Andreas Boe - cp's OEIS frontend

A343538 Longest sequence of distinct two-digit numbers (10-99) such that any two or three consecutive terms can be concatenated into a prime number, even when connected into a loop. Smallest number is listed first.

13, 61, 63, 67, 19, 99, 31, 69, 91, 87, 79, 37, 39, 43, 27, 49, 73, 51

Offset: 1

Andreas Boe, Apr 18 2021

A total of 164 sequences exist from two to eighteen terms. Checked to a depth of 50 terms with two different programs. 11262 sequences with two to twenty-one terms exist if the demand on connecting the sequence into a loop is removed.

			13,61,63,67 -> 1361, 6163, 6367, 136163, 616367 are all prime numbers.

Andreas Boe, The Toy Robot, Amazon, 2014, Afterword Quiz.

Andreas Boe, All 2-digit looped sequences
Illustration for 11,17,33,47,93,71,51,53,81,23,41,39

A274227 Primes in A274226.

Original entry on oeis.org

29, 53, 61, 109, 157, 277, 397

Offset: 1

Andreas Boe, Jun 14 2016

If a(8) exists it must be larger than 10^8.

From a proof outline of Wagner, the discriminants of Q(sqrt(-p)) with class number 6 end at -1588, ending this sequence at 397. - Travis Scott, Feb 09 2023

			29 is a term because 2^2 + 3^2 + 4^2 = 29 is the only representation of 29 as a sum of 3 positive squares, and those squares are distinct.
41 is not a term because, even though it can be represented in just one way as a sum of 3 distinct squares (1^2 + 2^2 + 6^2) it can also be represented as 3^2 + 4^2 + 4^2.

Andreas Boe, List of values with values of x, y and z attached
C. Wagner, Class number 5, 6 and 7, Math. Comput. 64 (1996), pp. 785-800.

Cf. A274226, A025339.

rp[n_] := Flatten@ IntegerPartitions[n, {3}, Range[Sqrt@n]^2]; Select[
Range[265] // Prime, Length[u = rp[#]] == 3 && Union[u] == Sort[u] &] (* Giovanni Resta, Jun 16 2016 *)
Select[Prime@Range@78,Sum[(-1)^Boole@Xor[Mod[t,4]==1,PowerMod[t,(#-1)/2,#]==1],{t,1,#-1,2}]==6&] (* Travis Scott, Feb 09 2023 *)

A274226 Numbers that have a unique representation as a sum of three nonzero squares, and furthermore in this representation the squares are distinct.

Original entry on oeis.org

14, 21, 26, 29, 30, 35, 42, 45, 46, 49, 50, 53, 56, 61, 65, 70, 78, 84, 91, 93, 104, 106, 109, 115, 116, 120, 133, 140, 142, 145, 157, 168, 169, 180, 184, 190, 196, 200, 202, 205, 212, 224, 235, 244, 253, 260, 265

Offset: 1

Andreas Boe, Jun 14 2016

nonn

The numbers in this sequence can be expressed as a sum of 3 positive squares in exactly one way, and those 3 squares are distinct. This is different from A025339.

			14 is a term because it can be expressed in just one way as a sum of 3 squares (1^2+2^2+3^2) and the 3 squares are different.
38 is not a term, because, even if it can be expressed as a sum of 3 distinct squares in just one way (2^2+3^2+5^2), it can also be expressed as a sum of 3 non-distinct squares (1^2+1^2+6^2). This makes 38 a member of A004432 and A025339.

Andreas Boe, Table of n, a(n) for n = 1..373
Andreas Boe, List of values with values of x, y and z

Cf. A025339, A004432, A274227 (the primes in this sequence).

rp[n_] := Flatten@ IntegerPartitions[n, {3}, Range[Sqrt@n]^2]; Select[
Range[265], Length[u = rp[#]] == 3 && Union[u] == Sort[u] &] (* Giovanni Resta, Jun 15 2016 *)

A259571 Numbers n whose square root contains at least one instance of n in the first n digits after the decimal point.

Original entry on oeis.org

3, 7, 8, 17, 23, 27, 38, 39, 43, 48, 58, 65, 66, 67, 68, 69, 74, 75, 76, 77, 78, 83, 84, 88, 89, 91, 92, 93, 98, 99, 105, 110, 113, 140, 156, 157, 170, 182, 186, 188, 191, 194, 205, 212, 215, 218, 222, 227, 229, 234, 235, 241, 246, 247, 252, 267, 287, 293, 294

Offset: 1

Andreas Boe, Jun 30 2015

Numbers beginning with a nine are most likely to have this property, and numbers beginning with a one are least likely. No squares have it.

			sqrt(17) = 4.12310562561766054... The first 17 digits after the decimal point contain "17".

Andreas Boe, Table of n, a(n) for n = 1..551

Select[Range@ 301, ! StringFreeQ[ToString@ FromDigits@ First@ RealDigits@ N[Sqrt@ #, #], ToString@ #] &] // Rest (* Michael De Vlieger, Jul 02 2015 *)

A248548 Sums of Pythagorean sextuples in increasing order: The sums of sets of six natural numbers which correspond to the lengths of the edges of a tetrahedron whose four faces are all different Pythagorean triangles.

Original entry on oeis.org

2491, 3616, 4385, 4450, 4783, 4982, 7232, 7473, 7974, 8770, 8900, 9566, 9964, 10848, 11784, 12455, 12503, 13155, 13350, 13565, 14086, 14141, 14349, 14464, 14778, 14946, 15948, 16389, 17394, 17437, 17540, 17800, 18080, 19132, 19453, 19928, 21696, 21925, 22250, 22419, 22821, 23568, 23915, 23922, 24079

Offset: 1

Andreas Boe, Oct 08 2014

nonn

A Pythagorean sextuple contains four Pythagorean triples.
The sequence is sorted on increasing sum of set.
See attached a-file to see the specific values in the sets.
The first sum shared by two sets is 956670.
The first set contains the values 104, 153, 672, 185, 680, 697.
The first set to have a ratio less than 2:1 between its highest and lowest value is 1680, 1925, 2052, 2555, 2652, 3277.
The method used for finding the sets was to connect two Pythagorean triangles along one edge and then calculate the length of the remaining edge.

			The first value in the sequence: 2491.
104^2 + 153^2 = 185^2, 104^2 + 672^2 = 680^2, 153^2 + 680^2 = 697^2, 672^2 + 185^2 = 697^2;
104 + 153 + 185 + 672 + 680 + 697 = 2491.

Andreas Boe, Table of n, a(n) for n = 1..2426
Andreas Boe, Sets of Pythagorean sextuples, cross indexed with the base sequence

Sums a + b + c + d + e + f where a^2 + b^2 = d^2, a^2 + c^2 = e^2, b^2 + e^2 = f^2, c^2 + d^2 = f^2.

A245616 Pythagorean Threesomes: triples of natural numbers defining the six legs of three Pythagorean triangles.

Original entry on oeis.org

44, 117, 240, 240, 252, 275, 88, 234, 480, 85, 132, 720, 160, 231, 792, 132, 351, 720, 480, 504, 550, 176, 468, 960, 170, 264, 1440, 220, 585, 1200, 720, 756, 825, 320, 462, 1584, 264, 702, 1440, 308, 819, 1680, 255, 396, 2160, 960, 1008, 1100, 352, 936, 1920, 480, 693, 2376, 396, 1053, 2160, 429, 880, 2340, 340, 528, 2880

Offset: 1

Andreas Boe, Nov 05 2014

nonn

The sequence is sorted by increasing sums of triples and secondly by increasing order of first term.
The three numbers in a Pythagorean Threesome define the lengths of three sides of a tetrahedron with all integer length edges and one right angle vertex.
The sequence was calculated for the science fiction novel "The Fifth Jack" by Andreas Boe, Amazon books, 2014.
I do not have that book, but this sequence is closely related to (and may be an erroneous version of) A268396. - Arkadiusz Wesolowski, Feb 03 2016

			(44,117,240)  sqrt(44^2+117^2)=125  sqrt(117^2+240^2)=267 sqrt(240^2+44^2)=244

Andreas Boe, Table of n, a(n) for n = 1..285

Same numbers sorted gives A195816.

Cf. A268396.

x,y,sqrt(x^2+y^2) y,z,sqrt(y^2+z^2) z,x,sqrt(z^2+x^2)

Edited by N. J. A. Sloane, Feb 11 2016

A248010 Non-primatic permutable numbers: All permutations of the number's digits except the ones resulting in leading zeros are nonprimes.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 33, 36, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 75, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 93, 94, 96, 99, 100, 102, 105, 108, 111, 114, 116, 117, 120, 122, 123, 126, 129, 132, 135, 138, 141, 144, 147

Offset: 1

Andreas Boe, Sep 29 2014

This sequence differs slightly from "absolute composite numbers". 30 is not an absolute composite since 03 is counted as a prime, but in this sequence permutations with leading zeros are disqualified as viable permutations.

			7000 qualifies since it is a composite and the only allowed permutation of its four digits.

Andreas Boe, Table of n, a(n) for n = 1..10000

Absolute composites: A067012, A067013.
Monoprimatic permutable numbers: A245808.
Biprimatic permutable numbers: A246043.

A246850 Even numbers which cannot be represented by the surface area of an n1 X n2 X n3 block.

Original entry on oeis.org

2, 4, 8, 12, 20, 36, 44, 60, 84, 116, 140, 156, 204, 260, 380, 420, 660, 924

Offset: 1

Andreas Boe, Sep 05 2014

Twice A025052, see there for further comments.

			A 1 X 1 X 1 block has surface area 6. A 1 X 1 X 2 block has surface area 10. No n1 X n2 X n3 block of intermediate size exists, so there is no way to create an n1 X n2 X n3 block with surface area 8.

Cf. A025052.

from sympy import integer_nthroot
def aupto(lim):
  e, r, lim2 = set(range(2, lim+1, 2)), set(), integer_nthroot(lim//2, 2)[0]
  for n1 in range(1, lim2):
    for n2 in range(n1, lim2):
      for n3 in range(n2, lim+1):
        r.add(2*(n1*n2 + n1*n3 + n2*n3))
  return sorted(e - r)
print(aupto(1000)) # Michael S. Branicky, Feb 04 2021

a(n) = 2 * A025052(n).

Surface area = 2*(n1*n2 + n1*n3 + n2*n3).

A246045 Biprimatic permutable primes: prime numbers whose digits can be rearranged to form exactly one other prime number. No leading zeros allowed.

Original entry on oeis.org

13, 17, 31, 37, 71, 73, 79, 97, 107, 127, 139, 181, 191, 193, 239, 241, 251, 271, 277, 281, 283, 293, 313, 331, 347, 349, 367, 421, 439, 457, 461, 463, 467, 479, 521, 547, 563, 569, 577, 587, 619, 641, 643, 647, 653, 659, 673, 683, 691, 701, 709, 727, 743, 757, 769, 787, 797, 811, 821, 823, 857, 863, 877, 907, 911, 947, 967

Offset: 1

Andreas Boe, Aug 23 2014

In base ten the numbers can be said to have a prime twin made up of the same digits.

			709 -> 079 (forbidden), 097 (forbidden), 709 (prime), 790 (even), 907 (prime), 970 (even) -> conclusion: Two primes.

Andreas Boe, Table of n, a(n) for n = 1..828

A245808: Monoprimatic permutable numbers
A246044: Monoprimatic permutable primes
A246043: Biprimatic permutable numbers

A246044 Monoprimatic permutable primes: Decimal prime numbers whose digits cannot be rearranged to form another prime number. No leading zeros allowed.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 101, 103, 109, 151, 211, 223, 227, 229, 233, 257, 263, 269, 307, 353, 383, 401, 409, 431, 433, 443, 449, 487, 499, 503, 509, 523, 541, 557, 599, 601, 607, 661, 677, 773, 809, 827, 829, 853, 859, 881, 883, 887, 929, 997, 1447, 1451, 1481, 2003, 2017, 2029, 2087

Offset: 1

Andreas Boe, Aug 23 2014

			859 -> 589 (composite), 598 (even), 859 (prime), 895 (composite), 958 (even), 985 (composite) -> conclusion: one prime number.

Giovanni Resta, Table of n, a(n) for n = 1..760 (terms < 10^13, terms 1..406 from Andreas Boe, terms 407..538 from Chai Wah Wu)

Cf. A245808 (monoprimatic permutable numbers)

Cf. A246043 (biprimatic permutable numbers), A246045 (biprimatic permutable primes).

mppQ[n_]:=Total[Boole[PrimeQ[Select[FromDigits/@Permutations[IntegerDigits[n]],IntegerLength[ #] == IntegerLength[ n]&]]]] ==1; Select[Prime[Range[500]],mppQ] (* Harvey P. Dale, Dec 06 2021 *)

from itertools import permutations
from sympy import prime, isprime
A246044 = []
for n in range(1,10**6):
    p = prime(n)
    for x in permutations(str(p)):
        if x[0] != '0':
            p2 = int(''.join(x))
            if p2 != p and isprime(p2):
                break
    else:
        A246044.append(p) # Chai Wah Wu, Aug 27 2014

cp's OEIS Frontend

User: Andreas Boe

A343538 Longest sequence of distinct two-digit numbers (10-99) such that any two or three consecutive terms can be concatenated into a prime number, even when connected into a loop. Smallest number is listed first.

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A274227 Primes in A274226.

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A274226 Numbers that have a unique representation as a sum of three nonzero squares, and furthermore in this representation the squares are distinct.

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A259571 Numbers n whose square root contains at least one instance of n in the first n digits after the decimal point.

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A248548 Sums of Pythagorean sextuples in increasing order: The sums of sets of six natural numbers which correspond to the lengths of the edges of a tetrahedron whose four faces are all different Pythagorean triangles.

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A245616 Pythagorean Threesomes: triples of natural numbers defining the six legs of three Pythagorean triangles.

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A248010 Non-primatic permutable numbers: All permutations of the number's digits except the ones resulting in leading zeros are nonprimes.

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A246850 Even numbers which cannot be represented by the surface area of an n1 X n2 X n3 block.

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A246045 Biprimatic permutable primes: prime numbers whose digits can be rearranged to form exactly one other prime number. No leading zeros allowed.

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