A006988 -id:A006988 - cp's OEIS frontend

A110896 Difference between prime(10^n) and prime(10^(n-1)).

27, 512, 7378, 96810, 1194980, 14186154, 163938810, 1858650070, 20763688746, 229296037134, 2508629501894, 27235496973316, 293784284670498, 3151605249578196, 33649122286540910, 357782405858669892, 3790389667563960340, 40026493653364473662, 421463674881774896218

Offset: 1

Cino Hilliard, Sep 20 2005

nonn

			The (10^1)th prime is 29. The (10^0)th prime is 2 and the difference is 27 the first entry.

Amiram Eldar, Table of n, a(n) for n = 1..24

Cf. A000040, A006988, A135371, A135373.

Table[Prime[10^n] - Prime[10^(n - 1)], {n, 1, 12}] (* Marvin Ray Burns, Dec 09 2007 *)
Differences[Table[Prime[10^n],{n,0,12}]] (* Harvey P. Dale, Jul 03 2017 *)

a(n) = prime(10^n) - prime(10^(n-1)); \\ Jinyuan Wang, Nov 14 2020; adapted to offset by Georg Fischer, Jun 22 2022

from sympy import prime
def a(n): return prime(10**n)-prime(10**(n-1))
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Jun 22 2022

a(n) = A006988(n) - A006988(n-1) = prime(10^n) - prime(10^(n-1)). - Marvin Ray Burns, Dec 09 2007; corrected by Michel Marcus, Jun 22 2022

Corrected and extended by Marvin Ray Burns, Dec 09 2007

More terms from Jinyuan Wang, Nov 14 2020

A120044 The 10^n-th 3-almost prime.

Original entry on oeis.org

8, 45, 412, 3918, 38991, 395085, 4046429, 41657362, 429891626, 4439956573, 45851698382, 473238120286, 4880292241955, 50280826966354

Offset: 0

Robert G. Wilson v, Feb 15 2006

Cf. A109251, A006988, A114125, A120045, A120046.

ThreeAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi@ Sqrt[n/Prime@i]}]; ThreeAlmostPrime[n_] := Block[{e = Floor[Log[2, n]], a, b}, a = 2^e; Do[b = 2^p; While[ThreeAlmostPrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Do[ Print@ThreeAlmostPrime[10^n], {n, 0, 13}]
ThreePrime[n_] := Block[{e = Floor[ Log[2, n] +2], a, b}, a = 2^e; Do[b = 2^p; While[ ThreePrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Table[ ThreePrime[n], {n, 0, 13}]

A120045 The (10^n)-th 4-almost prime.

Original entry on oeis.org

16, 88, 693, 5958, 54328, 511725, 4922511, 47997635, 472514554, 4683086217, 46636297326, 466032880556

Offset: 0

Robert G. Wilson v, Feb 15 2006

Cf. A006988, A014613, A114106, A114125, A120044, A120046.

FourAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@Sqrt[n/(Prime@i*Prime@j)]}];
FourAlmostPrime[n_] := Block[{e = Floor[Log[2, n] +3], a, b}, a = 2^e; Do[b = 2^p; While[FourAlmostPrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Do[ Print@FourAlmostPrime[10^n], {n, 0, 11}]

A120046 The 10^n-th 5-almost prime.

Original entry on oeis.org

32, 176, 1272, 10374, 89896, 810220, 7475818, 70185558, 667561977, 6411296283, 62037096770, 603813941738

Offset: 0

Robert G. Wilson v, Feb 15 2006

Cf. A014614, A114453, A006988, A114125, A120044, A120045.

FiveAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k*Prime@l)] - l + 1, {i, PrimePi[n^(1/5)]}, {j, i, PrimePi[(n/Prime@i)^(1/4)]}, {k, j, PrimePi[(n/(Prime@i*Prime@j)^(1/3))]}, {l, k, PrimePi@Sqrt[(n/(Prime@i*Prime@j*Prime@k))]}];
FiveAlmostPrime[n_] := Block[{e = Floor[Log[2, n] +4], a, b}, a = 2^e; Do[b = 2^p; While[FiveAlmostPrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Do[ Print@FiveAlmostPrime[10^n], {n, 0, 13}]

lista(nmax) = {my(pow = 1, c = 0, n = 0); for(k = 1, oo, if(bigomega(k) == 5, c++; if(c == pow, print1(k, ", "); if(n == nmax, break); pow *= 10; n++)));} \\ Amiram Eldar, Apr 29 2024

a(n) = A014614(10^n). - Amiram Eldar, Apr 29 2024

a(6) corrected and a(7)-a(9) added by Amiram Eldar, Apr 29 2024

a(10)-a(11) from David A. Corneth, Apr 29 2024

A134182 Difference between the sums of the first 10^n odd primes and the first 10^n odd positive integers > 1.

Original entry on oeis.org

38, 14478, 2688838, 396250152, 52261798440, 6472980453364, 770530574266708, 89262852894258444, 10138479465982004008, 1134379338819040693132, 125436174619351016716668, 13738971133578180130155676, 1493061976858459711065006050, 161191473337955042966337346114

Offset: 1

Enoch Haga, Oct 13 2007

nonn

Original name: 10^n-th difference between cumulative prime and odd sums.

Beginning at 3, compute the sums of the prime and odd sequences at 10^n and take the difference.

			a(1) = (3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31) - (3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21) = 158 - 120 = 38.
a(2) = 14478 because at 10^2, 100 sums of primes and odds, the prime sum is 24678, the odd sum is 10200 and the difference is 14478.

Chai Wah Wu, Table of n, a(n) for n = 1..23 (using terms from A006988 and A099824)

Cf. A006988, A071148, A005563, A099824, A134181.

10 N=1: A=2
20 A=nxtprm(A): B=B+A
30 N=N+2: D=D+N
40 if C=9 then print A;N;B;D;B-D: stop
50 C=C+1: if C<10 then 20

a(n) = A134181(10^n).
a(n) = A099824(n) + prime(10^n+1) - (10^n*(10^n+2)) - 2. - Chai Wah Wu, Mar 30 2020
a(n) = A071148(10^n) - (10^n+1)^2 + 1, where A071148 are the partial sums of odd primes, and N^2 is the sum of the first N odd integers. - M. F. Hasler, Aug 08 2025

Edited by M. F. Hasler, Aug 08 2025

A049035 Number of pairs of twin primes whose smaller element is <= 10^n-th prime.

Original entry on oeis.org

5, 25, 174, 1270, 10250, 86027, 738597, 6497407, 58047180, 524733511, 4789919653, 44073509102, 408231310520

Offset: 1

Dennis S. Kluk (mathemagician(AT)ameritech.net)

a(0) = 0. - Eduard Roure Perdices, Dec 23 2022

			a(1) = 5 since the 10th prime is 29 and the first 5 twin primes are {3,5}, {5,7}, {11,13}, {17,19} and {29,31}.

Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See Table 5 p. 40.
Index entries for sequences related to numbers of primes in various ranges

See A093683 for another version.

Cf. A001359, A006988.

from sympy import prime, sieve # use primerange for larger terms
def afind(terms):
  c, prevp = 0, 1
  for n in range(1, terms+1):
    for p in sieve.primerange(prevp+1, prime(10**n)+3):
      if prevp == p - 2: c += 1
      prevp = p
    print(c, end=", ")
afind(6) # Michael S. Branicky, Apr 25 2021

More terms from Labos Elemer, Oct 13 2000; Jud McCranie, Dec 30 2000
a(9) from Michael S. Branicky, Apr 25 2021
a(10) from Eduard Roure Perdices, May 08 2021
a(11) from Eduard Roure Perdices, Feb 03 2022
a(12) from Eduard Roure Perdices, Dec 23 2022
a(13) from Eduard Roure Perdices, Jan 24 2024

A049040 Number of Sophie Germain primes <= prime(10^n).

Original entry on oeis.org

1, 6, 26, 167, 1222, 9668, 82237, 711154, 6284076, 56325540, 510628554, 4672267109

Offset: 0

Dennis S. Kluk (mathemagician(AT)ameritech.net)

Sophie Germain primes are primes p such that 2p+1 is also prime.

			The first 10 primes are 2,3,5,7,11,13,17,23,29 and 31. 6 of these are Sophie Germain primes, namely: 2,3,5,11,23 and 29.

Index entries for sequences related to numbers of primes in various ranges.

Cf. A005384, A006988.

cnt = 0; currentPrime = 1; For[ i = 1, i == i, i ++, currentPrime = NextPrime[ currentPrime ]; If[ PrimeQ[ 2*currentPrime + 1 ], cnt++ ]; If[ IntegerQ[ Log[ 10, i ] ], Print[ cnt ] ]; ]

list(len) = {my(c = 0, k = 0, pow = 1, pmax = prime(pow)); forprime(p = 1, , if(isprime(2*p+1), c++); if(p == pmax, print1(c, ", "); k++; if(k == len, break); pow *= 10; pmax = prime(pow)));} \\ Amiram Eldar, Jul 25 2025

More terms from Alexander D. Healy, Mar 19 2001

Offset changed to 0, a(0) prepended and a(9)-a(11) added by Amiram Eldar, Jul 25 2025

A049386 Binary order of 2^n-th prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 0

Labos Elemer

nonn

			The 549755813888th = (2^39)th prime is 16149760533341, whose binary order is 44: it is ceiling(43.87657801)=44, so a(39)=44;
a(0)=1 is the binary order of the (2^0)th = 1st prime (= 2), which is log_2(2) = 1.

Jinyuan Wang, Table of n, a(n) for n = 0..78

Cf. A006988, A029837, A033844.

a(n) = ceil(log(prime(2^n))/log(2)); \\ Michel Marcus, Aug 07 2021

a(n) = A029837(A033844(n)).

a(n) = ceiling(log_2(prime(2^n))).

a(40)-a(57) from Michel Marcus, Aug 25 2019

More terms from Jinyuan Wang, Aug 07 2021

A074383 Difference between (1+10^n)-th and (10^n)-th primes.

Original entry on oeis.org

1, 2, 6, 8, 14, 12, 4, 18, 8, 24, 6, 42, 18, 48, 60, 52, 48, 8, 22, 20, 60, 38, 54, 48, 58

Offset: 0

Labos Elemer, Aug 22 2002

			a(18)=22 because the (10^18)th prime is 44211790234832169331, the (1+10^18)th prime is 44211790234832169353.

Cf. A006988, A074325, A038833, A074382.

a(n) = prime(1+10^n) - prime(10^n).

a(n) = A151800(A006988(n)) - A006988(n).

a(19) from Max Alekseyev, May 08 2009
a(20)-a(22) from Max Alekseyev, Dec 04 2014
a(23)-a(24) from Chai Wah Wu using the terms in A006988, Sep 18 2018

A092251 Lesser of the smallest twin prime pair on or after the 10^n-th prime.

Original entry on oeis.org

3, 29, 569, 7949, 104759, 1299917, 15486257, 179424797, 2038074947, 22801763729, 252097801757, 2760727302647, 29996224276019, 323780508946379, 3475385758524917, 37124508045066647, 394906913903736539, 4185296581467696419, 44211790234832169821, 465675465116607069659

Offset: 0

Cino Hilliard, Feb 19 2004

nonn

			a(1) = 29 since the 10th prime is 29 and (29, 31) is the first twin prime pair on or after the 10th prime.

Amiram Eldar, Table of n, a(n) for n = 0..24

Cf. A006988.

More terms from Amiram Eldar, Jan 01 2020

cp's OEIS Frontend

A110896 Difference between prime(10^n) and prime(10^(n-1)).

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A120044 The 10^n-th 3-almost prime.

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A120045 The (10^n)-th 4-almost prime.

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A120046 The 10^n-th 5-almost prime.

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A134182 Difference between the sums of the first 10^n odd primes and the first 10^n odd positive integers > 1.

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UBASIC

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A049035 Number of pairs of twin primes whose smaller element is <= 10^n-th prime.

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Python

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A049040 Number of Sophie Germain primes <= prime(10^n).

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A049386 Binary order of 2^n-th prime.

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