A176465 -id:A176465 - cp's OEIS frontend

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

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

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

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, ...

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

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.

K. D. Bajpai, Table of n, a(n) for n = 1..1570

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

fHQ[n_]:=Plus@@IntegerDigits@n==Plus@@IntegerDigits@PrimePi@n;Select[Range[10000]^2+1, PrimeQ[#] && fHQ[#] &]  (* K. D. Bajpai, Apr 06 2021 *)

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

Comments tightened by R. J. Mathar, Jun 07 2010

a(21)-a(29) from K. D. Bajpai, Apr 06 2021

A177678 Palindromic primes p = q//r//q such that q and r are also palindromic primes.

Original entry on oeis.org

353, 373, 727, 757, 11311, 31013, 31513, 33533, 37273, 37573, 39293, 71317, 71917, 77977, 1175711, 1178711, 1317131, 1513151, 1917191, 3103013, 3106013, 3127213, 3135313, 3155513, 3160613, 3166613, 3181813, 3193913, 3198913, 3304033

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), May 10 2010

p = palprime(i), q = palprime(j), r = palprime(k) (see A002385).
Indices (i,j,k): (12,2,3), (13,2,4), (15,4,1), (16,4,3), (24,5,2), (52,2,6), (53,2,8), (56,2,12), (65,2,15), (66,2,16), (70,2,20), (74,4,7), (75,4,10), (88,4,18), (140,11,16), (142,11,17), (174,7,4), (206,8,2), (282,10,4), (318,2,21), (319,2,23), (320,2,27), (321,11,3), (323,2,35), (325,2,36), (326,2,39), (327,2,42), (329,2,44), (331,2,45), (354,2,49), (356,2,50), (357,2,52), (358,2,53), (365,2,61), (366,2,63), (368,2,64), (370,2,67), (372,2,70), (424,2,76), (426,2,79), (430,2,84), (434,2,86), (435,2,87), (437,2,89), (439,2,92), (440,2,94), (464,2,97), (467,2,102), (468,2,103), (469,2,107).
Note an example that the description with q and r is not (necessarily) unique: (2388,2,179) or (2388,11,14) for 313383313 = 3//1338331//3 = 313//383//313.

			3 = palprime(2), 5 = palprime(3), 3//5//3 = 353 = palprime(12) is first term.
3 = palprime(2), 7 = palprime(4), 3//7//3 = 373 = palprime(13) is 2nd term.

M. Gardner: Mathematischer Zirkus, Ullstein Berlin-Frankfurt/Main-Wien, 1988
K. G. Kroeber: Ein Esel lese nie. Mathematik der Palindrome, Rowohlt Tb., Hamburg, 2003

Cf. A002385, A176465.

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

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

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.

			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.

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

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

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

Edited by D. S. McNeil, Nov 21 2010

a(43)-a(51) from Jason Yuen, Oct 13 2024

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

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A177678 Palindromic primes p = q//r//q such that q and r are also palindromic primes.

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Comments

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

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Extensions