Sergey Pavlov - cp's OEIS frontend

A330673 The possible v-factors for any A202018(k) (while A202018(k) = v * w, v and w are integers, w >= v >= 41, v = w iff w = 41, all such v-factors form the set V).

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 163, 167, 173, 179, 197, 199, 223, 227, 251, 263, 281, 307, 313, 347, 359, 367, 373, 379, 383, 397, 409, 419, 421, 439, 457, 461, 487, 499, 503, 523, 547, 563, 577, 593, 607, 641, 647, 653, 661, 673, 677, 691, 701, 709, 733

Offset: 0

Sergey Pavlov, Dec 23 2019

nonn

This is different from A257362: a(n) = A257362(n+1) for n=0..109, but a(110) = 1468 != 1471 = A257362(111). - Alois P. Heinz, Mar 02 2020
A kind of prime number sieve for the numbers of form x^2+x+41 (for so-called Euler primes, or A005846).
A set of all composite Euler numbers of form x^2+x+41 could be written as a 4-dimensional matrix m(i,j,t,u); a set of all terms of a(n) could be written as a 3-dimensional matrix v(i,j,t), since, for any integer u > -1, and for any w-factor that has the same values for i, j, t, we have the same v-factor (u = -1 iff w = 41); see formulas below.
Theorem. Let m be a term of A202018. Then m is composite iff m == 0 (mod v), where v is a term of a(n), v <= sqrt(m) (v = sqrt(m) iff m = 1681); otherwise, m is prime. Moreover, while m == 0 (mod p) (p is prime, p <= sqrt(m), p = sqrt(m) iff m = 1681), p is a term of a(n).
While i = 1, any v(i,t,j) is a term of both A202018 and a(n) (trivial).
Any w is a term of V and of a(n) which is the superset of V.

			Let  i = 3, t = 1, j = -1. Then v(i,t,j) = m(j) * i^2 + b + ja = 41 * 3^2 + 4 - 6 = 41 * 9 - 2 = 367, and 367 is a term of a(n).
We could find all terms of a(n) v < 10^n and then all Euler primes p < 10^(2n) (for n > 1, number of all numbers m such that are terms of A202018 (and any m < 10^(2n)) is 10^n; trivial).
Let 2n = 10; it's easy to establish that, while i > 49, any v(i,t,j)^2 > 10^10; thus, we can use 0 < i < 50 to find all numbers v < 10^5. While m is a term of A202018, m < 10^10, m is composite iff there is at least one v such that m == 0 (mod v); otherwise, m is prime. We could easily remove all "false" numbers v that cannot be divisors of any m. Let p' be a regular prime (p' is a term of A000040, but not of a(n)) such that any 3p' < UB(i); in our case, any 3p' < 50. Thus, we could try any v with p' = {2,3,5,7,11,13}; if v == 0 (mod p'), it is "false"; otherwise, there is at least one m < 10^10 such that m == 0 (mod v).

Sergey Pavlov, Table of n, a(n) for n = 0..312
Sergey Pavlov, The Euler primes sieve (for the primes of form x^2+x+41; draft paper)

Cf. A005846, A145292, A202018, A257362.

Let j = {-1;0;-2;1;-3;2;...;-(n+1);n}, m(-1) = 41, m(0) = 41, etc. (while j is negative, m(j) = A202018(-(j+1)); while j is nonnegative, m(j) = A202018(j)). Any term of a(n) could be written at least once as v(i,t+1,j) = m(j) * i^2 + b + ja, where i, t, and j are integers (j could be negative), i > 2; a = (i^2 - 2i) - 2i(t - 1), b = a - ((i^2 - 4)/4 - ((t - 1)^2 + 2(t - 1))), 0 < t < (i/2), while i is even; a = (i^2 - i) - 2i(t - 1), b = a - ((i^2 - 1)/4 - ((t - 1)^2 + (t - 1))), 0 < t < ((i + 1)/2), while i is odd (Note: v(i,1,j) = v(i,i/2,j), while i is even; v(i,1,j) = v(i,(i + 1)/2,j), while i is odd); at i = 2, v(2,1,j) = 4 * m(j) + 3 + 4j (at i = 2, we use only j < 0); at i = 1, v(1,1,j) = m(j) (at i = 1, we use only j >= 0; trivial).

A284279 Primitive terms of A131884.

Original entry on oeis.org

276, 306, 396, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098

Offset: 1

Sergey Pavlov, Mar 24 2017

In other words, terms of A131884 that cannot be written as k*t where k is an integer, k > 1, t is a term of A131884.

			Since 276 is the first term of A131884, the number is in the sequence. But 131884(4) = 552 is not, because A131884(4) = 2 * A131884(1) = 2 * 276.

Cf. A131884.

A284277 Primitive terms of A216072.

Original entry on oeis.org

276, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992, 2232, 2340, 2360, 2514, 2664, 2712, 2982, 3270, 3366, 3408, 3432, 3564, 3678, 3774, 3876, 3906, 4116, 4224, 4290, 4350, 4380, 4788, 4800, 4842

Offset: 1

Sergey Pavlov, Mar 24 2017

In other words, such terms of A216072 that cannot be written as k*t where k is an integer, k > 1, t is a term of A216072.

			Since 276 is the first term of A216072, the number is in the sequence. But A216072(2) = 552 is not, because A216072(2) = 2 * A216072(1) = 2 * 276.

Cf. A216072.

A280428 a(n) = 1729*n^3.

Original entry on oeis.org

1729, 13832, 46683, 110656, 216125, 373464, 593047, 885248, 1260441, 1729000, 2301299, 2987712, 3798613, 4744376, 5835375, 7081984, 8494577, 10083528, 11859211, 13832000, 16012269, 18410392, 21036743, 23901696, 27015625, 30388904, 34031907, 37955008, 42168581, 46683000

Offset: 1

Sergey Pavlov, Mar 03 2017

Previous name was: Taxi-cab numbers of form n^3*1729; in other words, taxi-cab numbers of form n^3*A001235(1).

			For n = 11, a(11) = 11^3 * 1729 = 2301299 and a(11) = A001235(67).

Sergey Pavlov, Table of n, a(n) for n = 1..303.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A001235, A017163, A017271, A017523.

Array[1729 #^3 &, 30] (* Michael De Vlieger, Mar 04 2017 *)

vector(30,n,1729*n^3) \\ Derek Orr, Mar 05 2017

a(n) = n^3 * 1729.
From Chai Wah Wu, Jan 19 2021: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
G.f.: x*(1729*x^2 + 6916*x + 1729)/(x - 1)^4. (End)
E.g.f.: 1729*exp(x)*x*(1 + 3*x + x^2). - Stefano Spezia, Apr 05 2025
From Alois P. Heinz, Apr 05 2025: (Start)
a(n) = A017163(n) + A017271(n).
a(n) = A000578(n) + A017523(n). (End)

New name from Joerg Arndt, Mar 04 2017

More terms from Sergey Pavlov, Mar 04 2017

A281424 Numbers k such that 16*(10^k - 1)/3 + 1 is prime.

Original entry on oeis.org

6, 23, 65, 82, 108, 188, 300, 342, 401, 584, 1570, 4119, 10030, 24870, 34710

Offset: 1

Sergey Pavlov, Jan 21 2017

All prime numbers of the form 16*(10^k - 1)/3 + 1 are terms of A002476.
For any k = a(n), the m-index of 16*(10^k - 1)/3 + 1 in sequence 6m+1 contains exactly a(n) digits, and each digit is 8. E.g., while k = a(1) = 6, 16*(10^6 - 1)/3 + 1 = 6*888888 + 1 = 5333329.
In any number of form 16*(10^k - 1)/3 + 1, its first digit is 5, its two last digits are 29, and each other digit that is between (5...29) is 3.
For k=1, k=2, k=3, the numbers of form 16*(10^k - 1)/3 + 1 are squares of the primes 7, 23, and 73, respectively (see A001248).
Equivalently defined as primes of the form (16*10^k-13)/3. - Tyler Busby, Apr 16 2024

			For k = a(1) = 6, 16*(10^6 - 1)/3 + 1 = 5333329 and 16*(10^6 - 1)/3 + 1 is prime.

Makoto Kamada, Search for 53w29.

Cf. A002476.

Select[Range@ 3000, PrimeQ[16 (10^# - 1)/3 + 1] &] (* Michael De Vlieger, Jan 23 2017 *)

from sympy import isprime
def afind(limit, startk=1):
    pow10 = 10**startk
    for k in range(startk, limit+1):
        if isprime(16*(pow10 - 1)//3 + 1): print(k, end=", ")
        pow10 *= 10
afind(600) # Michael S. Branicky, Aug 17 2021

a(12) from Michael S. Branicky, Aug 17 2021
a(13)-a(14) from Michael S. Branicky, Apr 06 2023
a(15) from Kamada data by Tyler Busby, Apr 16 2024

A280828 Numbers k of the form 2*10^m + 2 such that 10^k + 9 is prime.

Original entry on oeis.org

4, 22, 202

Offset: 1

Sergey Pavlov, Jan 08 2017

Let k=2*10^(n-1)+2, then a(n)=10^k+9. For all k>4, k is a term of A058441.
The only known terms from A088275 (Numbers n such that 10^n + 9 is prime) that are of the form 2*10^j + 2 are 4, 22, and 202; given the lower bound given for that sequence's next term, a(4) >= 200002. - Jon E. Schoenfield, Jan 11 2017
For n<4, let k=a(n) and p=(10^k-9)/10^(k/2)+3=10^(k/2)+3, then p is prime. - Sergey Pavlov, Jan 13 2017

			For n=1, a(1)=4 and 10^4 + 9 is prime.

Cf. A043037, A058441, A088275, A144249, A205529.

Numbers k of the form 2*10^m + 2 such that 10^k + 9 is prime.

A280427 a(n) is a prime, such that a(n) = p^d-2 where p is a prime and d is the number of digits of p.

Original entry on oeis.org

3, 5, 167, 359, 839, 1367, 1847, 2207, 3719, 5039, 7919, 1295027, 3442949, 9393929, 13997519, 21253931, 49430861, 84604517, 95443991, 237176657, 329939369, 384240581, 487443401, 633839777, 904231061, 1078193566319, 1427186233199, 1556727840719, 1985193642959

Offset: 1

Sergey Pavlov, Jan 02 2017

These numbers (proved for all p < 500) are a subset of A007528. For all even p, such numbers are a subset of A007528. The sequence is a subset of all numbers f(i) such that f(i) = i^d-2 (d - number of digits of integer i) and f(i) is a prime: e.g., f(15) is prime while f(15) = 15^2-2 = 223.

			If p=5, then d=1 and a(1)=3; if p=7, then d=1 and a(2)=5; if p=13, then d=2 and a(3)=167; etc.

Cf. A007528.

Select[Array[#^IntegerLength@ # - 2 &@ Prime@ # &, 200], PrimeQ] (* Michael De Vlieger, Jan 03 2017 *)

a(n) = p^d-2, a(n) is prime, p is a prime and d is the number of digits of p.

More terms from Michael De Vlieger, Jan 03 2017

A280419 Primes in A132934.

Original entry on oeis.org

1468910121415161820212224252627283032333435363839, 14689101214151618202122242526272830323334353638394042444546484950515254555657586062636465666869

Offset: 1

Sergey Pavlov, Jan 02 2017

Primes that can be obtained from concatenation of the nonprime numbers.
In other words: primes that can be obtained from concatenation of 1 and the next consecutive composite numbers.
For a(n), the smallest prime divisor is a(n) and a(n) is a term of A241845.

Prime Curios!, 39

Cf. A002808, A018252, A132934, A241845.

A261191 40-gonal numbers: a(n) = 38n(n-1)/2 + n.

Original entry on oeis.org

0, 1, 40, 117, 232, 385, 576, 805, 1072, 1377, 1720, 2101, 2520, 2977, 3472, 4005, 4576, 5185, 5832, 6517, 7240, 8001, 8800, 9637, 10512, 11425, 12376, 13365, 14392, 15457, 16560, 17701, 18880, 20097, 21352, 22645, 23976, 25345, 26752, 28197, 29680, 31201

Offset: 0

Sergey Pavlov, Aug 11 2015

According to the common formula for the polygonal numbers: (s-2)*n*(n-1)/2 + n (here s = 40).
The 4th number of the sequence, 117, is also the 10th pentagonal number (see A000326). The next number of the series, 232, is also the 9th decagonal number (see A001107), while 576 is the 25th square number (see A000290). The 12th number of the sequence, 2101, is the 23rd 11-gonal number (see A051682).
From Bruno Berselli, Aug 21 2015: (Start)
a(n) and a(n) - 2*n + 1 provide the numbers m such that 19*m + 81 is a square.
Partial sums of the numbers of the type 38*h + 1 (quadrisections of A113541 and A151979). (End)

Colin Barker, Table of n, a(n) for n = 0..1000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

JavaScript
```
function a(n){return 38*n*(n-1)/2+n}
```

[n*(19*n-18): n in [0..45]]; // Vincenzo Librandi, Aug 12 2015

A261191:=n->38*n*(n-1)/2+n: seq(A261191(n), n=0..50); # Wesley Ivan Hurt, Aug 15 2015

Table[n (19 n - 18), {n, 0, 45}] (* Bruno Berselli, Aug 21 2015 *)

concat(0, Vec(-x*(37*x+1)/(x-1)^3 + O(x^100))) \\ Colin Barker, Aug 11 2015

first(m)=my(v=vector(m,i,i--;38*i*(i-1)/2+i));v; \\ Anders Hellström, Aug 13 2015

a(n) = n*(19*n - 18).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), for n > 2. - Colin Barker, Aug 11 2015
G.f.: -x*(37*x+1) / (x-1)^3. - Colin Barker, Aug 11 2015
E.g.f.: exp(x)*(x + 19*x^2). - Nikolaos Pantelidis, Feb 10 2023

A261343 50-gonal numbers: a(n) = 48n(n-1)/2 + n.

Original entry on oeis.org

0, 1, 50, 147, 292, 485, 726, 1015, 1352, 1737, 2170, 2651, 3180, 3757, 4382, 5055, 5776, 6545, 7362, 8227, 9140, 10101, 11110, 12167, 13272, 14425, 15626, 16875, 18172, 19517, 20910, 22351, 23840, 25377, 26962, 28595, 30276, 32005, 33782, 35607, 37480

Offset: 0

Sergey Pavlov, Aug 15 2015

According to the common formula for the polygonal numbers: (s-2)*n*(n-1)/2 + n (here s = 50).

96*a(n) + 23^2 is a square.

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001107, A051872, A254474.

JavaScript
```
function a(n){return 48*n*(n-1)/2+n}
```

[n*(24*n-23): n in [0..40]]; // Vincenzo Librandi, Aug 17 2015

A261343:=n->n*(24*n-23): seq(A261343(n), n=0..40); # Wesley Ivan Hurt, Aug 20 2015

PolygonalNumber[50,Range[0,40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 11 2019 *)

first(m)=vector(m,n,n--;n*(24*n-23)) \\ Anders Hellström, Aug 15 2015

a(n) = n*(24*n - 23).
G.f.: x*(1+47*x)/(1-x)^3. - Vincenzo Librandi, Aug 17 2015
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Aug 17 2015
E.g.f.: exp(x)*(x + 24*x^2). - Nikolaos Pantelidis, Feb 10 2023

cp's OEIS Frontend

User: Sergey Pavlov

A330673 The possible v-factors for any A202018(k) (while A202018(k) = v * w, v and w are integers, w >= v >= 41, v = w iff w = 41, all such v-factors form the set V).

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A284279 Primitive terms of A131884.

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A284277 Primitive terms of A216072.

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A280428 a(n) = 1729*n^3.

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A281424 Numbers k such that 16*(10^k - 1)/3 + 1 is prime.

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A280828 Numbers k of the form 2*10^m + 2 such that 10^k + 9 is prime.

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A280427 a(n) is a prime, such that a(n) = p^d-2 where p is a prime and d is the number of digits of p.

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A280419 Primes in A132934.

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A261191 40-gonal numbers: a(n) = 38*n*(n-1)/2 + n.

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A261191 40-gonal numbers: a(n) = 38n(n-1)/2 + n.

A261343 50-gonal numbers: a(n) = 48n(n-1)/2 + n.