Search: a104842 -id:a104842
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A104824
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Primes from merging of 4 successive digits in decimal expansion of Pi.
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+10
31
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4159, 5897, 9323, 8419, 1693, 8209, 9749, 5923, 2089, 2803, 4211, 7253, 8111, 1117, 7019, 193, 8521, 6229, 1097, 6659, 8233, 7867, 1201, 9091, 5669, 4603, 4861, 3607, 4127, 631, 5881, 5209, 9209, 4091, 3643, 5903, 11, 113, 6521, 1511, 1609, 9433
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OFFSET
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1,1
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COMMENTS
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LINKS
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MATHEMATICA
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With[{pi = FromDigits/@Partition[RealDigits[Pi, 10, 500][[1]], 4, 1]}, Select[pi, PrimeQ]] (* Vincenzo Librandi, Apr 21 2013 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A104826
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Primes from merging of 6 successive digits in decimal expansion of Pi.
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+10
31
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314159, 358979, 589793, 462643, 971693, 169399, 592307, 348253, 534211, 808651, 844609, 822317, 725359, 502841, 102701, 288109, 612847, 337867, 104543, 815209, 925409, 917153, 665213, 951941
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OFFSET
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1,1
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COMMENTS
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Leading zeros are not permitted, so each term is 6 digits in length. - Harvey P. Dale, Oct 23 2011
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LINKS
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MATHEMATICA
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With[{len=6}, FromDigits/@Select[Partition[RealDigits[Pi, 10, 1000][[1]], len, 1], PrimeQ[FromDigits[#]]&&IntegerLength[FromDigits[#]]==len&]] (* Harvey P. Dale, Oct 23 2011 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A104825
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Primes from merging of 5 successive digits in decimal expansion of Pi.
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+10
24
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14159, 35897, 58979, 38327, 97169, 71693, 39937, 9749, 30781, 20899, 34211, 64709, 47093, 82231, 84811, 46229, 81097, 56659, 66593, 86783, 85669, 66923, 34603, 93607, 60631, 9209, 25409, 54091, 25903, 113, 33053, 65213, 13841, 51941, 94151
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OFFSET
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1,1
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COMMENTS
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Because leading zeros are permitted, some terms have fewer than 5 digits. - Harvey P. Dale, Mar 01 2023
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LINKS
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MATHEMATICA
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With[{pi=FromDigits/@Partition[RealDigits[Pi, 10, 2000][[1]], 5, 1]}, Select[pi, PrimeQ]] (* Harvey P. Dale, Oct 18 2011 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A104841
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The first n-digit prime occurring in the decimal expansion of Pi, A000796.
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+10
7
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3, 31, 653, 4159, 14159, 314159, 1592653, 28841971, 795028841, 5926535897, 93238462643, 141592653589, 9265358979323, 23846264338327, 841971693993751, 8628034825342117, 89793238462643383, 348253421170679821, 3832795028841971693, 89793238462643383279
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OFFSET
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1,1
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COMMENTS
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For the next n-digit primes, see the b-file (link).
Sequence A198344 gives the position of these primes withing the digits of Pi.
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LINKS
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PROG
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(PARI) default(realprecision, 2000); A104841(n)={for( c=0, default(realprecision)-n-2, Pi\.1^c%10 & ispseudoprime(p=Pi\.1^(n+c-1)%10^n) & return(p)); error("Please increase default(realprecision) to calculate A104841("n").")} \\ M. F. Hasler, Oct 23 2011
(Python)
from sympy import S, isprime
pi = "3"+str(S.Pi.n(10**5))[2:] # or load data from file
def A104841_A198344(n): return next(((p, i+1) for i in range(len(pi)-n) if pi[i]!="0" and isprime(p:=int(pi[i:i+n]))), "not enough digits")
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CROSSREFS
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KEYWORD
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nonn,base
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STATUS
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approved
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A198344
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Position of the first n-digit prime occurring in the decimal expansion of Pi, A000796.
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+10
4
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1, 1, 8, 3, 2, 1, 4, 34, 30, 5, 15, 2, 6, 17, 36, 82, 12, 87, 26, 12, 25, 215, 35, 18, 17, 3, 41, 17, 234, 17, 167, 92, 251, 15, 9, 12, 31, 1, 57, 290, 4, 99, 218, 502, 48, 164, 198, 201, 128, 7, 363, 143, 11, 138, 487, 32, 230, 82, 355, 515, 334, 186, 176, 223
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OFFSET
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1,3
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COMMENTS
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Differs from A104842 in a(22), a(43), a(55),..., because here, leading zeros are not allowed.
The corresponding primes are listed in A104841.
Among the first 99 terms, even though values up to 825 occur, the values 1 and 17 occur 4 times, 12 and 57 occur 3 times, and numbers as large as 82, 164, 167 and 234 occur twice.
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LINKS
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EXAMPLE
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a(1)=1 because the initial digit "3" of Pi is prime.
a(2)=a(6)=a(38)=1 because the first 2, 6, and 38 digits of Pi (including the initial 3) also form the primes 31, 314159 and 31415926535897932384626433832795028841, cf. A005042 and A060421.
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MATHEMATICA
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With[{pd=RealDigits[Pi, 10, 1000][[1]]}, Table[Position[Partition[pd, n, 1], _?(PrimeQ[FromDigits[#]]&&#[[1]]!=0&), {1}, 1, Heads->False], {n, 60}]]// Flatten (* Harvey P. Dale, Apr 25 2016 *)
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PROG
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(PARI) A198344(n)=for(c=0, 9e9, ispseudoprime(Pi\.1^(n+c-1)%10^n)&Pi\.1^c%10&return(c+1)) /* Replace upper limit 9e9 by "default(realprecision)-n" to avoid an error message and return 0 in case no n-digit prime is found */
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A104830
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Primes from merging of 10 successive digits in decimal expansion of Pi.
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+10
3
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5926535897, 4197169399, 1693993751, 7510582097, 348253421, 4825342117, 5822317253, 812848111, 2841027019, 8521105559, 8954930381, 4756482337, 2712019091, 5432664821, 3266482133, 6072602491, 5588174881, 8815209209
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OFFSET
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1,1
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COMMENTS
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Leading zeros are permitted, so some terms are less than 10 digits in length. - Bruno Berselli, May 01 2013
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LINKS
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MATHEMATICA
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With[{pi = FromDigits/@Partition[RealDigits[Pi, 10, 500][[1]], 10, 1]}, Select[pi, PrimeQ]] (* Vincenzo Librandi, Apr 21 2013 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A104823
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Primes from merging of three successive digits in decimal expansion of Pi.
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+10
1
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653, 643, 433, 383, 419, 197, 971, 937, 751, 97, 307, 89, 421, 211, 67, 821, 823, 647, 709, 223, 317, 359, 811, 701, 19, 193, 521, 211, 229, 881, 109, 97, 659, 593, 461, 823, 233, 337, 271, 19, 821, 607, 491, 127
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graph;
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listen;
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OFFSET
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1,1
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COMMENTS
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Leading zeros are permitted, so not every term is 3 digits in length. - Harvey P. Dale, Oct 24 2011
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LINKS
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MATHEMATICA
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With[{pi = FromDigits/@Partition[RealDigits[Pi, 10, 500][[1]], 3, 1]}, Select[pi, PrimeQ]] (* Vincenzo Librandi, Apr 23 2013 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A104827
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Primes from merging of 7 successive digits in decimal expansion of Pi.
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+10
1
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1592653, 6535897, 2643383, 5028841, 6939937, 3993751, 348253, 1170679, 8086513, 5822317, 1725359, 4930381, 2881097, 4612847, 3165271, 2712019, 1201909, 4914127, 917153, 1133053, 3841469, 1469519, 6951941, 9433057
(list;
graph;
refs;
listen;
history;
text;
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OFFSET
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1,1
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COMMENTS
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Leading zeros are permitted so some terms may have fewer than 7 digits. For example, 223, 907, 3137, and 4229 are all terms. - Harvey P. Dale, May 24 2020
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LINKS
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MATHEMATICA
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With[{pi = FromDigits/@Partition[RealDigits[Pi, 10, 500][[1]], 7, 1]}, Select[pi, PrimeQ]] (* Vincenzo Librandi, Apr 21 2013 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A104828
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Primes from merging of 8 successive digits in decimal expansion of Pi.
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+10
1
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28841971, 41971693, 82534211, 42117067, 30664709, 5822317, 31725359, 49303819, 75648233, 37867831, 71201909, 48566923, 26648213, 13393607, 25409171, 1133053, 9433057
(list;
graph;
refs;
listen;
history;
text;
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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With[{pi = FromDigits/@Partition[RealDigits[Pi, 10, 500][[1]], 8, 1]}, Select[pi, PrimeQ]] (* Vincenzo Librandi, Apr 21 2013 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A104829
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Primes from merging of 9 successive digits in decimal expansion of Pi.
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+10
1
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795028841, 502884197, 28841971, 884197169, 971693993, 348253421, 421170679, 306647093, 812848111, 659334461, 233786783, 648566923, 346034861, 326648213, 829254091, 678925903, 1133053, 959195309, 530921861, 938183011
(list;
graph;
refs;
listen;
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text;
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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With[{pi = FromDigits/@Partition[RealDigits[Pi, 10, 500][[1]], 9, 1]}, Select[pi, PrimeQ]] (* Vincenzo Librandi, Apr 21 2013 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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