A173977 -id:A173977 - cp's OEIS frontend

A173979 a(n) is the smallest number m from A173977 for which A020639(2m-1) = prime(n).

5, 13, 25, 127, 85, 196, 181, 472, 421, 946, 685, 1210, 925, 1105, 1882, 3157, 1861, 2446, 2521, 3541, 4306, 4690, 3961, 6160, 5707, 5305, 5725, 6922, 9436, 8065, 8581, 10207, 9661, 13336, 12307, 12796, 14752, 18955, 14965

Offset: 2

Vladimir Shevelev, Mar 04 2010

nonn

If the requirement that m be an element of A173977 is dropped, the sequence becomes A006254. - R. J. Mathar, Nov 02 2011

Amiram Eldar, Table of n, a(n) for n = 2..1000

Cf. A006254, A020639, A173977.

A020639 := proc(n) if n = 1 then 1; else min(op(numtheory[factorset](n)) ) ; end if; end proc:
isA173977 := proc(n) A020639(2*n-1) < A020639(2*n-3) ; end proc:
A173979 := proc(n) local p,m ; p := ithprime(n) ; for m from 1 do if A020639(2*m-1) = p and isA173977(m) then return m ; end if; end do: end proc:
seq(A173979(n),n=2..40) ; # R. J. Mathar, Sep 02 2011

lpf[n_] := lpf[n] = FactorInteger[n][[1, 1]]; q[n_] := lpf[2*n-1] < lpf[2*n-3]; seq[len_] := Module[{s = Table[0, {Prime[len+1]}], k = 2, c = 0, p}, While[c < len, If[q[k], p = lpf[2*k-1]; If[p <= Length[s] && s[[p]] == 0, c++; s[[p]] = k]]; k++]; Select[s, # > 0 &]]; seq[100] (* Amiram Eldar, Oct 25 2024 *)

Name corrected by Vladimir Shevelev, Mar 15 2010

A173978 Numbers n such that the least prime factor of 2n - 3 is less than that of 2n - 1, unless 2n - 3 and 2n - 1 are (twin) primes.

Original entry on oeis.org

2, 6, 9, 12, 15, 18, 19, 21, 24, 27, 30, 33, 34, 36, 39, 40, 42, 45, 48, 49, 51, 54, 57, 60, 61, 63, 64, 66, 69, 72, 75, 78, 79, 81, 82, 84, 87, 90, 93, 94, 96, 99, 102, 105, 106, 108, 109, 111, 112, 114, 117, 120, 123, 124

Offset: 1

Vladimir Shevelev, Mar 04 2010

nonn

Integers > 1 for which 2n - 3 is not in A001359 and A020639(2n-3) < A020639(2n-1).

Every multiple of 3 greater than 3 is in the sequence.

			a(3) = 9 because 2*9 - 3 = 15, the least prime factor of which is 3 and that is smaller than the least prime factor of 2*9 - 1 = 17.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A020639, A001359, A173977.

Select[Range[200], Not[PrimeQ[2# - 3] && PrimeQ[2# - 1]] && TrueQ[FactorInteger[2# - 3][[1, 1]] < FactorInteger[2# - 1][[1, 1]]] &] (* Alonso del Arte, Jun 05 2011 *)

More terms from Alonso del Arte, Jun 05 2011

A173980 a(n) is the smallest term m in A173978 for which A020639(2m-3) = prime(n), n > 1.

Original entry on oeis.org

6, 19, 40, 106, 112, 265, 220, 427, 625, 730, 871, 1252, 1141, 1717, 2095, 2332, 2716, 2380, 3445, 6097, 4465, 4027, 6187, 6646, 6415, 7675, 6796, 7141, 15991, 8701, 9106, 12400, 12025, 11251, 12610, 14995, 14101, 16117, 16696, 16201, 21631, 19006, 22486, 21967

Offset: 2

Vladimir Shevelev, Mar 04 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 2..1000
Vladimir Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A020639, A173978, A173977, A173979.

m = 50; v = Table[0, {m}]; lpf[n_] := FactorInteger[n][[1, 1]]; aQ[n_] := (! PrimeQ[2 n - 3] || ! PrimeQ[2 n - 1]) && lpf[2 n - 3] < lpf[2 n - 1]; c = 0; n = 1; While[c < m - 1, If[aQ[n], s = PrimePi[lpf[2 n - 3]]; If[s > 1 && s <= m && v[[s]] == 0, v[[s]] = n; c++]]; n++]; Rest[v] (* Amiram Eldar, Sep 12 2019 *)

More terms from Amiram Eldar, Sep 12 2019

A174453 a(n) is the smallest k >= 1 for which gcd(m + (-1)^m, m + n - 4) > 1, where m = n + k - 1.

Original entry on oeis.org

1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 12, 1, 2, 1, 1, 1, 18, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 30, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 42, 1, 2, 1, 1, 1, 6, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 60, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 72, 1, 2, 1, 1, 1, 9, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 6, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 102

Offset: 5

Vladimir Shevelev, Mar 20 2010

nonn

If a(n) > sqrt(n), then n-3 is the larger of twin primes. In these cases we have a(10)=5 and, for n > 10, a(n) = n-4. For odd n and for n == 2 (mod 6), a(n)=1; for n == 0 (mod 6), a(n)=2; for {n == 4 (mod 6)} & {n == 8 (mod 10)}, a(n)=4, etc. The problem is to develop this sieve for the excluding n for which a(n) <= sqrt(n) and to obtain nontrivial lower estimates for the counting function of the larger of twin primes.

Paul Tek, Table of n, a(n) for n = 5..10000
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A173980, A020639, A173978, A173977, A173979, A174217, A174216, A174214, A174215, A166945, A167495.

A174453 := proc(n) local k,m ; for k from 1 do m := n+k-1 ; if igcd(m+(-1)^m,m+n-4) > 1 then return k; end if; end do: end proc: seq(A174453(n),n=5..120); # R. J. Mathar, Nov 04 2010

a[n_] := For[k=1, True, k++, m=n+k-1; If[GCD[m+(-1)^m, m+n-4]>1, Return[k]] ];
Table[a[n], {n, 5, 106}] (* Jean-François Alcover, Nov 29 2017 *)

Terms beyond a(34) from R. J. Mathar, Nov 04 2010

cp's OEIS Frontend

A173979 a(n) is the smallest number m from A173977 for which A020639(2m-1) = prime(n).

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A173978 Numbers n such that the least prime factor of 2n - 3 is less than that of 2n - 1, unless 2n - 3 and 2n - 1 are (twin) primes.

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A173980 a(n) is the smallest term m in A173978 for which A020639(2m-3) = prime(n), n > 1.

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A174453 a(n) is the smallest k >= 1 for which gcd(m + (-1)^m, m + n - 4) > 1, where m = n + k - 1.

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