A198165 -id:A198165 - 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

A198162 Primes from merging of 2 successive digits in decimal expansion of sqrt(2).

Original entry on oeis.org

41, 13, 23, 37, 73, 67, 71, 53, 37, 73, 31, 17, 67, 79, 97, 73, 37, 79, 73, 47, 53, 43, 41, 73, 13, 23, 29, 97, 83, 73, 37, 41, 97, 83, 31, 41, 13, 59, 59, 79, 11, 71, 47, 59, 97, 71, 59, 97, 53, 59, 47, 17, 41, 89, 19, 23, 29, 23, 43, 71, 43, 83, 97, 79, 79

Offset: 1

Harvey P. Dale, Oct 21 2011

Leading zeros are not permitted, so each term is 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, A198161, A198163, A198164, A198165, A198166, A198167, A198168, A198169, A198170, A198171, A198172, A198173, A198174, A198175, A104851, A198177.

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

A198163 Primes from merging of 3 successive digits in decimal expansion of sqrt(2).

Original entry on oeis.org

421, 373, 887, 569, 967, 769, 317, 797, 379, 907, 107, 503, 641, 157, 727, 229, 149, 709, 659, 557, 571, 701, 109, 599, 997, 971, 919, 523, 839, 397, 251, 463, 331, 829, 523, 239, 547, 457, 877, 599, 617, 983, 557, 337, 857, 701, 113, 997, 503, 277, 823, 929

Offset: 1

Harvey P. Dale, Oct 21 2011

Leading zeros are not permitted, so each term is 3 digits in length.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A103773, A103789, A103793, A103808, A103809, A103810, A103811, A103812, A104824, A104825, A104826, A104843, A104844, A104845, A104846, A104847, A104848, A104849, A104850, A198161, A198162, A198164, A198165, A198166, A198167, A198168, A198169, A198170, A198171, A198172, A198173, A198174, A198175, A104851, A198177.

With[{len=3},Select[FromDigits/@Partition[RealDigits[Sqrt[2],10,1000][[1]], len,1],IntegerLength[#]==len&&PrimeQ[#]&]]

A198164 Primes from merging of 4 successive digits in decimal expansion of sqrt(2).

Original entry on oeis.org

5623, 7309, 6967, 7187, 8753, 7537, 3769, 6679, 9907, 4621, 8753, 4327, 4157, 2309, 1229, 2297, 3583, 6659, 5927, 5927, 5011, 7027, 2851, 1741, 8609, 4079, 7253, 7457, 7759, 3557, 2203, 5701, 5437, 4603, 8689, 6899, 8999, 7069, 4027, 7823, 9293, 3691, 6311

Offset: 1

Harvey P. Dale, Oct 21 2011

Leading zeros are not permitted, so each term is 4 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, A198161, A198162, A198163, A198165, A198166, A198167, A198168, A198169, A198170, A198171, A198172, A198173, A198174, A198175, A104851, A198177.

With[{len=4},Select[FromDigits/@Partition[RealDigits[Sqrt[2],10,1000][[1]], len,1],IntegerLength[#]==len&&PrimeQ[#]&]]

A198169 Primes from merging of 9 successive digits in decimal expansion of sqrt(2).

Original entry on oeis.org

213562373, 488016887, 688724209, 807856967, 718753769, 376948073, 501384623, 470109559, 609552329, 292304843, 260362799, 396546331, 523950547, 877599617, 172983557, 220337531, 570113543, 160386899, 603868999, 782306849, 684929369, 861249497, 124949771

Offset: 1

Harvey P. Dale, Oct 21 2011

Leading zeros are not permitted, so each term is 9 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, A198161, A198162, A198163, A198164, A198165, A198166, A198167, A198168, A198170, A198171, A198172, A198173, A198174, A198175, A104851, A198177.

With[{len=9},Select[FromDigits/@Partition[RealDigits[Sqrt[2],10,1000][[1]],len,1],IntegerLength[#]==len&&PrimeQ[#]&]]

A198166 Primes from merging of 6 successive digits in decimal expansion of sqrt(2).

Original entry on oeis.org

135623, 569671, 480731, 850387, 157273, 384623, 585073, 970999, 927557, 275579, 950501, 686201, 450839, 514079, 989687, 872533, 583523, 750287, 759961, 961729, 983557, 752203, 531857, 857011, 570113, 374603, 340849, 868999, 997069, 970699, 900481, 277903

Offset: 1

Harvey P. Dale, Oct 21 2011

Leading zeros are not permitted, so each term is 6 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, A198161, A198162, A198163, A198164, A198165, A198167, A198168, A198169, A198170, A198171, A198172, A198173, A198174, A198175, A104851, A198177.

With[{len=6},Select[FromDigits/@Partition[RealDigits[Sqrt[2],10,1000][[1]], len,1],IntegerLength[#]==len&&PrimeQ[#]&]]

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

A198167 Primes from merging of 7 successive digits in decimal expansion of sqrt(2).

Original entry on oeis.org

3562373, 5048801, 2420969, 5038753, 7534327, 6415727, 5073721, 2126441, 2644121, 9709993, 9935831, 2226659, 9275579, 8206057, 5714701, 7027453, 2851741, 8640889, 2145083, 5835239, 3868999, 8689997, 9970699, 9900481, 2779031, 6311159, 6668713, 6871301

Offset: 1

Harvey P. Dale, Oct 21 2011

Leading zeros are not permitted, so each term is 7 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, A198161, A198162, A198163, A198164, A198165, A198166, A198168, A198169, A198170, A198171, A198172, A198173, A198174, A198175, A104851, A198177.

With[{len=7},Select[FromDigits/@Partition[RealDigits[Sqrt[2],10,1000][[1]],len,1],IntegerLength[#]==len&&PrimeQ[#]&]]

cp's OEIS Frontend

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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A198162 Primes from merging of 2 successive digits in decimal expansion of sqrt(2).

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A198163 Primes from merging of 3 successive digits in decimal expansion of sqrt(2).

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A198164 Primes from merging of 4 successive digits in decimal expansion of sqrt(2).

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A198169 Primes from merging of 9 successive digits in decimal expansion of sqrt(2).

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A198166 Primes from merging of 6 successive digits in decimal expansion of sqrt(2).

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

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