References

  • M. du Sautoy: Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Beck, 4. Auflage, 2005

Crossrefs

Cf. A000040, A033548, A176012

Extensions

4137 replaced by 4357, 8821 removed, Extensive list of auxiliary prime indices reduced - R. J. Mathar, Nov 01 2010

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

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
Showing 1-3 of 3 results.

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