A198784 -id:A198784 - cp's OEIS frontend

A198161 Primes from merging of 10 successive digits in decimal expansion of sqrt(2).

4142135623, 8872420969, 9698078569, 7537694807, 7973799073, 7846210703, 2644121497, 9935831413, 6592750559, 7010955997, 1472851741, 5251407989, 2533965463, 5339654633, 6152583523, 1525835239, 3950547457, 5750287759, 5996172983, 4084988471, 6668713013

Offset: 1

Harvey P. Dale, Oct 21 2011

Leading zeros are not permitted, so each term is 10 digits in length.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

For sqrt(2), see also A198162, A198163, A198164, A198165,A198166, A198167, A198168, A198169, A198161 (this sequence).
For e, see A104843, A104844, A104845, A104846, A104847, A104848, A104849, A104850, A104851.
For Pi, see A198175, A198170, A104824, A104825, A104826, A198171, A198172, A198173, A198174.
For the Golden Ratio, see A198177, A103773, A103789, A103793, A103808, A103809, A103810, A103811, A103812.
For the Euler-Mascheroni constant gamma, see A198776, A198777, A198778, A198779, A198780, A198781, A198782, A198783, A198784.

With[{len=10},Select[FromDigits/@Partition[RealDigits[Sqrt[2],10,1000][[1]],len,1],IntegerLength[#]==len&&PrimeQ[#]&]]

A198161(n, x=sqrt(2), m=10, silent=0)={m=10^m; for(k=1, default(realprecision), (isprime(p=x\.1^k%m)&&p*10>m)||next; silent||print1(p", "); n--||return(p))} \\ The optional arguments can be used to produce other sequences of this series (cf. Crossrefs). Use e.g. \p999 to set precision to 999 digits. - M. F. Hasler, Nov 02 2014

A198177 10-digit primes found in the decimal expansion of the Golden Ratio phi, in the order of occurrence.

Original entry on oeis.org

1772030917, 4189391137, 6222353693, 7931800607, 5959395829, 5829056383, 3832266131, 6131992829, 6892501711, 9250171169, 1043216269, 3136144381, 7587012203, 7954454749, 8509874339, 4487706647, 1240076521, 7780531531, 5315317141, 1704666599, 7046665991

Offset: 1

Harvey P. Dale, Oct 21 2011

Leading zeros are not permitted, so each term is 10 digits in length.

The sequence A103752 has erroneously the same definition; the actual definition of the terms is unknown. - M. F. Hasler, Nov 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A103773, A103789, A103793, A103808, A103809, A103810, A103811, A103812; A103752.

See also, for e: A104843, A104844, A104845, A104846, A104847, A104848, A104849, A104850, A104851; for Pi: A198175, A198170, A104824, A104825, A104826, A198171, A198172, A198173, A198174; for sqrt(2): A198162, A198163, A198164, A198165, A198166, A198167, A198168, A198169, A198161; for the Euler-Mascheroni constant gamma: A198776, A198777, A198778, A198779, A198780, A198781, A198782, A198783, A198784.

With[{len=10},Select[FromDigits/@Partition[RealDigits[GoldenRatio,10,1000][[1]],len,1],IntegerLength[#]==len&&PrimeQ[#]&]]

default(realprecision,N=1000);m=10^10;phi=sqrt(5/4)+.5;for(k=9,N,isprime(phi\.1^k%m)||next;(p=phi\.1^k%m)>10^9&&print1(p",")) \\ M. F. Hasler, Oct 31 2014

A103808 Primes from merging of 6 successive digits in decimal expansion of the Golden Ratio; (1+sqrt(5))/2.

Original entry on oeis.org

339887, 458683, 638117, 628189, 902449, 418939, 189391, 386891, 235369, 693179, 607667, 595939, 613199, 171169, 631361, 497587, 864449, 987433, 544877, 647809, 217057, 705751, 427621, 410117, 666599, 979873, 731761, 874807, 530567, 228911

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 29 2005

Leading zeros are not permitted, so each term is 6 digits in length. - Harvey P. Dale, Oct 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.
Eric Weisstein's World of Mathematics, The Golden Ratio.
Expansion of the Golden Ratio to 20,000 digits as part of project Gutenberg.

Cf. A198177, A103773, A103789, A103793, A103808 (this), A103809, A103810, A103811, A103812; A103752.

See also, for e: A104843, A104844, A104845, A104846, A104847, A104848, A104849, A104850, A104851; for Pi: A104824, A104825, A104826, A198170, A198171, A198172, A198173, A198175; for sqrt(2): A198161, A198162, A198163, A198164, A198165, A198166, A198167, A198168, A198169; for the Euler-Mascheroni constant gamma: A198776, A198777, A198778, A198779, A198780, A198781, A198782, A198783, A198784 and A104944.

With[{len=6},FromDigits/@Select[Partition[RealDigits[GoldenRatio,10, 1000][[1]],len,1],PrimeQ[FromDigits[#]] &&IntegerLength[ FromDigits[#]] ==len&]] (* Harvey P. Dale, Oct 23 2011 *)

A103808(n,x=(sqrt(5)+1)/2, m=6,silent=0)={m=10^m; for(k=1, default(realprecision), (isprime(p=x\.1^k%m)&&p*10>m)||next;silent||print1(p", ");n--||return(p))} \\ The optional arguments can be used to produce other sequences of this series (cf. Crossrefs). Use, e.g., \p999 to set precision to 999 digits. - M. F. Hasler, Nov 01 2014

Offset changed from 0 to 1 by Vincenzo Librandi, Apr 22 2013

A104851 Primes from merging of 10 successive digits in decimal expansion of e.

Original entry on oeis.org

7427466391, 7413596629, 6059563073, 3490763233, 2988075319, 1573834187, 7021540891, 5408914993, 6480016847, 9920695517, 1838606261, 6062613313, 3845830007, 1692836819, 4425056953, 2505695369, 5490598793, 1782154249, 8215424999, 9229576351, 9519366803

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 27 2005

Scan decimal expansion of e from left to right, recording any 10-digit primes seen. - N. J. A. Sloane, Feb 05 2012
All the primes listed here must have 10 digits, i.e., "leading zeros are not allowed". Otherwise, one would also have some terms as 297606737 or 865746377 or 98793127 from A104850. - M. F. Hasler, Nov 01 2014
The original version read (1185790117, 1180978417, 1573834187, 1838606261, 1308008771, 1692836819, 1782154249, 1825288693, 1525971943, 1730123819, 1332069811, 1881593041, 1934580727, 1978623209, 1164218399, 1574862173, 1635834619, 1311914371, ...). These terms are obtained when using signed 32-bit integers, i.e., take the 10-digit numbers modulo 2^32, and select the primes between 10^9 and 2^31. - M. F. Hasler, Nov 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

For e, see also A104843, A104844, A104845, A104846, A104847, A104848, A104849, A104850, A104851 (this).
For Pi, see A198175, A198170, A104824, A104825, A104826, A198171, A198172, A198173, A198174.
For sqrt(2), see A198162, A198163, A198164, A198165, A198166, A198167, A198168, A198169, A198161.
For the Golden Ratio, see A198177, A103773, A103789, A103793, A103808, A103809, A103810, A103811, A103812; A103752.
For the Euler-Mascheroni constant gamma: A198776, A198777, A198778, A198779, A198780, A198781, A198782, A198783, A198784.

With[{de=FromDigits/@Partition[RealDigits[E,10,10000][[1]],10,1]}, Select[de,#>10^9&&PrimeQ[#]&]] (* Harvey P. Dale, Feb 05 2012 *)

list_A104851(x=exp(1), m=10)=m=10^m; for(k=1, default(realprecision), isprime(p=x\.1^k%m)&&p*10>m&&print1(p", ")) \\ The optional arguments can be used to produce other sequences of this series (cf. Crossrefs). Use e.g. \p999 to set precision to 999 digits. - M. F. Hasler, Nov 01 2014

Corrected by Harvey P. Dale, Feb 05 2012

Offset changed from 0 to 1 by Vincenzo Librandi, Apr 21 2013

A198174 Primes from merging of 10 successive digits in decimal expansion of Pi, in the order of appearance.

Original entry on oeis.org

5926535897, 4197169399, 1693993751, 7510582097, 4825342117, 5822317253, 2841027019, 8521105559, 8954930381, 4756482337, 2712019091, 5432664821, 3266482133, 6072602491, 5588174881, 8815209209, 6282925409, 2540917153, 5903600113, 8204665213, 3841469519

Offset: 1

Harvey P. Dale, Oct 21 2011

Leading zeros are not permitted, so each term is 10 digits in length.

See A104830 for the variant without this restriction. - M. F. Hasler, Nov 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf., for Pi: A198175, A198170, A104824, A104825, A104826, A198171, A198172, A198173, A198174 (this) and A104830 (a variant).
Cf., for sqrt(2): A198162, A198163, A198164, A198165,A198166, A198167, A198168, A198169, A198161.
Cf., for e: A104843, A104844, A104845, A104846, A104847, A104848, A104849, A104850, A104851.
Cf., for the Golden Ratio: A198177, A103773, A103789, A103793, A103808, A103809, A103810, A103811, A103812.
Cf., for the Euler-Mascheroni constant gamma: A198776, A198777, A198778, A198779, A198780, A198781, A198782, A198783, A198784.

With[{len=10},Select[FromDigits/@Partition[RealDigits[Pi,10,1000][[1]], len,1],IntegerLength[#]==len&&PrimeQ[#]&]]

A198174(n, x=Pi, m=10, silent=0)={m=10^m; for(k=1, default(realprecision), (isprime(p=x\.1^k%m)&&p*10>m)||next; silent||print1(p", "); n--||return(p))} \\ The optional arguments can be used to produce other sequences of this series (cf. Crossrefs). Use e.g. \p999 to set precision to 999 digits. - M. F. Hasler, Nov 01 2014

A198776 Primes from merging of 2 successive digits in decimal expansion of Euler-Mascheroni constant.

Original entry on oeis.org

53, 2, 43, 31, 59, 59, 23, 59, 5, 67, 23, 67, 67, 67, 47, 29, 17, 67, 31, 47, 7, 5, 83, 41, 17, 73, 97, 23, 53, 53, 3, 37, 29, 37, 73, 37, 73, 37, 67, 73, 79, 59, 47, 73, 3, 67, 53, 23, 31, 17, 61, 11, 11, 19, 7, 79, 47, 79, 37, 5, 2, 29, 13, 47, 61, 2, 29

Offset: 1

Harvey P. Dale, Oct 29 2011

Leading zeros are permitted, so some terms may be less than 2 digits in length.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A103773, A103789, A103793, A103808, A103809, A103810, A103811,
A103812, A104824, A104825, A104826, A104843, A104844, A104845,
A104846, A104847, A104848, A104849, A104850, A104936, A198161, A198162,
A198163, A198164, A198165, A198166, A198167, A198168, A198169,
A198170, A198171, A198172, A198173, A198175, A104851, A198177,
A198777, A198778, A198779, A198780, A198781, A198782,
A198783, A198784.

  egp[len_]:=Module[{egterms=FromDigits/@Partition[RealDigits[EulerGamma, 10, 1000][[1]],len,1]},Select[egterms,PrimeQ[#]&]];egp[2]
Select[FromDigits/@Partition[RealDigits[EulerGamma,10,500][[1]],2,1],PrimeQ] (* Harvey P. Dale, Mar 19 2020 *)

A198777 Primes from merging of 3 successive digits in decimal expansion of Euler-Mascheroni constant.

Original entry on oeis.org

577, 431, 421, 593, 359, 593, 359, 677, 677, 467, 709, 947, 467, 463, 631, 809, 401, 283, 241, 173, 739, 997, 449, 353, 3, 337, 293, 937, 373, 733, 337, 773, 673, 739, 709, 491, 853, 233, 331, 151, 661, 211, 199, 79, 479, 937, 857, 2, 29, 547, 29, 43, 421

Offset: 1

Harvey P. Dale, Oct 29 2011

Leading zeros are permitted, so some terms are less than 3 digits in length.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A103773, A103789, A103793,A103808,A103809,A103810, A103811, A103812, A104824, A104825, A104826, A104843, A104844, A104845, A104846, A104847, A104848, A104849, A104850, A104937, A198161, A198162, A198163, A198164, A198165, A198166, A198167, A198168, A198169, A198170, A198171, A198172, A198173, A198175, A104851, A198177, A198776, A198778, A198779, A198780, A198781, A198782, A198783, A198784.

egp[len_]:=Module[{egterms=FromDigits/@Partition[RealDigits[EulerGamma, 10, 1000][[1]],len,1]},Select[egterms,PrimeQ[#]&]]; egp[3]

A198778 Primes from merging of 4 successive digits in decimal expansion of Euler-Mascheroni constant A001620.

Original entry on oeis.org

577, 421, 3359, 3593, 5939, 9923, 8677, 2677, 6709, 6947, 6329, 2917, 4951, 1447, 401, 4283, 2417, 6449, 5003, 3733, 3767, 7673, 9491, 2039, 853, 5323, 6211, 4793, 7937, 857, 7057, 29, 3547, 6043, 587, 6733, 7331, 3313, 1399, 7541, 5413, 4139, 8423, 4877, 503, 8431, 3109, 1093, 9973, 3613, 8893, 8933, 17, 7247

Offset: 1

Harvey P. Dale, Oct 29 2011

In contrast to A104938, leading zeros are allowed here, which explains the terms having fewer than 4 digits; e.g., a(32)=29 comes from consecutive digits "...0029..." starting at the 268th decimal digit of gamma (if the initial "0." counts as the first digit). - M. F. Hasler, Oct 31 2011

			The first four decimal digits of gamma = 0.5772... form the prime 577=a(1).

Cf. A103773, A103789, A103793, A103808-A103812, A104824-A104826, A104843-A104850, A198161-A198177, A198776-A198784.

Digits := 420 ;
for sh from 3 do
        p := floor(gamma*10^sh) mod 10000 ;
        if isprime(p) then
                printf("%d,",p);
        end if;
end do: # R. J. Mathar, Oct 31 2011

(* see A104938 for Mmca code *)
Join[{577},Select[FromDigits/@Partition[RealDigits[EulerGamma,10,1000][[1]],4,1],PrimeQ]] (* Harvey P. Dale, May 07 2019 *)

L=10^4;for(i=3,999,isprime(p=Euler\.1^i%L)&print1(p",")) \\ M. F. Hasler, Oct 31 2011

A198779 Primes from merging of 5 successive digits in decimal expansion of Euler-Mascheroni constant.

Original entry on oeis.org

64901, 59399, 48677, 77267, 26777, 66467, 36947, 6329, 32917, 17467, 49807, 24809, 92353, 50033, 74293, 42937, 37337, 33773, 79259, 24709, 70949, 9491, 16567, 70853, 53233, 33151, 31517, 28621, 62119, 79847, 98479, 84793, 50857, 29921, 14669, 96043, 35267, 52673, 40129, 12967

Offset: 1

Harvey P. Dale, Oct 29 2011

In contrast to A104939, leading zeros are permitted, so this sequence contains all elements of A104939 and additional primes having fewer than 5 digits.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A186206, A103773, A103789, A103793, A103808-A103812, A104824-A104826, A104843-A104850, A198161-A198177, A198776-A198784.

egp[len_] := Module[{egterms = FromDigits /@ Partition[RealDigits[EulerGamma, 10, 400][[1]], len, 1]}, Select[egterms, PrimeQ[#] &]]; egp[5] (* Vincenzo Librandi, Apr 20 2013 *)

A198780 Primes from merging of 6 successive digits in decimal expansion of Euler-Mascheroni constant.

Original entry on oeis.org

939923, 992359, 746749, 241739, 644923, 350033, 500333, 374293, 937337, 773767, 160087, 670853, 532331, 199501, 79847, 847937, 29921, 299213, 325421, 526733, 673313, 331399, 12967, 375413, 395491, 954911, 38431, 93997, 939973, 60889, 271351, 349291, 79843, 984301, 341777

Offset: 1

Harvey P. Dale, Oct 29 2011

In contrast to A104940, leading zeros are permitted, so this sequence contains all terms of A104940 plus additional primes with fewer than 6 digits.

Cf. A186206, A103773, A103789, A103793, A103808-A103812, A104824-A104826, A104843-A104850, A198161-A198177, A198776-A198784.

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A198161 Primes from merging of 10 successive digits in decimal expansion of sqrt(2).

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A198177 10-digit primes found in the decimal expansion of the Golden Ratio phi, in the order of occurrence.

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A103808 Primes from merging of 6 successive digits in decimal expansion of the Golden Ratio; (1+sqrt(5))/2.

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A104851 Primes from merging of 10 successive digits in decimal expansion of e.

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A198174 Primes from merging of 10 successive digits in decimal expansion of Pi, in the order of appearance.

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A198776 Primes from merging of 2 successive digits in decimal expansion of Euler-Mascheroni constant.

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A198777 Primes from merging of 3 successive digits in decimal expansion of Euler-Mascheroni constant.

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A198778 Primes from merging of 4 successive digits in decimal expansion of Euler-Mascheroni constant A001620.

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