A038511 -id:A038511 - cp's OEIS frontend

A140461 Numbers in A008364 but not in A038511.

1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281

Offset: 1

Juri-Stepan Gerasimov, Jul 22 2008

isA008364 := proc(n) if n = 1 then true; else min( op(numtheory[factorset](n)) ); if % > 7 then true; else false; end if; end if; end proc:
isA038511 := proc(n) if n = 1 then false; else min( op(numtheory[factorset](n)) ); if % >= 11 and not isprime(n) then true; else false; end if; end if; end proc:
for n from 1 to 300 do if isA008364(n) and not isA038511(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Apr 25 2010

a(n)=if(n==1,1,prime(n+3)) \\ Charles R Greathouse IV, Aug 26 2011

Entries checked by R. J. Mathar, Apr 25 2010

A141864 Frequency of A038511 in the interval [(n-1)357,n357].

Original entry on oeis.org

1, 5, 5, 8, 6, 9, 8, 8, 10, 8, 9, 10, 11, 10, 8, 9, 11, 10, 13, 8, 13, 9, 9, 14, 11, 7, 11, 11, 12, 14, 11, 10, 12, 10, 10, 11, 12, 12, 10, 14, 8, 14, 12, 13, 10, 12, 14, 8, 14, 12, 13, 11, 11, 11, 14, 7, 16, 12, 14, 9, 9, 16, 12, 11, 14, 11, 10, 14, 12, 14, 17, 8, 12, 10, 15, 12, 15, 12

Offset: 1

Juri-Stepan Gerasimov, Sep 15 2008

nonn

A value of A038511(0)=1 is artificially introduced which leads to a(1)=1 by definition.

			In the 2nd interval from 105 to 210 we find A038511(1)=121, A038511(2)=143, A038511(3)=169,
A038511(4)=187 and A038511(5)=209, that is a(2)=5 entries.
In the 3rd interval from 210 to 315 we find A038511(6)=221, A038511(7)=247, A038511(8)=253,
A038511(9)=289 and A038511(10)=299, that is a(3)=5 entries.

Cf. A002808, A141468, A038511.

isA038511 := proc(n) local pfs; if isprime(n) then false; else pfs := numtheory[factorset](n) ; RETURN( pfs intersect {2,3,5,7} = {}) ; fi; end:
A141864 := proc(n) local k,a ; if n = 1 then 1; else a := 0 ; for k from (n-1)*105 to n*105 do if isA038511(k) then a := a+1 ; fi; od: a; fi; end:
seq(A141864(n),n=1..90) ; # R. J. Mathar, Aug 03 2009

Edited, a(13) corrected by R. J. Mathar, Aug 03 2009

A038510 Composite numbers with smallest prime factor >= 7.

Original entry on oeis.org

49, 77, 91, 119, 121, 133, 143, 161, 169, 187, 203, 209, 217, 221, 247, 253, 259, 287, 289, 299, 301, 319, 323, 329, 341, 343, 361, 371, 377, 391, 403, 407, 413, 427, 437, 451, 469, 473, 481, 493, 497, 511, 517, 527, 529, 533, 539, 551, 553, 559, 581, 583

Offset: 1

Jeff Burch

nonn

Let A = set of numbers of form 6n + 1, B = numbers of form 6n - 1. Eliminating numbers of form 25 + 30s from A and those of form 35 + 30s from B we obtain sets A* and B*. Removing all terms of the sequence from the union of A* and B*, only prime numbers remain. - Hisanobu Shinya (ilikemathematics(AT)hotmail.com), Jul 14 2002
Divide n by a*b*c where a = 2^(A001511(n)-1), b = 3^(A051064(n)-1) and c = 5^(A055457(n) -1). Then the resulting sequence includes only primes and a(n). - Alford Arnold, Sep 08 2003
Composite numbers not divisible by 2, 3 or 5. - Lekraj Beedassy, Jun 30 2004
Composite numbers k such that k^4 mod 30 = 1. - Gary Detlefs, Dec 09 2012
Composite numbers congruent to 1, 7, 11, 13, -13, -11, -7, -1 (mod 30). Since asymptotically, 100% of integers are composite, we have a(n)/n ~ 30/phi(30) = 30/8 = 3.75. - Daniel Forgues, Mar 16 2013
"John [Conway] recommends the more refined partition [of the positive numbers]: 1, prime, trivially composite, or nontrivially composite. Here, a composite integer is trivially composite if it is divisible by 2, 3, or 5." See link to (van der Poorten, Thomsen, and Wiebe; 2006) pp. 73-74. - Daniel Forgues, Jan 30 2015, Feb 04 2015
For the eight congruences coprime to 30, we can use one byte to encode the "primality/non-primality (unit or composite)" for each [30*n, 30*(n+1)[, n >= 0, closed-open interval, either as little endian binary sequence {01111111, 11111011, 11110111, 01111110, ...}, or as big endian binary sequence {11111110, 11011111, 11101111, 01111110, ...}, which we may then express in base 10. - Daniel Forgues, Feb 05 2015

J. H. Silverman, A Friendly Introduction to Number Theory, 2nd Edn. "Appendix A: Factorization of Small Composite Integers", Prentice Hall NY 2001.

T. D. Noe, Table of n, a(n) for n = 1..10000
Alf van der Poorten, Kurt Thomsen, and Mark Wiebe, A Curious Cubic Identity and Self-similar Sums of Squares, 2006, pp. 73-74.

Cf. A001511, A051064, A055457, A038509, A071904.
Cf. A070884, A038511.
Intersection of A002808 and A007775.

for n from 1 to 583 do if n^4 mod 30 = 1 and not isprime(n) then print(n)fi od; # Gary Detlefs, Dec 09 2012

Select[Range[1000], ! PrimeQ[#] && FactorInteger[#][[1, 1]] >= 7 &] (* T. D. Noe, Mar 16 2013 *)

is(n)=gcd(n,30)==1 && !isprime(n) \\ Charles R Greathouse IV, Dec 09 2012

a(n) ~ 3.75n. - Charles R Greathouse IV, Dec 09 2012

Corrected by Ralf Stephan, Apr 04 2003

A008367 Composite but smallest prime factor >= 17.

Original entry on oeis.org

289, 323, 361, 391, 437, 493, 527, 529, 551, 589, 629, 667, 697, 703, 713, 731, 779, 799, 817, 841, 851, 893, 899, 901, 943, 961, 989, 1003, 1007, 1037, 1073, 1081, 1121, 1139, 1147, 1159, 1189, 1207, 1219, 1241, 1247, 1271, 1273, 1333, 1343, 1349, 1357, 1363, 1369, 1387, 1403, 1411, 1457

Offset: 1

N. J. A. Sloane

nonn

Composite numbers k such that k^720 mod 30030 = 1. - Gary Detlefs, May 02 2012

The asymptotic density of this sequence is 192/1001. - Amiram Eldar, Mar 22 2021

T. D. Noe, Table of n, a(n) for n = 1..10000

Intersection of A002808 and A008366.
Cf. A038511, A084969, A084970.
Cf. A287391.

Filtered([17..1500],n->PowerMod(n,720,30030)=1 and not IsPrime(n)); # Muniru A Asiru, Nov 24 2018

for i from 1 to 2000 do if gcd(i,30030) = 1 and not isprime(i) then print(i); fi; od;

Select[ Range[ 1500 ], (GCD[ #1, 30030 ]==1&&!PrimeQ[ #1 ])& ]
Select[Range[2000], ! PrimeQ[#] && FactorInteger[#][[1, 1]] >= 17 &] (* T. D. Noe, Mar 16 2013 *)

is(n)={gcd(n,30030)==1 && !ispseudoprime(n)} \\ M. F. Hasler, Oct 04 2018

For 1 <= n < 107, a(n) = A287391(n+2); then a(107) = 2329, a(108) = 2363 are not in A287391, but again a(n) = A287391(n) for 108 < n < 120. - M. F. Hasler, Oct 04 2018

A100319 Even numbers m such that at least one of m-1 and m+1 is composite.

Original entry on oeis.org

8, 10, 14, 16, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 62, 64, 66, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 104, 106, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 140, 142, 144, 146, 148

Offset: 1

Rick L. Shepherd, Nov 13 2004

nonn

Subsequence of A100318. For each k >= 0, a(k+1) = a(k) + 2 unless a(k) + 1 and a(k) + 3 are twin primes, in which case a(k+1) = a(k) + 4 (as a(k) - 1 and a(k) + 5 are divisible by 3).

The even nonisolated primes(n+1). - Juri-Stepan Gerasimov, Nov 09 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A100318 (supersequence containing odd and even n), A045718 (n such that at least one of n-1 and n+1 is prime).
Cf. A167692(the even nonisolated nonprimes). - Juri-Stepan Gerasimov, Nov 09 2009
Cf. A005818, A038179, A007310, A038511, A025584.
Complement of A014574 (average of twin prime pairs) w.r.t. A005843 (even numbers), except for missing term 2.

Select[2*Range[100], CompositeQ[#-1] || CompositeQ[#+1] &]  (* G. C. Greubel, Mar 09 2019 *)

forstep(n=4,300,2,if(isprime(n-1)+isprime(n+1)<=1,print1(n,",")))

[n for n in (3..250) if mod(n,2)==0 and (is_prime(n-1) + is_prime(n+1)) < 2] # G. C. Greubel, Mar 09 2019

a(n) = A167692(n+1). - Juri-Stepan Gerasimov, Nov 09 2009

A121764 Single (or isolated or non-twin) primes of form 6n + 1.

Original entry on oeis.org

37, 67, 79, 97, 127, 157, 163, 211, 223, 277, 307, 331, 337, 367, 373, 379, 397, 409, 439, 457, 487, 499, 541, 547, 577, 607, 613, 631, 673, 691, 709, 727, 733, 739, 751, 757, 769, 787, 853, 877, 907, 919, 937, 967, 991, 997, 1009, 1039, 1069, 1087, 1117

Offset: 1

Lekraj Beedassy, Aug 20 2006

nonn

For the first 30000 terms a(n) > A121762(n), see plot A121764(n) - A121762(n). But is it so for all n? - Zak Seidov, Apr 25 2015

Subsequence of A002476. - Michel Marcus, Apr 26 2015

Zak Seidov, Table of n, a(n) for n = 1..10000
Zak Seidov, Plot of A121764(n) - A121762(n)

Cf. A007510, A121765, A002476, A121762, A038179, A007310, A038511.

[n: n in [1..1150] | (n mod 6 eq 1) and not IsPrime(n-2) and  IsPrime(n)]; // G. C. Greubel, Feb 26 2019

Select[Table[6n + 1, {n, 200}], PrimeQ[#] && !PrimeQ[#-2] &] (* Ray Chandler, Aug 22 2006 *)
Select[Prime[Range[200]],Mod[#,6]==1&&NoneTrue[#+{2,-2},PrimeQ]&] (* Harvey P. Dale, Jul 16 2021 *)

{is(n)=n%6==1 && isprime(n) && !isprime(n-2)}; \\ G. C. Greubel, Feb 26 2019

[n for n in (1..1150) if mod(n,6)==1 and not is_prime(n-2) and is_prime(n)] # G. C. Greubel, Feb 26 2019

Extended by Ray Chandler, Aug 22 2006

A054550 Composite numbers whose least prime factor is either 5 or 7.

Original entry on oeis.org

25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 125, 133, 145, 155, 161, 175, 185, 203, 205, 215, 217, 235, 245, 259, 265, 275, 287, 295, 301, 305, 325, 329, 335, 343, 355, 365, 371, 385, 395, 413, 415, 425, 427, 445, 455, 469, 475, 485, 497, 505, 511, 515, 535

Offset: 1

Stuart M. Ellerstein (ellerstein(AT)aol.com), May 15 2000

Original definition: Union of 4 AP's: 25+30n, 35+30n, 49+42n, 77+42n.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A002808, A038509, A038511, A047229, A067793.

Union[Flatten[Table[{30n+{25,35},42n+{49,77}},{n,0,20}]]] (* Harvey P. Dale, Feb 19 2016 *)

select( is_A054550(n)=vecsum((n=factor(n,0))[,2])>1&&n[1,1]>=5, [0..550]) \\ M. F. Hasler, Nov 18 2018

Vec(x*(25 + 10*x + 14*x^2 + 6*x^3 + 10*x^4 + 12*x^5 + 8*x^6 + 6*x^7 + 4*x^8 + 20*x^9 + 4*x^10 + 6*x^11 + 8*x^12 + 12*x^13 + 10*x^14 + 6*x^15 + 14*x^16 + 10*x^17 + 18*x^18 + 2*x^19 + 10*x^20 + 2*x^21 - 7*x^22) / ((1 - x)^2*(1 + x)*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10)) + O(x^60)) \\ Colin Barker, Nov 19 2018

a(n) = a(n-1) + a(n-22) - a(n-23). - Charles R Greathouse IV, Jun 01 2018

G.f.: x*(25 + 10*x + 14*x^2 + 6*x^3 + 10*x^4 + 12*x^5 + 8*x^6 + 6*x^7 + 4*x^8 + 20*x^9 + 4*x^10 + 6*x^11 + 8*x^12 + 12*x^13 + 10*x^14 + 6*x^15 + 14*x^16 + 10*x^17 + 18*x^18 + 2*x^19 + 10*x^20 + 2*x^21 - 7*x^22) / ((1 - x)^2*(1 + x)*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10)). - Colin Barker, Nov 19 2018

More terms from R. J. Mathar, Sep 30 2008

New name suggested by Andrew Howroyd, Nov 19 2018

A271630 Composite numbers n coprime to all number that can be obtained by changing just one digit of n.

Original entry on oeis.org

121, 143, 169, 187, 209, 221, 247, 253, 289, 299, 319, 323, 341, 343, 361, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 529, 533, 551, 553, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703, 713, 731, 737, 767, 779, 781, 793, 799, 803, 817, 841, 851

Offset: 1

Paolo P. Lava, Apr 14 2016

Only numbers ending in 1, 3, 7 and 9.
Apart from the first 10 terms, A078972 is a subset of this sequence.
Subsequence of A038510. - Altug Alkan, Apr 15 2016
Least squareless numbers with increasing number of primes:
143 = 11 * 13;
2431 = 11 * 13 * 17;
45353 = 7 * 11 * 19 * 31;
1062347 = 11 * 13 * 17 * 19 * 23;
30808063 = 11 * 13 * 17 * 19 * 23 * 29;
955049953 = 11 * 13 * 17 * 19 * 23 * 29 * 31;
35336848261 = 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37;
1448810778701 = 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41; etc.

			343 is coprime to:
43, 143, 243, 443, 543, 643, 743, 843, 943 (where the MSD has been changed);
303, 313, 323, 333, 353, 363, 373, 383, 393 (where the '4' in the middle has been changed);
340, 341, 342, 344, 345, 346, 347, 348, 349 (where the LSD has been changed) .

Paolo P. Lava, Table of n, a(n) for n = 1..10000
Paolo P. Lava, First 500 palindromes in the sequence
Paolo P. Lava, First 10000 non palindromes whose reverses are terms of the sequence

Cf. A038510, A038511, A078972.

with(numtheory); P:=proc(q) local a,j,k,n,ok;
for n from 2 to q do if not isprime(n) then ok:=1; j:=0;
while ok=1 and j<9 do j:=j+1; for k from 1 to ilog10(n)+1 do
a:=trunc(n/10^k)*10^k+((trunc((n mod 10^k)/10^(k-1))-j) mod 10)*10^(k-1)+(n mod 10^(k-1));
if gcd(n,a)>1 then ok:=0; break; fi; od; od;
if ok=1 then print(n); fi; fi; od; end: P(10^5);

Select[Range[10^3], Function[n, And[CompositeQ@ n, AllTrue[Flatten@ Function[w, Map[Function[k, Map[FromDigits[ReplacePart[w, k -> #]] &, Range[0, 9]]], Range@ Length@ w] /. m_ /; m == n -> Nothing]@ IntegerDigits@ n, CoprimeQ[#, n] &]]]] (* Michael De Vlieger, Apr 15 2016 *)

A329100 Composite palindromes whose divisors > 1 are all nontrivial palindromes (i.e., palindromes with at least two digits).

Original entry on oeis.org

121, 1111, 1331, 1441, 1661, 1991, 3443, 3883, 7997, 10201, 12221, 13231, 14641, 15251, 15851, 18281, 19291, 31613, 35653, 37673, 37873, 38683, 112211, 113311, 115511, 116611, 124421, 125521, 134431, 136631, 139931, 145541, 146641, 157751, 167761, 169961, 176671

Offset: 1

Maxim Veselov, Nov 04 2019

This is the intersection of A062687 and A038511.
From Chai Wah Wu, Nov 08 2019 : (Start)
All terms start and end with the digits 1,3,7 or 9.
First term with 3 prime factors: 1331 = 11^3.
First term with 3 distinct prime factors: 145541 = 11*101*131.
First term with 4 prime factors: 14641 = 11^4.
First term with 5 prime factors: 1478741 = 11^4*101.
No term with more than 3 distinct prime factors or more than 5 prime factors among first 10000 terms.
(End)

			For k = 1331, its divisors > 1 are 11, 121 and 1331, all of which are palindromes with at least two digits, so 1331 is a term.
For k = 167761, its divisors > 1 are 11, 101, 151, 1111, 1661, 15251 and 167761, all of which are palindromes with at least two digits, so 167761 is a term.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A008364, A038511, A062687.

aQ[n_] := CompositeQ[n] && AllTrue[Rest @ Divisors[n], # > 10 && PalindromeQ @ IntegerDigits[#] &]; Select[Range[200000], aQ] (* Amiram Eldar, Nov 06 2019 *)

isA329100(n) = if((n>1) && !isprime(n) && gcd(n,210)==1, {d = divisors(n); rd = vector(#d, i, subst(Polrev(digits(d[i])), x, 10)); (d == rd); }, 0) \\ Jianing Song, Nov 06 2019, based on the program of A062687

More terms from Jianing Song, Nov 06 2019

cp's OEIS Frontend

A140461 Numbers in A008364 but not in A038511.

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Maple

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A141864 Frequency of A038511 in the interval [(n-1)*3*5*7,n*3*5*7].

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Maple

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A038510 Composite numbers with smallest prime factor >= 7.

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Maple

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A008367 Composite but smallest prime factor >= 17.

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GAP

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A100319 Even numbers m such that at least one of m-1 and m+1 is composite.

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Mathematica

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Sage

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A121764 Single (or isolated or non-twin) primes of form 6n + 1.

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Magma

Mathematica

PARI

Sage

Extensions

A054550 Composite numbers whose least prime factor is either 5 or 7.

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Mathematica

A141864 Frequency of A038511 in the interval [(n-1)357,n357].