Displaying 1-10 of 23 results found.
Primes from merging of 10 successive digits in decimal expansion of sqrt(2).
+10
34
4142135623, 8872420969, 9698078569, 7537694807, 7973799073, 7846210703, 2644121497, 9935831413, 6592750559, 7010955997, 1472851741, 5251407989, 2533965463, 5339654633, 6152583523, 1525835239, 3950547457, 5750287759, 5996172983, 4084988471, 6668713013
COMMENTS
Leading zeros are not permitted, so each term is 10 digits in length.
MATHEMATICA
With[{len=10}, Select[FromDigits/@Partition[RealDigits[Sqrt[2], 10, 1000][[1]], len, 1], IntegerLength[#]==len&&PrimeQ[#]&]]
PROG
(PARI) 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
10-digit primes found in the decimal expansion of the Golden Ratio phi, in the order of occurrence.
+10
31
1772030917, 4189391137, 6222353693, 7931800607, 5959395829, 5829056383, 3832266131, 6131992829, 6892501711, 9250171169, 1043216269, 3136144381, 7587012203, 7954454749, 8509874339, 4487706647, 1240076521, 7780531531, 5315317141, 1704666599, 7046665991
COMMENTS
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
MATHEMATICA
With[{len=10}, Select[FromDigits/@Partition[RealDigits[GoldenRatio, 10, 1000][[1]], len, 1], IntegerLength[#]==len&&PrimeQ[#]&]]
PROG
(PARI) 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
CROSSREFS
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.
Primes from merging of 6 successive digits in decimal expansion of the Golden Ratio; (1+sqrt(5))/2.
+10
30
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
COMMENTS
Leading zeros are not permitted, so each term is 6 digits in length. - Harvey P. Dale, Oct 23 2011
LINKS
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.
MATHEMATICA
With[{len=6}, FromDigits/@Select[Partition[RealDigits[GoldenRatio, 10, 1000][[1]], len, 1], PrimeQ[FromDigits[#]] &&IntegerLength[ FromDigits[#]] ==len&]] (* Harvey P. Dale, Oct 23 2011 *)
PROG
(PARI) 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
CROSSREFS
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.
AUTHOR
Andrew G. West (WestA(AT)wlu.edu), Mar 29 2005
Primes from merging of 2 successive digits in decimal expansion of sqrt(2).
+10
27
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
COMMENTS
Leading zeros are not permitted, so each term is 2 digits in length.
MATHEMATICA
With[{len=2}, Select[FromDigits/@Partition[RealDigits[Sqrt[2], 10, 1000][[1]], len, 1], IntegerLength[#]==len&&PrimeQ[#]&]]
CROSSREFS
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.
Primes from merging of 3 successive digits in decimal expansion of sqrt(2).
+10
27
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
COMMENTS
Leading zeros are not permitted, so each term is 3 digits in length.
MATHEMATICA
With[{len=3}, Select[FromDigits/@Partition[RealDigits[Sqrt[2], 10, 1000][[1]], len, 1], IntegerLength[#]==len&&PrimeQ[#]&]]
CROSSREFS
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.
Primes from merging of 4 successive digits in decimal expansion of sqrt(2).
+10
27
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
COMMENTS
Leading zeros are not permitted, so each term is 4 digits in length.
MATHEMATICA
With[{len=4}, Select[FromDigits/@Partition[RealDigits[Sqrt[2], 10, 1000][[1]], len, 1], IntegerLength[#]==len&&PrimeQ[#]&]]
CROSSREFS
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.
Primes from merging of 9 successive digits in decimal expansion of sqrt(2).
+10
27
213562373, 488016887, 688724209, 807856967, 718753769, 376948073, 501384623, 470109559, 609552329, 292304843, 260362799, 396546331, 523950547, 877599617, 172983557, 220337531, 570113543, 160386899, 603868999, 782306849, 684929369, 861249497, 124949771
COMMENTS
Leading zeros are not permitted, so each term is 9 digits in length.
MATHEMATICA
With[{len=9}, Select[FromDigits/@Partition[RealDigits[Sqrt[2], 10, 1000][[1]], len, 1], IntegerLength[#]==len&&PrimeQ[#]&]]
CROSSREFS
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.
Primes from merging of 5 successive digits in decimal expansion of sqrt(2).
+10
25
56237, 37309, 78569, 67187, 48073, 76679, 66797, 97379, 79907, 50387, 34327, 64157, 15727, 91229, 70249, 73721, 12149, 70999, 35831, 65927, 55927, 55799, 11527, 55997, 59971, 86201, 20147, 28517, 88919, 30871, 14321, 45083, 50839, 62603, 51407, 87253, 72533
COMMENTS
Leading zeros are not permitted, so each term is 5 digits in length.
MATHEMATICA
With[{len=5}, Select[FromDigits/@Partition[RealDigits[Sqrt[2], 10, 1000][[1]], len, 1], IntegerLength[#]==len&&PrimeQ[#]&]]
CROSSREFS
Cf. A103773, A103789, A103793, A103808, A103809, A103810, A103811, A103812, A104824, A104825, A104826, A104843, A104844, A104845, A104846, A104847, A104848, A104849, A104850, A198161, A198162, A198163, A198164, A198166, A198167, A198168, A198169, A198170, A198171, A198172, A198173, A198174, A198175, A104851, A198177.
Primes from merging of 6 successive digits in decimal expansion of sqrt(2).
+10
25
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
COMMENTS
Leading zeros are not permitted, so each term is 6 digits in length.
MATHEMATICA
With[{len=6}, Select[FromDigits/@Partition[RealDigits[Sqrt[2], 10, 1000][[1]], len, 1], IntegerLength[#]==len&&PrimeQ[#]&]]
CROSSREFS
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.
Primes from merging of 10 successive digits in decimal expansion of e.
+10
23
7427466391, 7413596629, 6059563073, 3490763233, 2988075319, 1573834187, 7021540891, 5408914993, 6480016847, 9920695517, 1838606261, 6062613313, 3845830007, 1692836819, 4425056953, 2505695369, 5490598793, 1782154249, 8215424999, 9229576351, 9519366803
COMMENTS
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
MATHEMATICA
With[{de=FromDigits/@Partition[RealDigits[E, 10, 10000][[1]], 10, 1]}, Select[de, #>10^9&&PrimeQ[#]&]] (* Harvey P. Dale, Feb 05 2012 *)
PROG
(PARI) 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
AUTHOR
Andrew G. West (WestA(AT)wlu.edu), Mar 27 2005
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