cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-19 of 19 results.

A083739 Pseudoprimes to bases 2, 3, 5 and 7.

Original entry on oeis.org

29341, 46657, 75361, 115921, 162401, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 721801, 852841, 1024651, 1152271, 1193221, 1461241, 1569457, 1615681, 1857241, 1909001, 2100901
Offset: 1

Views

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

Keywords

Examples

			a(1)=29341 since it is the first number such that 2^(k-1) = 1 (mod k), 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Links

Crossrefs

Cf. A001567, A005935, A005936, A005938.
Proper subset of A083737.

Programs

  • Maple
    a001567 := [] : f := fopen("b001567.txt",READ) : bfil := readline(f) : while StringTools[WordCount](bfil) > 0 do if StringTools[FirstFromLeft]("#",bfil ) <> 0 then ; else bfil := sscanf(bfil,"%d %d") ; a001567 := [op(a001567), op(2,bfil) ] ; fi ; bfil := readline(f) ; od: fclose(f) : isPsp := proc(n,b) if n>3 and not isprime(n) and b^(n-1) mod n = 1 then true; else false; fi; end: isA001567 := proc(n) isPsp(n,2) ; end: isA005935 := proc(n) isPsp(n,3) ; end: isA005936 := proc(n) isPsp(n,5) ; end: isA005938 := proc(n) isPsp(n,7) ; end: isA083739 := proc(n) if isA001567(n) and isA005935(n) and isA005936(n) and isA005938(n) then true ; else false ; fi ; end: n := 1: for psp2 from 1 do i := op(psp2,a001567) ; if isA083739(i) then printf("%d %d ",n,i) ; n :=n+1 ; fi ; od: # R. J. Mathar, Feb 07 2008
  • Mathematica
    Select[ Range[2113920], !PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, 1 - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 && PowerMod[7, 1 - 1, # ] == 1 & ]
  • PARI
    is(n)=!isprime(n)&&Mod(2,n)^(n-1)==1&&Mod(3,n)^(n-1)==1&&Mod(5,n)^(n-1)==1&&Mod(7,n)^(n-1)==1 \\ Charles R Greathouse IV, Apr 12 2012

Formula

a(n) = n-th positive integer k(>1) such that 2^(k-1) = 1 (mod k), 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).
A005938 INTERSECT A083737. - R. J. Mathar, Feb 07 2008

Extensions

Edited by Robert G. Wilson v, May 06 2003

A243010 Pseudoprimes to base 5 that are not squarefree.

Original entry on oeis.org

4, 124, 11476, 59356, 80476, 91636, 250876, 261964, 482516, 1385836, 1926676, 2428084, 2589796, 3743476, 4101796, 6797764, 9155476, 10701076, 10743436, 11263396, 13799836, 13859956, 15570556, 20396476
Offset: 1

Views

Author

Felix Fröhlich, Aug 18 2014

Keywords

Comments

Any term is divisible by the square of a base 5 Wieferich prime (A123692).
Intersection of A005936 and A013929. - Michel Marcus, Aug 21 2014

Links

Crossrefs

Cf. A005936, A123692, A158358, A243089, A243090, A244065.

Programs

  • PARI
    forcomposite(n=1, 1e9, if(Mod(5, n)^(n-1)==1, if(!issquarefree(n), print1(n, ", "))))

A083732 Pseudoprimes to bases 2 and 5.

Original entry on oeis.org

561, 1729, 2821, 5461, 6601, 8911, 12801, 13981, 15841, 29341, 41041, 46657, 52633, 63973, 68101, 75361, 101101, 113201, 115921, 126217, 137149, 162401, 172081, 188461, 252601, 294409, 314821, 334153, 340561, 399001, 401401, 410041, 488881
Offset: 1

Views

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

Keywords

Examples

			a(1)=561 since 561 is the first positive integer k(>1) which satisfies 2^(k-1) = 1 (mod k) and 5^(k-1) = 1 (mod k).

Links

Crossrefs

Intersection of A001567 and A005936. - R. J. Mathar, Apr 05 2011

Programs

  • Mathematica
    Select[Range[1, 10^5, 2], CompositeQ[#] &&  PowerMod[2, #-1,#] == PowerMod[5, #-1,#] == 1 &] (* Amiram Eldar, Jun 29 2019 *)
  • PARI
    lista(nn) = forcomposite(n=1, nn, if ((Mod(2, n)^(n-1)==1) && (Mod(5, n)^(n-1)==1), print1(n, ", "));); \\ Michel Marcus, Sep 08 2016

Formula

a(n) = n-th positive integer k(>1) such that 2^(k-1) = 1 (mod k) and 5^(k-1) = 1 (mod k).

A083734 Pseudoprimes to bases 3 and 5.

Original entry on oeis.org

1541, 1729, 1891, 2821, 6601, 8911, 15841, 29341, 41041, 46657, 52633, 63973, 75361, 88831, 101101, 112141, 115921, 126217, 146611, 162401, 172081, 188461, 218791, 252601, 294409, 314821, 334153, 340561, 342271, 399001, 410041, 416641
Offset: 1

Views

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

Keywords

Examples

			a(1)=1541 since it is the first nonprime number such that 3^(k-1) = 1 (mod k) and 5^(k-1) = 1 (mod k). - clarified by _Harvey P. Dale_, Jan 29 2013

Links

Crossrefs

Intersection of A005935 and A005936.

Programs

  • Mathematica
    Select[Range[420000],!PrimeQ[#]&&PowerMod[3,#-1,#]==PowerMod[5,#-1,#]==1&] (* Harvey P. Dale, Jan 29 2013 *)

Formula

a(n) = n-th positive integer k(>1) such that 3^(k-1) = 1 (mod k) and 5^(k-1) = 1 (mod k).

A371729 The number of pseudoprimes to base n that are smaller than n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 4, 0, 4, 0, 3, 1, 1, 0, 5, 3, 1, 2, 5, 0, 4, 1, 4, 3, 2, 1, 7, 0, 1, 1, 8, 0, 6, 2, 3, 3, 1, 0, 9, 2, 3, 1, 8, 0, 6, 3, 6, 1, 2, 0, 9, 3, 1, 7, 7, 1, 6, 2, 4, 1, 9, 0, 11, 2, 1, 7, 6, 1, 7, 3, 10, 5, 3, 0, 8, 4, 1, 1
Offset: 2

Views

Author

Amiram Eldar, Apr 05 2024

Keywords

Examples

			a(2) = 0 since the smallest pseudoprime to base 2 (A001567) is 341 which is larger than 2.
a(5) = 1 since there is one pseudoprime to base 5 (A005936) that is smaller than 5: 4.
a(9) = 2 since there are 2 pseudoprimes to base 9 (A020138) that are smaller than 9: 4 and 8.

Links

Crossrefs

Cf. A001567, A005936, A020138, A090086, A316504, A371730.

Programs

  • Mathematica
    a[n_] := Count[Range[4, n-1], _?(CompositeQ[#] && PowerMod[n, # - 1, #] == 1 &)]; Array[a, 100, 2]
  • PARI
    a(n) = {my(c = 0); forcomposite(k = 4, n-1, if(Mod(n, k)^(k-1) == 1, c++)); c;}

Formula

a(n) = 0 if and only if A090086(n) > n, or equivalently, n-1 is in A316504.

A083736 Pseudoprimes to bases 2,5 and 7.

Original entry on oeis.org

561, 29341, 46657, 75361, 115921, 162401, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 656601, 658801, 710533, 721801, 852841, 1024651, 1141141, 1152271, 1168513, 1193221, 1461241, 1569457, 1615681
Offset: 1

Views

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

Keywords

Examples

			a(1)=561 since it is the first number such that 2^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Links

Crossrefs

Intersection of A083732 and A005938. Intersection of A083733 and A005936. - R. J. Mathar, Apr 05 2011

Programs

  • Mathematica
    Select[Range[1, 10^5, 2], CompositeQ[#] &&  PowerMod[2, #-1,#] == PowerMod[5, #-1,#] == PowerMod[7, #-1,#] == 1&] (* Amiram Eldar, Jun 29 2019 *)

Formula

a(n) = n-th positive integer k(>1) such that 2^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

A083740 Pseudoprimes to bases 3,5 and 7.

Original entry on oeis.org

29341, 46657, 75361, 88831, 115921, 146611, 162401, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 721801, 852841, 954271, 1024651, 1152271, 1193221, 1314631, 1461241, 1569457, 1615681
Offset: 1

Views

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

Keywords

Examples

			a(1)=29341 since it is the first number such that 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Links

Crossrefs

Cf. A005935, A005936, A005938, A083734, A083735.

Programs

  • Mathematica
    Select[Range[1, 10^5, 2], CompositeQ[#] &&  PowerMod[3, #-1, #] == PowerMod[5, #-1, #] == PowerMod[7, #-1, #] == 1&]

Formula

a(n) = n-th positive integer k(>1) such that 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).
Intersection of A083734 and A005938. Intersection of A083735 and A005936. - R. J. Mathar, Apr 05 2011

A374976 Odd k with p^k mod k != p for all primes p.

Original entry on oeis.org

1, 9, 27, 63, 75, 81, 115, 119, 125, 189, 207, 209, 215, 235, 243, 279, 299, 319, 323, 387, 407, 413, 423, 515, 517, 531, 535, 551, 567, 575, 583, 611, 621, 623, 667, 675, 707, 713, 729, 731, 747, 767, 779, 783, 799, 815, 835, 851, 869, 893, 899, 917, 923, 927
Offset: 1

Views

Author

Francois R. Grieu, Jul 26 2024

Keywords

Comments

Alternatively: 1, and odd composites not a pseudoprime to any prime base.
The sequence contains no primes, no pseudoprimes to any prime base (A001567, A005935, A005936, A005938, A020139, A020141...), and no Carmichael numbers (A002997).

Examples

			k=3 (resp. 5, 7) is not in the sequence because for prime p=2 it holds p^k mod k = 2 which is p.
k=9 is in the sequence because for prime p=2 (resp. 3, 5, 7) it holds p^k mod k = 8 (resp. 0, 8, 1) which is not p, and for all other primes p it holds p>=k therefore p^k mod k can't be p.

Links

Crossrefs

Cf. A000040, A001567, A005935, A005936, A005938, A020139, A020141, A020145, A002997.

Programs

  • Mathematica
    Cases[Range[1, 930, 2], k_/; (For[p=2, p=k)]

A247906 a(n) = n-th pseudoprime to base n.

Original entry on oeis.org

561, 286, 341, 781, 1105, 1105, 133, 364, 703, 793, 1105, 1099, 1891, 6541, 1271, 3991, 1649, 1849, 3059, 7363, 2047, 1738, 4537, 1128, 3145, 2993, 5365, 4069, 4097, 7421, 2465, 11305, 2937, 16589, 4495, 2044, 6601, 26885, 13073, 6892, 22945, 3885, 8695, 10879
Offset: 2

Views

Author

Felix Fröhlich, Sep 26 2014

Keywords

Examples

			a(2) = A001567(2) = 561.
a(3) = A005935(3) = 286.

Crossrefs

Cf. A007535, A098650.
Cf. Pseudoprimes to base b: A001567 (b=2), A005935 (b=3), A020136 (b=4), A005936 (b=5), A005937 (b=6), A005938 (b=7), A020137 (b=8), A020138 (b=9).

Programs

  • PARI
    for(n=2, 20, i=0; forcomposite(c=2, 1e9, if(Mod(n, c)^(c-1)==1, i++; if(i==n, print1(c, ", "); i=0; break({1}))); if(c==1e9, print1(">1e9, "))))
Previous Showing 11-19 of 19 results.

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