1,2

No a(n) can be even, since a(n)+2 must be prime. If a(n) is a prime, then it is a Sophie Germain twin prime (A045536). The only square is 9. Let the degree of n be the sum of the exponents in its prime factorization. By convention, degree(1)=0. Then every a(n) has degree less than or equal to 3. Let the weight of n be the number of its distinct prime factors. By convention, weight(1)=0. Clearly, w<=d is always true, with d=w only when the number is square-free. Let [w,d] be the set of all integers with weight w and degree d. Then only the following possibilities occur: 1. [0,0] => a(1)=1. 2. [1,1] => Sophie Germain twin prime: 3, 5, 11, 29, A005384, A045536. 3. [1,2] => a(4)=9 is the only occurrence. 4. [1,3] => 5^3, 71^3 and 303839^3 are the first few cubes, A000578, A120808. 5. [2,2] => 5*7, 3*13 and 5*13 are the first few semiprimes, A001358, A120807. 6. [2,3] => 11*13^2, 61^2*89 and 13^2*12671 are the first few examples, A014612, A054753, A120809. 7. [3,3] => 5*11*17, 5*53*1151, 5*11*42533 are the first few 3-almost primes, A007304, A120810.

a(n) = n-th number such that n+d+1 is prime for all divisors d of n.

a(11)=125 since divisors(125)={1,5,25,125} and the set of all n+d+1 is {127,131,151,251} and these are all prime.

with(numtheory); L:=[1]: for w to 1 do for k from 1 to 12^6 while nops(L)<=1000 do x:=2*k+1; if andmap(isprime, [x+2, 2*x+1]) then S:=divisors(x) minus {1, x}; Q:=map(z-> x+z+1, S); if andmap(isprime, Q) then L:=[op(L), x]; print(nops(L), ifactor(x)); fi; fi; od od; L;

Cf. A000578, A001358, A005384, A007304, A014612, A054753, A120776, A120807, A120808, A120809, A120810.

Sequence in context: A092917 A163778 A160358 this_sequence A020946 A091785 A003075

Adjacent sequences: A120803 A120804 A120805 this_sequence A120807 A120808 A120809

nonn

Walter Kehowski (wkehowski(AT)cox.net), Jul 06 2006

1,2

a(1)=1. a(n) = p^3 where p is the (n-1)st prime such that a(n)+d+1 is prime for all divisors d of a(n).

a(3)=357911 since n=357911=71^3, divisors(n)={1,71,71^2,71^3} and n+d+1={357913,357983,362953,715823} are all prime.

L:=[]: for w to 1 do for k from 1 while nops(L)<=50 do p:=ithprime(k); x:=p^3; if p mod 6 = 5 and andmap(isprime, [x+2, 2*x+1]) then S:={p, p^2}; Q:=map(z-> x+z+1, S); if andmap(isprime, Q) then L:=[op(L), x]; print(nops(L), p, x); fi; fi; od od;

Cf. A120806, A120808.

Sequence in context: A050640 A161354 A003751 this_sequence A013836 A048563 A110825

Adjacent sequences: A120804 A120805 A120806 this_sequence A120808 A120809 A120810

nonn

Walter Kehowski (wkehowski(AT)cox.net), Jul 06 2006

1,2

a(1)=1. a(n) = p where p is the (n-1)st prime such that x=p^3 is in A120806: x+d+1 is prime for all divisors d of x.

a(3)=71 since x=71^3=357911, divisors(x)={1,71,71^2,71^3} and x+d+1={357913,357983,362953,715823} are all prime.

L:=[]: for w to 1 do for k from 1 while nops(L)<=50 do p:=ithprime(k); x:=p^3; if p mod 6 = 5 and andmap(isprime, [x+2, 2*x+1]) then S:={p, p^2}; Q:=map(z-> x+z+1, S); if andmap(isprime, Q) then L:=[op(L), x]; print(nops(L), p, x); fi; fi; od od;

Cf. A120806, A120807.

Sequence in context: A033507 A092250 A133990 this_sequence A092204 A079874 A079340

Adjacent sequences: A120805 A120806 A120807 this_sequence A120809 A120810 A120811

nonn

Walter Kehowski (wkehowski(AT)cox.net), Jul 06 2006

1,1

a(n) = n-th element of A120806 of the form p^2*q where p and q are distinct odd primes.

a(1)=1859 since x=11*13^2, divisors(x)={1,11,13,11*13,13^2,11*13^2} and x+d+1={1861,1871,1873,2003,2029,3719} are all prime.

with(numtheory); is3almostprime := proc(n) local L; if n in [0, 1] or isprime(n) then return false fi; L:=ifactors(n)[2]; if nops(L) in [1, 2, 3] and convert(map(z-> z[2], L), `+`) = 3 then return true else return false fi; end; L:=[]: for w to 1 do for k from 1 while nops(L)<=50 do x:=2*k+1; y:=simplify(x^(1/3)); if x mod 6 = 5 and not type(y, integer) #clunky and not issqrfree(x) and is3almostprime(x) and andmap(isprime, [x+2, 2*x+1]) then S:=divisors(x); Q:=map(z-> x+z+1, S); if andmap(isprime, Q) then L:=[op(L), x]; print(nops(L), ifactor(x)); fi; fi; od od;

Cf. A120806, A054753.

Sequence in context: A051355 A064978 A159212 this_sequence A022062 A107526 A132202

Adjacent sequences: A120806 A120807 A120808 this_sequence A120810 A120811 A120812

nonn

Walter Kehowski (wkehowski(AT)cox.net), Jul 06 2006

1,1

a(n) = n-th element of A120806 of the form p*q*r where p, q and r are distinct odd primes.

a(1)=935 since x=5*11*17, divisors(x)={1,5,11,17,5*11,5*17,11*17,5*11*17} and x+d+1={937, 941, 947, 953, 991, 1021, 1123, 1871} are all prime.

with(numtheory); is3almostprime := proc(n) local L; if n in [0, 1] or isprime(n) then return false fi; L:=ifactors(n)[2]; if nops(L) in [1, 2, 3] and convert(map(z-> z[2], L), `+`) = 3 then return true else return false fi; end; L:=[]: for w to 1 do for k from 1 while nops(L)<=50 do x:=2*k+1; if x mod 6 = 5 and issqrfree(x) and is3almostprime(x) and andmap(isprime, [x+2, 2*x+1]) then S:=divisors(x); Q:=map(z-> x+z+1, S); if andmap(isprime, Q) then L:=[op(L), x]; print(nops(L), ifactor(x)); fi; fi; od od;

Cf. A120806, A007304, A120809, A120808, A120807.

Sequence in context: A171348 A013545 A029570 this_sequence A027554 A031902 A104917

Adjacent sequences: A120807 A120808 A120809 this_sequence A120811 A120812 A120813

nonn

Walter Kehowski (wkehowski(AT)cox.net), Jul 06 2006

1,2

Equivalently, products of primes == 1 (mod 6).

Positive integers n such that n+d+1 is divisible by 3 for all divisors d of n. For example, a(13)=91 since 91=7*13, 91+1+1=93=3*31, 91+7+1=99=9*11, 91+13+1=105=3*7*5, 91+91+1=183=3*61. The only prime p such that x+d+1 is divisible by p for all divisors d of x is p=3. The sequence consists of 1 and all integers whose prime divisors are of the form 6k+1. - Walter Kehowski (wkehowski(AT)cox.net), Aug 09 2006.

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

J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).

Walter Kehowski, D Numbers.

with(numtheory): for n from 1 to 1801 by 6 do it1 := ifactors(n)[2]: it2 := 1: for i from 1 to nops(it1) do if it1[i][1] mod 6 > 1 then it2 := 0; break fi: od: if it2=1 then printf(`%d, `, n) fi: od:

with(numtheory): cnt:=0: L:=[]: for w to 1 do for n from 1 while cnt<100 do dn:=divisors(n); Q:=map(z-> n+z+1, dn); if andmap(z-> z mod 3 = 0, Q) then cnt:=cnt+1; L:=[op(L), [cnt, n]]; fi; od od; L; - Walter Kehowski (wkehowski(AT)cox.net), Aug 09 2006.

Cf. A120806.

Sequence in context: A101324 A129904 A088513 this_sequence A133290 A038590 A129389

Adjacent sequences: A004608 A004609 A004610 this_sequence A004612 A004613 A004614

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Oct 30 2000

Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Andrew Plewe, May 31 2007

1,1

This sequence (A120811) is a generalization of twin prime (A001359), the sequence A120776 is a generalization of Sophie Germain prime (A005384), while A120806 is the generalization of Sophie germain twin prime (A045536). The same observations apply to A120811 as to A120806: the elements are (a) twin primes, (b) semiprimes pq, (c) 3-almost-primes, (d) 4-almost-primes. Moreover, the sequence includes all twin primes but in (b), (c) and (d) the containments are proper. The first occurrence of (d) is A120811(3980)=3^3*13147. Any others? A120811 CONJECTURE: These are all the elements, that is, no element of A120811 has more than 3 prime factors with no degree (sum of exponents) higher than 4.

a(n)=nth number such that n+d+1 is prime for all proper divisors d of n.

a(6)=27 since proper divisors={1,3,3^2} and 27+d+1={29,31,37} are all prime.

a(3980)=3^3*13147 since proper divisors={1,3,3^2,3^3,13147,3*13147,3^2*13147} and a(3980)+d+1={354971,354973,354979,354997,368117,394411,473293} are all prime.

with(numtheory); L:=[]: for w to 1 do for k from 1 while nops(L)<=5000 do x:=2*k+1; if isprime(x+2) then S:=divisors(x) minus {x}; Q:=map(z-> x+z+1, S); if andmap(isprime, Q) then fd:=fopen("C:/temp/n+d+1=prime-lower.txt", APPEND); fprintf(fd, "%d", x); fclose(fd); L:=[op(L), x]; print(nops(L), ifactor(x)); fi; #Q fi; #x od od;

Cf. A120806, A120776, A120806, A001359, A005384, A045536.

Sequence in context: A007952 A145819 A094509 this_sequence A123069 A100456 A059819

Adjacent sequences: A120808 A120809 A120810 this_sequence A120812 A120813 A120814

nonn

Walter Kehowski (wkehowski(AT)cox.net), Jul 07 2006

1,1

D Numbers.

a(n)=nth number x such that x+d+1 is composite for all divisors d of x.

a(9)=46=2*23 since 46+1+1=48=16*3, 46+2+1=49=7*7, 46+23+1=70=2*5*7, 46+46+1=93=3*31.

with(numtheory): cnt:=0: L:=[]: for w to 1 do for n from 1 while cnt<100 do dn:=divisors(n); Q:=map(z-> n+z+1, dn); if andmap(z-> not isprime(z), Q) then cnt:=cnt+1; L:=[op(L), [cnt, n]]; fi; od od; L;

Cf. A120806.

Sequence in context: A024606 A074628 A031194 this_sequence A007921 A092409 A124095

Adjacent sequences: A121055 A121056 A121057 this_sequence A121059 A121060 A121061

nonn

Walter Kehowski (wkehowski(AT)cox.net), Aug 09 2006