A029709 -id:A029709 - cp's OEIS frontend

A001223 Prime gaps: differences between consecutive primes.

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12

Offset: 1

N. J. A. Sloane

There is a unique decomposition of the primes: provided the weight A117078(n) is > 0, we have prime(n) = weight * level + gap, or A000040(n) = A117078(n) * A117563(n) + a(n). - Rémi Eismann, Feb 14 2008
Let rho(m) = A179196(m), for any n, let m be an integer such that p_(rho(m)) <= p_n and p_(n+1) <= p_(rho(m+1)), then rho(m) <= n < n + 1 <= rho(m + 1), therefore a(n) = p_(n+1) - p_n <= p_rho(m+1) - p_rho(m) = A182873(m). For all rho(m) = A179196(m), a(rho(m)) < A165959(m). - John W. Nicholson, Dec 14 2011
A solution (modular square root) of x^2 == A001248(n) (mod A000040(n+1)). - L. Edson Jeffery, Oct 01 2014
There exists a constant C such that for n -> infinity, Cramer conjecture a(n) < C log^2 prime(n) is equivalent to (log prime(n+1)/log prime(n))^n < e^C. - Thomas Ordowski, Oct 11 2014
a(n) = A008347(n+1) - A008347(n-1). - Reinhard Zumkeller, Feb 09 2015
Yitang Zhang proved lim inf_{n -> infinity} a(n) is finite. - Robert Israel, Feb 12 2015
lim sup_{n -> infinity} a(n)/log^2 prime(n) = C <==> lim sup_{n -> infinity}(log prime(n+1)/log prime(n))^n = e^C. - Thomas Ordowski, Mar 09 2015
a(A038664(n)) = 2*n and a(m) != 2*n for m < A038664(n). - Reinhard Zumkeller, Aug 23 2015
If j and k are positive integers then there are no two consecutive primes gaps of the form 2+6j and 2+6k (A016933) or 4+6j and 4+6k (A016957). - Andres Cicuttin, Jul 14 2016
Conjecture: For any positive numbers x and y, there is an index k such that x/y = a(k)/a(k+1). - Andres Cicuttin, Sep 23 2018
Conjecture: For any three positive numbers x, y and j, there is an index k such that x/y = a(k)/a(k+j). - Andres Cicuttin, Sep 29 2018
Conjecture: For any three positive numbers x, y and j, there are infinitely many indices k such that x/y = a(k)/a(k+j). - Andres Cicuttin, Sep 29 2018
Row m of A174349 lists all indices n for which a(n) = 2m. - M. F. Hasler, Oct 26 2018
Since (6a, 6b) is an admissible pattern of gaps for any integers a, b > 0 (and also if other multiples of 6 are inserted in between), the above conjecture follows from the prime k-tuple conjecture which states that any admissible pattern occurs infinitely often (see, e.g., the Caldwell link). This also means that any subsequence a(n .. n+m) with n > 2 (as to exclude the untypical primes 2 and 3) should occur infinitely many times at other starting points n'. - M. F. Hasler, Oct 26 2018
Conjecture: Defining b(n,j,k) as the number of pairs of prime gaps {a(i),a(i+j)} such that i < n, j > 0, and a(i)/a(i+j) = k with k > 0, then
  lim_{n -> oo} b(n,j,k)/b(n,j,1/k) = 1, for any j > 0 and k > 0, and
  lim_{n -> oo} b(n,j,k1)/b(n,j,k2) = C with C = C(j,k1,k2) > 0. - Andres Cicuttin, Sep 01 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 92.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 186-192.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vojtech Strnad, First 100000 terms [First 10000 terms from N. J. A. Sloane]
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Anonymous ["TheHereticAnthem20"], Prime gaps mapped to sounds, Youtube video (2018).
B. Apostol, L. Panaitopol, L Petrescu, and L. Toth, Some Properties of a Sequence Defined with the Aid of Prime Numbers, J. Int. Seq. 18 (2015) # 15.5.5.
S. Ares and M. Castro, Hidden structure in the randomness of the prime number sequence?, arXiv:cond-mat/0310148 [cond-mat.stat-mech], 2003-2005.
József Beck, Inevitable randomness in discrete mathematics, University Lecture Series, 49. American Mathematical Society, Providence, RI, 2009. xii+250 pp. ISBN: 978-0-8218-4756-5; MR2543141 (2010m:60026). See page 7.
Chris K. Caldwell, Prime k-tuple conjecture, Prime Pages' Glossary entry.
Joel E. Cohen and Dexter Senft, Gaps of size 2, 4, and (conditionally) 6 between successive odd composite numbers occur infinitely often, Notes on Number Theory and Discrete Mathematics, Volume 31, Number 3, 494-503 (2025). See p. 495.
Péter L. Erdős, Gergely Harcos, Shubha R. Kharel, Péter Maga, Tamás Róbert Mezei and Zoltán Toroczkai, The sequence of prime gaps is graphic, Mathematische Annalen 388 (2024), 2195-2215.
D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, Small gaps between primes and almost primes, arXiv:math/0506067 [math.NT], 2005.
D. A. Goldston and A. H. Ledoan, On the differences between consecutive prime numbers, I", arXiv:1111.3380v1 [math.NT], Nov 14, 2011.
D. A. Goldston, J. Pintz, and C. Y. Yildirim, Positive Proportion of Small Gaps Between Consecutive Primes, arXiv:1103.3986 [math.NT], Mar 21, 2011.
D. R. Heath-Brown and H. Iwaniec, On the difference between consecutive primes, Bull. Amer. Math. Soc. 1 (1979), 758-760.
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
The Polymath project, Bounded gaps between primes
Carlos Rivera, Conjecture 82. Average of log Dn / log(logPn) equal R = 0,877 08..., The Prime Puzzles & Problems Connection.
Hisanobu Shinya, On the density of prime differences less than a given magnitude which satisfy a certain inequality, arXiv:0809.3458 [math.GM], 2008-2011.
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Eric Weisstein's World of Mathematics, Andrica's Conjecture
Eric Weisstein's World of Mathematics, Prime Difference Function
Yasuo Yamasaki and Aiichi Yamasaki, On the Gap Distribution of Prime Numbers, Kyoto University Research Information Repository, October 1994. MR1370273 (97a:11141).
Yitang Zhang, Bounded gaps between primes, Annals of Mathematics 179 (2014), 1121-1174.
Index entries for primes, gaps between

Cf. A000040 (primes), A001248 (primes squared), A000720, A037201, A007921, A030173, A036263-A036274, A167770, A008347.
Second difference is A036263, first occurrence is A000230.
For records see A005250, A005669.
Cf. A038664, A031131, A031165, A031166, A031167, A031168, A031169, A031170, A031171, A031172.
Cf. A174349, A029707, A029709, A320701, ..., A320720.
Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556-A330561.

a001223 n = a001223_list !! (n-1)
a001223_list = zipWith (-) (tail a000040_list) a000040_list
-- Reinhard Zumkeller, Oct 29 2011

[(NthPrime(n+1) - NthPrime(n)): n in [1..100]]; // Vincenzo Librandi, Apr 02 2011

with(numtheory): for n from 1 to 500 do printf(`%d,`,ithprime(n+1) - ithprime(n)) od:

Differences[Prime[Range[100]]] (* Harvey P. Dale, May 15 2011 *)

diff(v)=vector(#v-1,i,v[i+1]-v[i]);
diff(primes(100)) \\ Charles R Greathouse IV, Feb 11 2011

forprime(p=1, 1e3, print1(nextprime(p+1)-p, ", ")) \\ Felix Fröhlich, Sep 06 2014

from sympy import prime
def A001223(n): return prime(n+1)-prime(n) # Chai Wah Wu, Jul 07 2022

differences(prime_range(1000)) # Joerg Arndt, May 15 2011

G.f.: b(x)*(1-x), where b(x) is the g.f. for the primes. - Franklin T. Adams-Watters, Jun 15 2006
a(n) = prime(n+1) - prime(n). - Franklin T. Adams-Watters, Mar 31 2010
Conjectures: (i) a(n) = ceiling(prime(n)*log(prime(n+1)/prime(n))). (ii) a(n) = floor(prime(n+1)*log(prime(n+1)/prime(n))). (iii) a(n) = floor((prime(n)+prime(n+1))*log(prime(n+1)/prime(n))/2). - Thomas Ordowski, Mar 21 2013
A167770(n) == a(n)^2 (mod A000040(n+1)). - L. Edson Jeffery, Oct 01 2014
a(n) = Sum_{k=1..2^(n+1)-1} (floor(cos^2(Pi*(n+1)^(1/(n+1))/(1+primepi(k))^(1/(n+1))))). - Anthony Browne, May 11 2016
G.f.: (Sum_{k>=1} x^pi(k)) - 1, where pi(k) is the prime counting function. - Benedict W. J. Irwin, Jun 13 2016
Conjecture: Limit_{N->oo} (Sum_{n=2..N} log(a(n))) / (Sum_{n=2..N} log(log(prime(n)))) = 1. - Alain Rocchelli, Dec 16 2022
Conjecture: The asymptotic limit of the average of log(a(n)) ~ log(log(prime(n))) - gamma (where gamma is Euler's constant). Also, for n tending to infinity, the geometric mean of a(n) is equivalent to log(prime(n)) / e^gamma. - Alain Rocchelli, Jan 23 2023
It has been conjectured that primes are distributed around their average spacing in a Poisson distribution (cf. D. A. Goldston in above links). This is the basis of the last two conjectures above. - Alain Rocchelli, Feb 10 2023

More terms from James Sellers, Feb 19 2001

A356224 Number of divisors of n whose prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 1, 4, 1, 2, 1, 5, 1, 2, 1, 5, 1, 4, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 4, 1, 6, 1, 2, 1, 7, 1, 2, 1, 4, 1, 3, 1, 3, 1, 2, 1, 9, 1, 2, 1, 3, 1, 5, 1, 4, 1, 2, 1, 7, 1, 2, 1, 7, 1, 3, 1, 3, 1, 2, 1, 10, 1, 2, 1, 3, 1, 3, 1, 5, 1, 2, 1, 5, 1, 2, 1

Offset: 1

Gus Wiseman, Aug 04 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(n) gapless divisors of n = 1..24:
  1  2  1  4  1  6  1  8  1  2  1  12  1  2  1  16  1  18  1  4  1  2  1  24
     1     2     2     4     1     6      1     8      6      2     1     12
           1     1     2           4            4      2      1           8
                       1           2            2      1                  6
                                   1            1                         4
                                                                          2
                                                                          1
For example, the divisors of 12 are {1,2,3,4,6,12}, of which {1,2,4,6,12} belong to A055932, so a(12) = 5.

These divisors belong to A055932, a subset of A073491 (complement A073492).
The complement is A356225.
A001223 lists the prime gaps.
A328338 has third-largest divisor prime.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A000005, A001222, A028334, A029709, A055874, A056239, A070824, A112798, A119313, A137921, A287170, A289509, A356223.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
Table[Length[Select[Divisors[n],normQ[primeMS[#]]&]],{n,100}]

A174349 Square array: row n gives the indices i for which prime(i+1) = prime(i) + 2n; read by falling antidiagonals.

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 7, 8, 11, 24, 10, 12, 15, 72, 34, 13, 14, 16, 77, 42, 46, 17, 19, 18, 79, 53, 47, 30, 20, 22, 21, 87, 61, 91, 62, 282, 26, 25, 23, 92, 68, 97, 66, 295, 99, 28, 27, 32, 94, 80, 114, 137, 319, 180, 154, 33, 29, 36, 124, 82, 121, 146, 331, 205, 259, 189

Offset: 1

Clark Kimberling, Mar 16 2010

It is conjectured that every positive integer except 1 occurs in the array.
From M. F. Hasler, Oct 19 2018: (Start)
The above conjecture is obviously true: the integer i appears in row (prime(i+1) - prime(i))/2.
Polignac's Conjecture states that all rows are of infinite length.
To ensure the sequence is well-defined in case the conjecture would not hold, we can use the convention that finite rows are continued by 0's. (End)

			Corner of the array:
   2    3    5    7    10    13 ...
   4    6    8   12    14    17 ...
   9   11   15   16    18    21 ...
  24   72   77   79    87    92 ...
  34   42   53   61    68    80 ...
  46   47   91   97   114   121 ...
  (...)
Row 1: p(2) = 3, p(3) = 5, p(5) = 11, p(7) = 17, ..., these being the primes for which the next prime is 2 greater, cf. A029707.
Row 2: p(4) = 7, p(6) = 13, p(8) = 19, ..., these being the primes for which the next prime is 4 greater, cf. A029709.

T. D. Noe, Falling antidiagonals 1..50 of the array, flattened
Fred B. Holt and Helgi Rudd, On Polignac's Conjecture, arxiv:1402.1970 [math.NT], 2014.
Index entries for primes, gaps between.

Cf. A000040, A000720, A001223, A174350.
Rows 1, 2, 3, ... are A029707, A029709, A320701, ..., A320720; A116493 (row 35), A116496 (row 50), A116497 (row 100), A116495 (row 105).
Column 1 is A038664.

rows = 10; t2 = {}; Do[t = {}; p = Prime[2]; While[Length[t] < rows - off + 1, nextP = NextPrime[p]; If[nextP - p == 2*off, AppendTo[t, p]]; p = nextP]; AppendTo[t2, t], {off, rows}]; t3 = Table[t2[[b, a - b + 1]], {a, rows}, {b, a}]; PrimePi /@ t3 (* T. D. Noe, Feb 11 2014 *)

a(n) = A000720(A174350(n)). - Michel Marcus, Mar 30 2016

Name corrected and other edits by M. F. Hasler, Oct 19 2018

A320701 Indices of primes followed by a gap (distance to next larger prime) of 6.

Original entry on oeis.org

9, 11, 15, 16, 18, 21, 23, 32, 36, 37, 39, 40, 51, 54, 55, 56, 58, 67, 71, 73, 74, 76, 84, 86, 96, 100, 102, 103, 105, 107, 108, 110, 111, 118, 119, 123, 129, 130, 133, 160, 161, 164, 165, 167, 170, 174, 179, 184, 185, 187, 188, 194, 195, 199, 200, 202, 208, 210, 216, 218, 219, 227, 231

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes given in A031924.

Subsequence of indices of sexy primes A023201.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between.

Equals A000720 o A031924.
Row 3 of A174349.
Indices of 6's in A001223.
Cf. A029707, A029709, A320702, A320703, ..., A320720 (analog for gaps 2, 4, 8, 10, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

Position[Differences[Prime[Range[250]]],6]//Flatten (* Harvey P. Dale, Oct 13 2022 *)

A(N=100,g=6,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031924(n)).

A320701 = { i > 0 | prime(i+1) = prime(i) + 6 } = A001223^(-1)({6}).

A320702 Indices of primes followed by a gap (distance to next larger prime) of 8.

Original entry on oeis.org

24, 72, 77, 79, 87, 92, 94, 124, 126, 128, 132, 135, 156, 158, 166, 186, 192, 196, 220, 228, 241, 246, 248, 270, 281, 299, 304, 325, 330, 334, 338, 364, 370, 379, 386, 393, 400, 413, 417, 421, 432, 436, 454, 456, 482, 488, 507, 517, 519, 538, 589, 594, 620, 640, 661, 676, 689, 691, 712, 736, 750, 759

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes given in A031926.

Vincenzo Librandi, Table of n, a(n) for n = 1..10340
Index entries for primes, gaps between

Equals A000720 o A031926.
Row 4 of A174349.
Indices of 8's in A001223.
Cf. A029707, A029709, A320701, A320703, ..., A320720 (analog for gaps 2, 4, 6, 10, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..800] | NthPrime(n+1) - NthPrime(n) eq 8]; // Vincenzo Librandi, Mar 21 2019

p:= 2: Res:= NULL: count:= 0:
for n from 1 while count < 100 do
  q:= nextprime(p);
  if q-p = 8 then count:= count+1; Res:= Res, n; fi;
  p:= q;
od:
Res; # Robert Israel, Oct 19 2018

Select[Range[800], Prime[#] + 8 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 21 2019 *)

A_vec(N=100,g=8,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)}

a(n) = A000720(A031926(n)) = A174349(4,n).

A320702 = { i > 0 | prime(i+1) = prime(i) + 8 } = A001223^(-1)({8}).

A320720 Indices of primes followed by a gap (distance to next larger prime) of 44.

Original entry on oeis.org

1831, 3861, 4009, 7499, 8937, 10328, 10427, 11725, 12904, 12926, 13011, 13051, 16596, 16915, 18280, 20055, 20160, 20352, 20619, 21458, 21465, 21550, 21659, 23752, 23934, 24107, 24384, 24445, 24651, 24871, 24933, 24992, 25027, 26089, 26166, 26483, 26923, 27038, 27048, 28898, 29343

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A134121.

Index entries for primes, gaps between.

Cf. A029707, A029709 (analog for gaps 2 and 4), A320701, A320702, ... A320719 (analog for gaps 6, 8, 10, ..., 42), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).
Equals A000720 o A134121.
Indices of 44's in A001223.
Row 22 of A174349.

A(N=100,g=44,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A134121(n)).

A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 2, 6, 2, 3, 4, 5, 2, 6, 2, 4, 3, 3, 2, 8, 3, 3, 4, 4, 2, 7, 2, 6, 3, 3, 4, 9, 2, 3, 3, 5, 2, 5, 2, 4, 6, 3, 2, 10, 3, 4, 3, 4, 2, 8, 3, 5, 3, 3, 2, 10, 2, 3, 4, 7, 3, 5, 2, 4, 3, 5, 2, 12, 2, 3, 6, 4, 4, 5, 2, 6, 5, 3, 2, 7, 3, 3

Offset: 1

Gus Wiseman, Aug 28 2022

nonn

First differs from A000005 at 10, 14, 20, 21, 22, ... = A307516.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(n) counted divisors of n = 1, 2, 4, 6, 12, 16, 24, 30, 36, 48, 72, 90:
  1   2   4   6  12  16  24  30  36  48  72  90
      1   2   3   6   8  12  15  18  24  36  45
          1   2   4   4   8   6  12  16  24  30
              1   3   2   6   5   9  12  18  18
                  2   1   4   3   6   8  12  15
                  1       3   2   4   6   9   9
                          2   1   3   4   8   6
                          1       2   3   6   5
                                  1   2   4   3
                                      1   3   2
                                          2   1
                                          1

These divisors belong to A073491, a superset of A055932, complement A073492.
The initial case is A356224.
The complement in the initial case is counted by A356225.
A000005 counts divisors.
A001223 lists the prime gaps.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A328338 has third-largest divisor prime.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A028334, A029709, A055874, A070824, A119313, A137921, A287170, A307516, A356223, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
Table[Length[Select[Divisors[n],nogapQ[primeMS[#]]&]],{n,100}]

A320703 Indices of primes followed by a gap (distance to next larger prime) of 10.

Original entry on oeis.org

34, 42, 53, 61, 68, 80, 82, 101, 106, 115, 125, 127, 138, 141, 145, 157, 172, 175, 177, 191, 193, 204, 222, 233, 258, 266, 269, 279, 289, 306, 308, 310, 316, 324, 369, 383, 397, 399, 403, 418, 422, 431, 443, 474, 491, 497, 500, 502, 518, 525, 531, 535, 575

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes given in A031928.

Vincenzo Librandi, Table of n, a(n) for n = 1..8870
Index entries for primes, gaps between.

Equals A000720 o A031928.
Row 5 of A174349.
Indices of 10's in A001223.
Subsequence of A107730: prime(n+1) ends in same digit as prime(n).
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..1000] | NthPrime(n+1) - NthPrime(n) eq 10]; // Vincenzo Librandi, Mar 21 2019

Select[Range[1000], Prime[#] + 10 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 21 2019 *)
Position[Partition[Prime[Range[600]],2,1],?(#[[2]]-#[[1]]==10&),1,Heads-> False]//Flatten (* _Harvey P. Dale, Mar 08 2020 *)

A320703_vec(N=100,g=10,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031928(n)).

A320703 = { i > 0 | prime(i+1) = prime(i) + 10 }.

A375712 Numbers k such that A013929(k+1) - A013929(k) = 4. In other words, the k-th nonsquarefree number is 4 less than the next nonsquarefree number.

Original entry on oeis.org

1, 4, 7, 11, 12, 13, 14, 22, 25, 26, 29, 32, 35, 39, 40, 41, 42, 50, 53, 54, 61, 64, 70, 71, 72, 75, 78, 81, 82, 83, 84, 87, 90, 98, 99, 102, 109, 110, 117, 120, 123, 124, 127, 135, 139, 140, 144, 151, 154, 155, 156, 157, 160, 163, 168, 169, 170, 173, 176, 179

Offset: 1

Gus Wiseman, Sep 09 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 4 after the first, fourth, and seventh terms.

For prime numbers we have A029709.
Positions of 4's in A078147.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A014689, A029707, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A375927.

Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&]],4]

Complement of A375709 U A375710 U A375711.

A356223 Position of n-th appearance of 2n in the sequence of prime gaps (A001223). If 2n does not appear at least n times, set a(n) = -1.

Original entry on oeis.org

2, 6, 15, 79, 68, 121, 162, 445, 416, 971, 836, 987, 2888, 1891, 1650, 5637, 5518, 4834, 9237, 8152, 10045, 21550, 20248, 20179, 29914, 36070, 24237, 53355, 52873, 34206, 103134, 90190, 63755, 147861, 98103, 117467, 209102, 206423, 124954, 237847, 369223

Offset: 1

Gus Wiseman, Aug 04 2022

nonn

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

			We need the first 15 prime gaps (1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6) before we reach the 3rd appearance of 6, so a(6) = 15.

The first appearances are at A038664, seconds A356221.
Diagonal of A356222.
A001223 lists the prime gaps.
A073491 lists numbers with gapless prime indices.
A356224 counts divisors with gapless prime indices, complement A356225.
A356226 = gapless interval lengths of prime indices, run-lengths A287170.
Cf. A000005, A028334, A029709, A060681, A119313, A137921, A193829, A274121.

nn=1000;
gaps=Differences[Array[Prime,nn]];
Table[Position[gaps,2*n][[n,1]],{n,Select[Range[nn],Length[Position[gaps,2*#]]>=#&]}]

cp's OEIS Frontend

A001223 Prime gaps: differences between consecutive primes.

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A356224 Number of divisors of n whose prime indices cover an initial interval of positive integers.

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A174349 Square array: row n gives the indices i for which prime(i+1) = prime(i) + 2n; read by falling antidiagonals.

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A320701 Indices of primes followed by a gap (distance to next larger prime) of 6.

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A320702 Indices of primes followed by a gap (distance to next larger prime) of 8.

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A320720 Indices of primes followed by a gap (distance to next larger prime) of 44.

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A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

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