A091178 -id:A091178 - cp's OEIS frontend

A276414 Index of the first prime which starts a run of n consecutive primes all congruent to each other mod 3 (or mod 6).

1, 9, 15, 54, 271, 271, 2209, 11199, 13717, 13717, 34369, 136456, 172146, 1254203, 1254203, 4308948, 12762142, 21357253, 25813464, 25813464, 39500857, 39500857, 947438659, 947438659, 947438659, 5703167678, 5703167678, 16976360924, 57446769091, 57446769091, 57446769091

Offset: 1

M. F. Hasler, Sep 02 2016

nonn

First term of runs of increasing length of consecutive integers in A270190. - M. F. Hasler, Sep 03 2016

			prime(9) = 23 starts the first run of 2 consecutive primes, {23, 29}, which are congruent to each other (mod 6). Therefore a(2) = 9.
prime(15) = 47 starts the first run of 3 consecutive primes, {47, 53, 59}, which are congruent to each other (mod 6). Therefore a(3) = 15.
prime(54) = 251 starts the first run of 4 consecutive primes, {251, 257, 263, 269}, which are congruent to each other (mod 6). Therefore a(4) = 54.
prime(271) = 1741 starts the first run of 5 consecutive primes, {1741, 1747, 1753, 1759, 1777}, which are congruent to each other (mod 6). Therefore a(5) = 271. This is the first case where the primes are of the form 3k+1.
prime(271) = 1741 also starts the first run of 6 consecutive primes, {1741, 1747, 1753, 1759, 1777, 1783}, which are all congruent to each other (mod 6). Therefore a(6) = 271, too.

J. K. Andersen, Consecutive Congruent Primes.

Cf. A054679; A247816, A247967; A091177, A091178; A000720, A000040.

Subsequence of A270190 (after discarding 1 and duplicates of other terms).

m=c=i=o=0;print1(1);forprime(p=1,,i++;(o-o=p)%3&&(!c||!c=0)&&next;c++>m||next;print1(",",i-m=c))

a(n) = A000720(A054679(n)).

a(30)-a(31) from and name clarified by Jinyuan Wang, Feb 24 2020

A223881 Denominators in the expression m!/(prime(m-1)+1) for m > 1 such that this expression is not an integer.

Original entry on oeis.org

3, 2, 19, 31, 37, 79, 41, 97, 53, 139, 71, 157, 83, 199, 211, 229, 131, 271, 137, 307, 331, 337, 173, 367, 379, 197, 439, 227, 499, 263, 547, 281, 577, 293, 197, 199, 601, 607, 619, 661, 227, 229, 691, 239, 727, 383, 269, 811, 829, 283, 431, 877, 467, 937, 313

Offset: 1

Alexander R. Povolotsky, May 01 2013

nonn

It appears that all terms are primes.
From Alexander R. Povolotsky, Apr 26 2025: (Start)
The scatter plot reveals four distinct, well-separated, monotonically increasing curves. It became possible to extract the integers (all conjectured to be primes) corresponding to each of the four subsets.
Additionally, the approximation formulas for each of the four subsets were derived.
These four approximation formulas, given in the exponential form y=C_k*x^m were found to have a common slope: m=1.197311990 while their displacement coefficients are: C_1≈6.86845, C_2≈3.42058, C_3≈2.28335, C_4≈1.70460.
Notably, these displacement coefficients values exhibit a clear pattern: C_2≈C_1/2, C_3≈C_1/3, C_4≈C_1/4. (For instance, 3.42058≈6.86845/2, and so on.)
Above approximations were derived using general separation and approximation methods and do not specifically account for the fact that these values correspond to the prime numbers.
It appears that all primes in the groups 4, 2 and 1 are generated by the 6*k+1 formula, and so primes in the above groups constitute three subsets of A002476 terms, while all primes in the group 3 are generated by the 2*k+1 formula, and so primes in that group constitute a subset of the terms presented in A000040.
Also it appears that:
1. The first group constitutes a sequence, such that for n>=1, a(n) = A005382(n+6).
2. The third group constitutes a sequence, such that for n>1, a(n) = A158015(n+20).
3. The fourth group constitutes a sequence, such that for n>=1, a(n) = A158016(n+32).
The text files containing the primes, corresponding to the above discussed four groups, where primes are indexed against their position in the complete primes listing (see OEIS's A000040), are viewable and downloadable at the below links section. (End)

Paolo P. Lava, Table of n, a(n) for n = 1..1000
Alexander R. Povolotsky, Mathematics StackExchange, Conjecture regarding this sequence, 2025.
Alexander R. Povolotsky, Reindexed group1 primes, 2025.
Alexander R. Povolotsky, Reindexed group2 primes, 2025.
Alexander R. Povolotsky, Reindexed group3 primes, 2025.
Alexander R. Povolotsky, Reindexed group4 primes, 2025.

Cf. A007528, A024898, A002476, A091178.

Denominator[Select[Table[m!/(Prime[m - 1] + 1), {m, 2, 300}], ! IntegerQ[#] &]] (* T. D. Noe, May 03 2013 *)

m=M=1;forprime(p=2,1e5,M*=m++;t=denominator(M/(p+1)); if(t>1, print1(t", "))) \\ Charles R Greathouse IV, May 08 2013

A269703 Numbers k such that prime(k) == 1 (mod 7).

Original entry on oeis.org

10, 14, 20, 30, 31, 45, 47, 52, 60, 68, 75, 82, 87, 90, 94, 101, 113, 115, 120, 122, 126, 132, 134, 144, 153, 156, 162, 163, 169, 177, 183, 192, 209, 213, 220, 226, 233, 239, 250, 251, 262, 267, 269, 288, 295, 304, 306, 315, 320, 324, 330, 337, 342, 344, 346

Offset: 1

Vincenzo Librandi, Mar 04 2016

nonn

The asymptotic density of this sequence is 1/6 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

			a(1) = 10 because prime(10) = 29 and 29 == 1 (mod 7).

The associated primes are in A004619.

Sequences of numbers n such that prime(n) == 1 (mod k): A091178 (k=3,6), A080147 (k=4), A049511 (k=5,10), this sequence (k=7), A269704 (k=8), A269705 (k=9).

[n: n in [1..500] | NthPrime(n) mod 7 eq 1];

Select[Range[500], Mod[Prime[#], 7] == 1 &]

lista(nn) = for(n=1, nn, if(Mod(prime(n),7)==1, print1(n, ", "))); \\ Altug Alkan, Mar 04 2016

a(n) ~ 6*n. - Charles R Greathouse IV, Sep 20 2016 [Corrected by Amiram Eldar, Mar 01 2021]

A171820 Numbers n such that the n-th prime is of the form 3k + 1/2 +- 1/2.

Original entry on oeis.org

2, 4, 6, 8, 11, 12, 14, 18, 19, 21, 22, 25, 27, 29, 31, 34, 36, 37, 38, 42, 44, 46, 47, 48, 50, 53, 58, 59, 61, 63, 65, 67, 68, 70, 73, 74, 75, 78, 80, 82, 84, 85, 88, 90, 93, 95, 99, 100, 101, 105, 106, 110, 111, 112, 114, 115, 117, 121, 122, 125, 127, 129, 130, 131, 133

Offset: 1

Juri-Stepan Gerasimov, Dec 19 2009

Numbers n such that the n-th prime is a generalized Cuban prime (A007645). A171820 UNION A091177 = A000027.

Essentially the same as A091178. - R. J. Mathar, Jan 28 2010

			2 is in the sequence because 2nd prime = 3*1 + 1/2 - 1/2;
4 is in the sequence because 4th prime = 3*2 + 1/2 + 1/2.

Cf. A000027, A000040, A007645, A091177.

list(lim)=my(v=List(),n); forprime(p=2,, if(n++>lim, return(Vec(v))); if(p%3<2, listput(v,n))) \\ Charles R Greathouse IV, Feb 07 2017

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A276414 Index of the first prime which starts a run of n consecutive primes all congruent to each other mod 3 (or mod 6).

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A223881 Denominators in the expression m!/(prime(m-1)+1) for m > 1 such that this expression is not an integer.

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A269703 Numbers k such that prime(k) == 1 (mod 7).

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A171820 Numbers n such that the n-th prime is of the form 3k + 1/2 +- 1/2.

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