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.
z = 400; c = Select[Range[2, z], ! PrimeQ@# &]; (* A002808 *) d = Select[Range[2, z], ! PrimeQ@# && OddQ@# &]; (* A014076 *) a[n_] := Length[Intersection[c, Prime[n] - Select[d, # < Prime[n] &]]]; Table[a[n], {n, 1, 120}] (* A024683 *) (* Clark Kimberling, Jul 21 2020 *) Table[Count[IntegerPartitions[Prime[n],{2}],?(AllTrue[#,CompositeQ]&&#[[1]]!= #[[2]]&)],{n,80}] (* _Harvey P. Dale, Feb 24 2023 *)
Select[4*(Range[54])^2-1, Not[PrimeQ[Sqrt[(#+ 1)]-1] && PrimeQ[Sqrt[(#+1)]+1]]&] Select[4*Range[100]^2-1,PrimeOmega[#]!=2&] (* Harvey P. Dale, Jul 24 2016 *)
15 is 1111_2 and 15=3*5 where 3 is 11_2, so 15 is a term.
f[n_] := Block[{nb = ToString@ FromDigits@ IntegerDigits[n, 2], psb = ToString@ FromDigits@ IntegerDigits[ #, 2] & /@ First@ Transpose@ FactorInteger@n, c = 0, k = 1}, lmt = 1 + Length@ psb; While[k < lmt, If[ StringCount[nb, psb[[k]]] > 0, c++ ]; k++ ]; c]; f[1] = 0; Select[ Range@ 286, !PrimeQ@ # && OddQ@ # && f@# > 0 &]
1 is an odd nonprime and has an odd sum of digits, so a(1)=1. 9 is an odd nonprime and has an odd sum of digits (and this is not true for any integers between 1 and 9), so a(2)=9. 21 is an odd nonprime, and the sum of its digits (2+1=3) is odd (and this is not true for any integers between 9 and 21), so a(3)=21, etc. 45 is in the sequence because it is odd, it is a nonprime and the sum of its digits (9) is odd. - _Emeric Deutsch_, Jan 21 2009
sd := proc (n) options operator, arrow: add(convert(n, base, 10)[j], j = 1 .. nops(convert(n, base, 10))) end proc: a := proc (n) if `mod`(n, 2) = 1 and isprime(n) = false and `mod`(sd(n), 2) = 1 then n else end if end proc: seq(a(n), n = 1 .. 400); # Emeric Deutsch, Jan 21 2009
Select[Complement[Range[1,501,2],Prime[Range[PrimePi[501]]]],OddQ[Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Dec 11 2010 *)
isok(n) = ! isprime(n) && (n % 2) && (sumdigits(n) % 2); \\ Michel Marcus, Sep 16 2016
Exactly 10 consecutive odd integers 1131(2)1149 are composite while 1151 is prime.
R:= NULL: count:= 0: k:= 1: state:= 0:
while count < 100 do
k:= k+2;
if isprime(k) then
if state >= 10 then R:= R,k - 20; count:= count + 1; fi;
state:= 0;
else state:= state + 1
fi;
od:
R; # Robert Israel, Feb 12 2025
Transpose[Select[Partition[Range[1,8001,2],11,1],PrimeQ/@ == {False,False,False,False,False,False,False,False,False,False,True}&]] [[1]] (* Harvey P. Dale, Nov 21 2011 *)
S(n) evaluated at n=1, 9, 15, 21, ... (taken from A014076) is 3, 15, 24, 33, 42, 51, etc., where only the even values (i.e., 24, 42, etc.) join the sequence.
A014076 := proc(n) option remember ; if n = 1 then 1; else for a from procname(n-1)+2 by 2 do if not isprime(a) then RETURN(a) ; fi; od: fi; end: S := proc(n) A064455(A014076(n)+1) ; end: for n from 1 to 200 do if S(n) mod 2 = 0 then printf("%d,",S(n)) ; fi; od: # R. J. Mathar, Jul 21 2009
a(1) = -0*(-1)^0 - 1*(-1)^1 = 0 + 1 = 1; a(2) = -4*(-1)^4 - 5*(-1)^5 - 6*(-1)6 - 7*(-1)^7 - 8*(-1)^8 - 9*(-1)^9 = -4 + 5 - 6 + 7 - 8 + 9 = 3.
A163300 := proc(n) if n <= 2 then op(n,[0,4]) ; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a; end if; end do; end if; end proc: A014076 := proc(n) if n = 1 then 1; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a ; end if; end do: end if; end proc: A001057 := proc(n) (1-(-1)^n*(2*n+1))/4; end proc: A163301 := proc(n) A001057( A014076(n)) - A001057(A163300(n)-1) ; end proc: seq(A163301(n),n=1..120) ; # R. J. Mathar, May 21 2010
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