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A387201 Numbers k such that 32 * 3^k + 1 is prime.

1, 4, 8, 9, 32, 36, 48, 74, 112, 186, 204, 364, 393, 572, 781, 1208, 2624, 2778, 4522, 4896, 5272, 32884

Offset: 1

Ken Clements, Aug 21 2025

a(23) > 10^5.

Conjecture: This sequence intersects with A387197 at k = 4 to form twin primes with center N = 2^5 * 3^4 = 2592 = A027856(10). Any such intersection has to be at an even k because if k is odd, either N-1 or N+1 has to be divisible by 5. A covering system can be constructed that eliminates all other intersections except where k == 4 (mod 60), and for k > 4 with k == 4 (mod 60), the search up to 10^5 makes the probability of another intersection in this residue class vanishingly small.

Ken Clements, Python program to calculate covering system.

Cf. A027856, A003306, A005537, A005538, A387060, A387197.

Select[Range[0, 5000], PrimeQ[32 * 3^# + 1] &] (* Amiram Eldar, Aug 21 2025 *)

from gmpy2 import is_prime
print([ k for k in range(4000) if is_prime(32 * 3**k + 1)])

A387197 Numbers k such that 32 * 3^k - 1 is prime.

Original entry on oeis.org

0, 3, 4, 6, 46, 59, 84, 94, 124, 239, 267, 366, 371, 424, 616, 2139, 2299, 3523, 3563, 3843, 3923, 7627, 12751, 34798, 39911, 56568, 58779

Offset: 1

Ken Clements, Aug 21 2025

a(28) > 10^5.

Conjecture: This sequence intersects with A387201 at k = 4 to form twin primes with center N = 2^5 * 3^4 = 2592 = A027856(10). Any such intersection has to be at an even k because if k is odd, either N-1 or N+1 has to be divisible by 5. A covering system can be constructed that eliminates all other intersections except where k = 4(mod 60), and for k > 4 with k = 4(mod 60), the search up to 10^5 makes the probability of another intersection in this residue class vanishingly small.

Ken Clements, Python program to calculate covering system.

Cf. A027856, A003307, A005540, A005541, A385115, A387201.

Select[Range[0, 4000], PrimeQ[32 * 3^# - 1] &] (* Amiram Eldar, Aug 21 2025 *)

from gmpy2 import is_prime
print([ k for k in range(4000) if is_prime(32 * 3**k - 1)])

A386515 a(n) is the largest number of distinct primes in a partition of prime(n) into primes.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Alexander R. Povolotsky, Aug 21 2025

For each prime number prime(n) find all sums of smaller prime numbers which add up to this prime number. Among those sums find the largest number of distinct primes.

			Examples of such partitions for n = 3..11:
  prime(3) = 5 = 2 + 3 which gives a(3) = 2;
  prime(4) = 7 = 2 + 5 which gives a(4) = 2;
  prime(5) = 11 = 2 + 2 + 2 + 2 + 3 = 3 + 3 + 5 which gives a(5)=2;
  prime(6) = 13 = 2 + 3 + 5 + 3 which gives a(6)=3;
  prime(7) = 17 = 2 + 3 + 5 + 7 which gives a(7)=4;
  prime(8) = 19 = 2 + 3 + 5 + 7 + 2 which gives a(8)=4;
  prime(9) = 23 = 2 + 3 + 5 + 13 which gives a(9)=4;
  prime(10) = 29 = 2 + 3 + 5 + 19 which gives a(10)=4;
  prime(11) = 31 = 2 + 3 + 5 + 7 + 7 + 7 which gives a(11)=4.

Cf. A000040, A229219, A321578.

a(n) <= A321578(n). - David A. Corneth, Aug 22 2025

More terms from David A. Corneth, Aug 22 2025

A387207 The maximal norm of an additively indecomposable element in the real quadratic field Q(sqrt(D)), where D = A005117(n) is the n-th squarefree number.

Original entry on oeis.org

2, 1, 1, 3, 2, 10, 5, 3, 2, 1, 4, 9, 1, 11, 2, 26, 7, 6, 10, 4, 2, 1, 9, 19, 13, 10, 7, 21, 9, 2, 25, 13, 9, 7, 58, 29, 15, 2, 16, 33, 33, 3, 14, 10, 18, 74, 1, 3, 2, 82, 41, 21, 43, 13, 22, 30, 7, 18, 5, 24, 25, 51, 34, 4, 106, 53, 27, 11, 37, 28, 57, 9, 59, 2, 122, 61, 42, 16, 130, 65, 11

Offset: 2

Robin Visser, Aug 21 2025

For any totally real field K, an additively indecomposable element of K is a totally positive element in the maximal order of K which cannot be written as the sum of two totally positive integral elements of K. Here, an element x of K is totally positive if all conjugates of x are positive real numbers.

Let K = Q(sqrt(D)) be a real quadratic field. By studying the continued fraction expansion of sqrt(D), Dress and Scharlau classified all additively indecomposable elements of K and showed that every such indecomposable element has its norm bounded by the discriminant of K.

			For n = 2, every additively indecomposable element in Q(sqrt(A005117(2))) = Q(sqrt(2)) has norm either 1 or 2, thus a(2) = 2.
For n = 3, every additively indecomposable element in Q(sqrt(A005117(3))) = Q(sqrt(3)) has norm 1, thus a(3) = 1.
For n = 4, every additively indecomposable element in Q(sqrt(A005117(4))) = Q(sqrt(5)) has norm 1, thus a(4) = 1.
For n = 5, every additively indecomposable element in Q(sqrt(A005117(5))) = Q(sqrt(6)) has norm either 1 or 3, thus a(5) = 3.

Andreas Dress and Rudolf Scharlau, Indecomposable totally positive numbers in real quadratic orders, J. Number Theory 14 (1982), no. 3, 292-306.
Se Wook Jang and Byeong Moon Kim, A refinement of the Dress-Scharlau theorem, J. Number Theory 158 (2016), 234-243.
Vítězslav Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193-207.
Vítězslav Kala, Universal quadratic forms and indecomposables in number fields: a survey, Commun. Math. 31 (2023), no. 2, 81-114.
Magdaléna Tinková and Paul Voutier, Indecomposable integers in real quadratic fields, J. Number Theory 212 (2020), 458-482.

Cf. A005117, A035015, A387203.

def a(n):
    D = [d for d in range(2*n) if Integer(d).is_squarefree()][n-1]
    K. = QuadraticField(D); OK = K.ring_of_integers(); ans = 0
    if (D%4==1): cf, d = continued_fraction((a-1)/2), (a+1)/2
    else: cf, d = continued_fraction(a), a
    s = len(cf.period())
    ai = [1]+[c.numer() + c.denom()*d for c in cf.convergents()[:2*s+1]]
    ind = [ai[i]+t*ai[i+1] for i in range(0, 2*s+1, 2) for t in range(cf[i+1])]
    return max([c.norm() for c in ind])

a(n) <= A005117(n) for all n >= 2 [Dress-Scharlau].

A387196 Integers k such that 1/k = (1/p - 1/q)*(1/r - 1/s) for distinct primes p < q and r < s.

Original entry on oeis.org

13, 17, 19, 20, 21, 25, 36, 37, 45, 49, 55, 91, 105, 127, 169, 181, 187, 247, 307, 361, 391, 429, 541, 577, 667, 811, 937, 961, 969, 1147, 1297, 1567, 1591, 1801, 1849, 1927

Offset: 1

Yuto Tsujino, Aug 21 2025

For any prime p, an exhaustive search with primes up to p finds all terms t in the sequence that satisfy t < next_prime(p).
If p and p+d are primes with d in {2,6}, then 6*p*(p+d)/d is in the sequence.
If p and p+2 are primes, then (p+2)^2 is in the sequence.
If p is a prime such that p = (b+1)*(c-1)+1 for some primes b and c with c-b also prime, then p is in the sequence.

			1/13 = (1/2 - 1/5)*(1/3 - 1/13),
1/17 = (1/3 - 1/5)*(1/2 - 1/17),
1/20 = (1/2 - 1/3)*(1/2 - 1/5),
1/36 = (1/2 - 1/3)*(1/2 - 1/3),
1/45 = (1/2 - 1/3)*(1/3 - 1/5).

a(30)-a(36) from Hugo Pfoertner, Aug 23 2025

A386528 Primes which remain primes after the mapping {1 -> 3, 3 -> 5, 5 -> 7, 7 -> 9, 9 -> 1} of its decimal digits.

Original entry on oeis.org

2, 3, 5, 19, 31, 37, 41, 59, 97, 131, 137, 151, 157, 181, 191, 199, 211, 227, 239, 271, 281, 307, 349, 359, 367, 409, 419, 457, 461, 479, 509, 541, 569, 619, 631, 641, 691, 727, 797, 821, 827, 829, 881, 907, 919, 947, 971, 977, 991, 1009, 1021, 1049, 1069, 1087, 1097, 1109, 1151

Offset: 1

Tristan J. Jones and Robert G. Wilson v, Aug 21 2025

Of the 10! possible nontrivial decimal digital mappings, this one was chosen for its inclusion of all the odd numbers and none of the even numbers.

			19 is a term since the mapping produces 31, which is prime;
31 is a term since the mapping produces 53, which is prime.

Cf. A000040, A198047.

fQ[n_] := PrimeQ[ FromDigits[ IntegerDigits[ n] /. {1 -> 3, 3 -> 5, 5 -> 7, 7 -> 9, 9 -> 1}]]; Select[ Prime@ Range@ 200, fQ]

from sympy import isprime
def ok(n): return isprime(n) and isprime(int(str(n).translate(str.maketrans("13579","35791"))))
print([k for k in range(1200) if ok(k)]) # Michael S. Branicky, Aug 24 2025

A387203 Number of additively indecomposable elements in the real quadratic field Q(sqrt(D)) up to multiplication by totally positive units, where D = A005117(n) is the n-th squarefree number.

Original entry on oeis.org

2, 1, 1, 2, 2, 6, 3, 3, 2, 1, 5, 7, 1, 6, 2, 10, 5, 2, 8, 4, 2, 1, 7, 6, 4, 11, 2, 13, 8, 2, 7, 7, 4, 7, 20, 9, 11, 2, 9, 8, 19, 2, 6, 6, 21, 20, 1, 2, 2, 18, 9, 9, 16, 3, 21, 12, 3, 12, 2, 27, 11, 10, 18, 3, 34, 13, 17, 2, 8, 23, 12, 5, 18, 2, 22, 11, 24, 15, 26, 15, 6, 22, 27, 2, 31, 4, 2

Offset: 2

Robin Visser, Aug 21 2025

For any totally real field K, an additively indecomposable element of K is a totally positive element in the maximal order of K which cannot be written as the sum of two totally positive integral elements of K. Here, an element x of K is totally positive if all conjugates of x are positive real numbers.

Let K = Q(sqrt(D)) be a real quadratic field. By studying the continued fraction expansion of sqrt(D), Dress and Scharlau classified all additively indecomposable elements of K and showed that every such indecomposable element has its norm bounded by the discriminant of K.

			For n = 2, every additively indecomposable element in Q(sqrt(A005117(2))) = Q(sqrt(2)) is of the form u or u*(2 + sqrt(2)), for some totally positive unit u. Thus a(2) = 2.
For n = 3, every additively indecomposable element in Q(sqrt(A005117(3))) = Q(sqrt(3)) is a totally positive unit, so a(3) = 1.
For n = 4, every additively indecomposable element in Q(sqrt(A005117(4))) = Q(sqrt(5)) is a totally positive unit, so a(4) = 1.
For n = 5, every additively indecomposable element in Q(sqrt(A005117(5))) = Q(sqrt(6)) is of the form u or u*(3 + sqrt(6)), for some totally positive unit u. Thus a(5) = 2.

Andreas Dress and Rudolf Scharlau, Indecomposable totally positive numbers in real quadratic orders, J. Number Theory 14 (1982), no. 3, 292-306.
Se Wook Jang and Byeong Moon Kim, A refinement of the Dress-Scharlau theorem, J. Number Theory 158 (2016), 234-243.
Vítězslav Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193-207.
Vítězslav Kala, Universal quadratic forms and indecomposables in number fields: a survey, Commun. Math. 31 (2023), no. 2, 81-114.
Magdaléna Tinková and Paul Voutier, Indecomposable integers in real quadratic fields, J. Number Theory 212 (2020), 458-482.

Cf. A005117, A035015, A387207.

def a(n):
    D = [d for d in range(2*n) if Integer(d).is_squarefree()][n-1]
    K. = QuadraticField(D); OK = K.ring_of_integers(); ans = 0
    if (D%4==1): cf, d = continued_fraction((a-1)/2), (a+1)/2
    else: cf, d = continued_fraction(a), a
    s = len(cf.period())
    ai = [1]+[c.numer() + c.denom()*d for c in cf.convergents()[:2*s+1]]
    ind = [ai[i]+t*ai[i+1] for i in range(0,2*s+1,2) for t in range(cf[i+1])]
    for i in range(len(ind)):
        for j in range(i):
            if ((ind[i]/ind[j]) in OK) and (OK(ind[i]/ind[j]).is_unit()): break
        else: ans += 1
    return ans

A387199 Numbers which are not themselves palindromes, but a single swap of two digits creates a palindrome.

Original entry on oeis.org

110, 112, 113, 114, 115, 116, 117, 118, 119, 122, 133, 144, 155, 166, 177, 188, 199, 211, 220, 221, 223, 224, 225, 226, 227, 228, 229, 233, 244, 255, 266, 277, 288, 299, 311, 322, 330, 331, 332, 334, 335, 336, 337, 338, 339, 344, 355, 366, 377, 388, 399, 411, 422

Offset: 1

James S. DeArmon, Aug 21 2025

Might be called "single-transposition palindromes".

Leading zeros are not allowed in either the initial number or the resultant palindrome.

			110 is a term since a swap of the second and third digits yields the palindrome 101.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A002113.

def pal(s): return s == s[::-1]
def swaps(s): yield from (t for i in range(len(s)-1) for j in range(i+1, len(s)) if (t:=s[:i]+s[j]+s[i+1:j]+s[i]+s[j+1:])[0]!='0')
def ok(n): return not pal(s:=str(n)) and any(pal(t) for t in swaps(s))
print([k for k in range(425) if ok(k)]) # Michael S. Branicky, Aug 22 2025

More terms from Michael S. Branicky, Aug 22 2025

A387202 a(n) is the number of dissections of a (4*n+2)-gon into hexagons using strictly disjoint diagonals.

Original entry on oeis.org

1, 5, 21, 87, 363, 1534, 6570, 28492, 124944, 553301, 2471373, 11122275, 50389695, 229643895, 1052093655, 4842863465, 22386911925, 103885321615, 483759492255, 2259888333445, 10587902977185, 49738841822400, 234235771140876, 1105609645231322, 5229610939919718

Offset: 1

Muhammed Sefa Saydam, Aug 21 2025

Muhammed Sefa Saydam, Table of n, a(n) for n = 1..100

Cf. A002212, A093128.

seq(n)={my(g=(1 - 3*x - sqrt(1 - 6*x + 5*x^2 + O(x*x^n)))/(2*x)); Vec((1 + 4*g + 3*g^2)*x + g^2)} \\ Andrew Howroyd, Aug 21 2025

G.f.: x*(1 + 4*B(x) + 3*B(x)^2) + B(x)^2, where 1 + B(x) is the g.f. of A002212. - Andrew Howroyd, Aug 21 2025

D-finite with recurrence -(n+2)*(2*n-3)*a(n) +3*(2*n+1)*(2*n-3)*a(n-1) -5*(2*n+1)*(n-3)*a(n-2)=0. - R. J. Mathar, Aug 28 2025

A387206 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(6k-5)^6 + (-1)^(k+1)/(6k-1)^6.

Original entry on oeis.org

1, 0, 0, 0, 0, 5, 5, 1, 6, 0, 7, 9, 4, 8, 6, 9, 4, 6, 3, 4, 7, 3, 7, 5, 6, 6, 1, 8, 8, 4, 4, 9, 2, 5, 6, 2, 5, 8, 7, 8, 1, 9, 7, 6, 9, 3, 6, 5, 2, 0, 6, 5, 1, 8, 5, 6, 3, 1, 0, 1, 8, 2, 5, 7, 0, 6, 1, 3, 0, 3, 5, 7, 9, 3, 8, 0, 0, 9, 9, 7, 1, 9, 2, 0, 7, 8, 2, 1, 6, 6, 6, 2, 7, 3, 2, 5, 0, 6, 8, 9

Offset: 1

Jason Bard, Aug 21 2025

			1.000055160794869463473756618844925625878197693652065...

Jason Bard, Table of n, a(n) for n = 1..1000
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21. (Misprint contains erroneous minus sign)

c:= Re(sum((-1)^(k+1)/(6*k-5)^6+(-1)^(k+1)/(6*k-1)^6, k=1..infinity)):
evalf(c, 140);  # Alois P. Heinz, Aug 21 2025

RealDigits[(1/358318080)*(PolyGamma[5, 1/12] + PolyGamma[5, 5/12] - PolyGamma[5, 7/12] - PolyGamma[5, 11/12]), 10, 100][[1]]

(365/1492992) * (zetahurwitz(6, 1/4) - zetahurwitz(6, 3/4))

Equals (1/358318080) * (PolyGamma(5, 1/12) + PolyGamma(5, 5/12) - PolyGamma(5, 7/12) - PolyGamma(5, 11/12)).

Equals (73/35831808) * (PolyGamma(5, 1/4) - PolyGamma(5, 3/4)). - Amiram Eldar, Aug 22 2025

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A387201 Numbers k such that 32 * 3^k + 1 is prime.

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A387197 Numbers k such that 32 * 3^k - 1 is prime.

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A386515 a(n) is the largest number of distinct primes in a partition of prime(n) into primes.

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A387207 The maximal norm of an additively indecomposable element in the real quadratic field Q(sqrt(D)), where D = A005117(n) is the n-th squarefree number.

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A387196 Integers k such that 1/k = (1/p - 1/q)*(1/r - 1/s) for distinct primes p < q and r < s.

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A386528 Primes which remain primes after the mapping {1 -> 3, 3 -> 5, 5 -> 7, 7 -> 9, 9 -> 1} of its decimal digits.

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A387203 Number of additively indecomposable elements in the real quadratic field Q(sqrt(D)) up to multiplication by totally positive units, where D = A005117(n) is the n-th squarefree number.

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A387199 Numbers which are not themselves palindromes, but a single swap of two digits creates a palindrome.

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Python

Extensions

A387206 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(6k-5)^6 + (-1)^(k+1)/(6k-1)^6.