Hemjyoti Nath has authored 7 sequences.
A356690
Product of the prime numbers that are between 10*n and 10*(n+1).
Original entry on oeis.org
210, 46189, 667, 1147, 82861, 3127, 4087, 409457, 7387, 97, 121330189, 113, 127, 2494633, 149, 23707, 27221, 30967, 181, 1445140189, 1, 211, 11592209, 55687, 241, 64507, 70747, 75067, 79523, 293, 307, 30857731, 1, 111547, 121103, 126727, 367, 141367, 148987, 397, 164009, 419, 421
Offset: 0
210 = 2*3*5*7, 46189 = 11*13*17*19, 667 = 23*29, 1147 = 31*37, 82861 = 41*43*47.
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a[n_] := Times @@ Select[Range[10 n + 1, 10 n + 9], PrimeQ]; Array[a, 43, 0]
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a(n) = vecprod(select(isprime, [10*n..10*(n+1)])); \\ Michel Marcus, Aug 24 2022
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from math import prod
from sympy import primerange
def a(n): return prod(primerange(10*n, 10*(n+1)))
print([a(n) for n in range(43)]) # Michael S. Branicky, Aug 23 2022
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from math import prod
from sympy import isprime
def A356690(n): return prod(m for i in (1,3,7,9) if isprime(m:=10*n+i)) if n else 210 # Chai Wah Wu, Sep 23 2022
A354831
Primes of the form 3^k + 5^k + 7^k + 11^k + 13^k.
Original entry on oeis.org
5, 373, 46309, 6732373, 26450599458469, 4317810550653973, 15647143198792684919908583741989, 6864681654384231304317569259724531213945845885866391974437116993829, 5599548608682504162062596274137068329320798013420534505888549721133699842789
Offset: 1
3^2 + 5^2 + 7^2 + 11^2 + 13^2 = 373, which is a prime.
3^4 + 5^4 + 7^4 + 11^4 + 13^4 = 46309, which is a prime.
A352393 gives the corresponding exponents.
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Select[Table[3^n + 5^n + 7^n + 11^n + 13^n,{n,0,1000}],PrimeQ]
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from sympy import isprime
from itertools import count, islice
def agen(): yield from (p for p in (3**k + 5**k + 7**k + 11**k + 13**k for k in count(0)) if isprime(p))
print(list(islice(agen(), 9))) # Michael S. Branicky, Jun 07 2022
A354829
Numbers k such that 2^k + 3^k + 6 is prime.
Original entry on oeis.org
1, 2, 3, 4, 5, 8, 9, 15, 18, 23, 24, 33, 34, 35, 44, 63, 88, 89, 120, 220, 228, 229, 570, 1095, 1863, 2094, 2718, 3598, 4658, 6056, 8819, 9485, 11220, 23656, 28762, 35664, 36544, 39779, 46868, 50098, 58853
Offset: 1
For k=1 we obtain f(1) = 2^1 + 3^1 + 6 = 11 which is a prime.
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Select[Range[1, 1000], PrimeQ[2^# + 3^# + 6] &]
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from sympy import isprime
from itertools import count, islice
def agen(): yield from (k for k in count(1) if isprime(2**k+3**k+6))
print(list(islice(agen(), 24))) # Michael S. Branicky, Jun 07 2022
A352393
Numbers k such that 3^k + 5^k + 7^k + 11^k + 13^k is prime.
Original entry on oeis.org
0, 2, 4, 6, 12, 14, 28, 60, 68, 2070, 7910, 10740
Offset: 1
For k=2 we obtain f(2) = 3^2 + 5^2 + 7^2 + 11^2 + 13^2 = 373 which is a prime.
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Select[Range[0, 1000], PrimeQ[3^# + 5^# + 7^# + 11^# +13^#] &]
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from sympy import isprime
from itertools import count, islice
def agen(): yield from (k for k in count(0) if isprime(3**k + 5**k + 7**k + 11**k + 13**k))
print(list(islice(agen(), 9))) # Michael S. Branicky, Jun 07 2022
A353102
Primes of the form 2^k + 3^k + 6.
Original entry on oeis.org
11, 19, 41, 103, 281, 6823, 20201, 14381681, 387682639, 94151567441, 282446313703, 5559069156490121, 16677198879535759, 50031579458738081, 984770919775797277303, 1144561273440060866922804472241, 969773729787523912361831763509149540341223, 2909321189362571427600485469182379896242601
Offset: 1
2^1 + 3^1 + 6 = 11, which is a prime.
2^2 + 3^2 + 6 = 19, which is a prime.
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Select[Table[2^n + 3^n + 6,{n,1,1000}],PrimeQ]
A352913
a(n) = largest prime of the form prime(n) + k! (k >= 0).
Original entry on oeis.org
3, 5, 29, 727, 3628811, 733, 39916817, 87178291219, 20922789888023, 2432902008176640029, 1124000727777607680031, 8683317618811886495518194401280000037, 15511210043330985984000041, 523022617466601111760007224100074291200000043, 2658271574788448768043625811014615890319638528000000047
Offset: 1
Editors of OEIS, based on a suggestion from Hemjyoti Nath, Apr 16 2022
These are the final entries in the rows of the triangle in
A352912. See also
A082470.
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from sympy import isprime, prime
from itertools import count, islice
def agen(): # generator of terms
for n in count(1):
pn, fk = prime(n), 1
for k in range(1, pn+1):
if isprime(pn + fk): yield pn + fk
fk *= k
print(list(islice(agen(), 51))) # Michael S. Branicky, Apr 16 2022
A352912
Irregular triangle read by rows: row n (n>=1) lists the primes of the form prime(n) + k! for k >= 0.
Original entry on oeis.org
3, 3, 5, 7, 11, 29, 13, 31, 127, 727, 13, 17, 131, 5051, 3628811, 19, 37, 733, 19, 23, 41, 137, 362897, 39916817, 43, 139, 739, 5059, 3628819, 39916819, 87178291219, 29, 47, 743, 40343, 362903, 20922789888023, 31, 53, 149, 39916829, 479001629, 2432902008176640029, 37, 151, 751, 40351, 362911, 39916831, 355687428096031, 51090942171709440031, 1124000727777607680031
Offset: 1
Editors of OEIS, based on a suggestion from Hemjyoti Nath, Apr 16 2022
The initial rows, prefixed by prime(n), are:
[2]: 3, 3,
[3]: 5,
[5]: 7, 11, 29,
[7]: 13, 31, 127, 727,
[11]: 13, 17, 131, 5051, 3628811,
[13]: 19, 37, 733,
[17]: 19, 23, 41, 137, 362897, 39916817,
[19]: 43, 139, 739, 5059, 3628819, 39916819, 87178291219,
[23]: 29, 47, 743, 40343, 362903, 20922789888023,
[29]: 31, 53, 149, 39916829, 479001629, 2432902008176640029,
[31]: 37, 151, 751, 40351, 362911, 39916831, 355687428096031, 51090942171709440031, 1124000727777607680031,
[37]: 43, 61, 157, 757, 5077, 40357, 39916837, 6402373705728037, 2432902008176640037, 51090942171709440037, 8683317618811886495518194401280000037,
...
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forprime(p=2,59,print1([p],": ");for(k=0,p,if(ispseudoprime(p+k!),print1(p+k!,", ")));print())
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from sympy import isprime, prime
from itertools import count, islice
def agen(): # generator of terms
for n in count(1):
pn, fk = prime(n), 1
for k in range(1, pn+1):
if isprime(pn + fk): yield pn + fk
fk *= k
print(list(islice(agen(), 51))) # Michael S. Branicky, Apr 16 2022
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