A198778 -id:A198778 - 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 *)

A198784 Primes from merging of 10 successive digits in decimal expansion of Euler-Mascheroni constant (in the order of appearance).

Original entry on oeis.org

7215664901, 1566490153, 3286060651, 6060651209, 9008240243, 4310421593, 2159335939, 9235988057, 8486772677, 8070824809, 2836224173, 3622417399, 3997644923, 33374293, 2582470949, 6008735203, 87352039, 3151776611, 5015079847, 7400299213, 3139925401, 3754139549

Offset: 1

Harvey P. Dale, Oct 29 2011

Leading zeros are permitted, so some terms are less than 10 digits in length.

See A104944 for the variant where no leading zeros are allowed. - M. F. Hasler, Nov 01 2014

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

For the Euler-Mascheroni constant, see also A198776, A198777, A198778, A198779, A198780, A198781, A198782, A198783, A198784 (this sequence) and A104944 (a variant).
For sqrt(2), see A198162, A198163, A198164, A198165, A198166, A198167, A198168, A198169, A198161.
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.

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

list_A198784(x=Euler,m=10)=m=10^m;for(k=1,default(realprecision),isprime(p=x\.1^k%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

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]

A104944 Primes from merging of 10 successive digits in decimal expansion of the Euler-Mascheroni constant.

Original entry on oeis.org

7215664901, 1566490153, 3286060651, 6060651209, 9008240243, 4310421593, 2159335939, 9235988057, 8486772677, 8070824809, 2836224173, 3622417399, 3997644923, 2582470949, 6008735203, 3151776611, 5015079847, 7400299213, 3139925401, 3754139549, 7984234877

Offset: 1

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

Leading zeros are not permitted, so each term is 10 digits in length. - Harvey P. Dale, Oct 30 2011

See A198784 for the variant without this restriction.-- The original version read (1566490153, 1290642131, 1386514643, 1851726733, 1383679133, 1706757499, 1072945781, 1015442651, 1403043203, 1100525291, 1332985747, 1866475913, 1834810931, 1887149587, 1197399197, 1956311131, 1449885007, 2137384231, ...). These terms are obtained when using signed 32-bit integers, i.e., take the 10-digit strings 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
Jon Borwein, 170,000 digits of Gamma [Wayback Machine copy]
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.

Cf. A198776, A198777, A198778, A198779, A198780, A198781, A198782, A198783, A198784.

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 golden ratio phi = (sqrt(5)-1)/2: A103773, A103789, A103793, A103808, A103809, A103810, A103811, A103812, A198177.

egp[len_]:=Module[{egterms=FromDigits/@Partition[RealDigits[EulerGamma, 10, 1000][[1]],len,1]},Select[egterms,IntegerLength[#]==len&&PrimeQ[#]&]];egp[10] (* Harvey P. Dale, Oct 30 2011 *)

list_A104944(x=Euler, 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 and extended by Harvey P. Dale, Oct 30 2011

A105383 Primes between 10^9 and 2^31 obtained from merging 10 successive digits in the decimal expansion of zeta(2) = Pi^2/6, taken modulo 2^32.

Original entry on oeis.org

1902619757, 1896233719, 2025479923, 1979084773, 1834487573, 2069040007, 1357689757, 1422433483, 1421193281, 1865610371, 1664088953, 1716574481, 1524418627, 2018846497, 2028620161, 1384352219, 1828868887, 1485949159

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Apr 03 2005

Erroneous version of A225143.
The author must have used signed 32-bit integers to store 10 successive digits of zeta(2). This is the sequence you get by taking the 10-digit numbers modulo 2^32 and then listing primes between 10^9 and 2^31 = 2147483648. - Jens Kruse Andersen, Sep 15 2014
In other words, primes p in (10^9, 2^31) such that either p, p + 2^32 or p + 2^32*2 is the concatenation of 10 successive digits in the decimal expansion of Pi^2/6. - Jianing Song, Mar 14 2021

			From _Jianing Song_, Mar 14 2021: (Start)
1902619757 is a term since 1902619757 + 2^32 = 6197587053 is the concatenation of A013661(92) to A013661(101).
1896233719 is a term since it is the concatenation of A013661(108) to A013661(117). (End)

Simon Plouffe, 10000 digits of Zeta(2).
Eric Weisstein, Riemann Zeta Function.

Cf. A013661 (decimal expansion of Pi^2/6).
Cf. A103752 (a similar erroneous version).
Cf. (for Pi) A198175, A198170, A104824, A104825, A104826, A198171, A198172, A198173, A198174 and A104830 (a variant).
Cf. (for e) A104843, A104844, A104845, A104846, A104847, A104848, A104849, A104850, A104851.
Cf. (for sqrt(2)) A198162, A198163, A198164, A198165, A198166, A198167, A198168,A198169, A198161.
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.

A105383(n, x=Pi^2/6, m=10, silent=0)={m=10^m; for(k=1, default(realprecision), (isprime(p=x\.1^k%m%2^32)&&p*10>m&&p<2^31)||next; silent||print1(p", "); n--||return(p))} \\  Use e.g. \p999 to set precision to 999 digits. - M. F. Hasler, Nov 01 2014

Definition updated by M. F. Hasler, Nov 01 2014

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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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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A198784 Primes from merging of 10 successive digits in decimal expansion of Euler-Mascheroni constant (in the order of appearance).

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

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