A108605 -id:A108605 - cp's OEIS frontend

A378748 Möbius transform of A378747.

1, 0, 1, 1, 2, 0, 4, 5, 7, 0, 5, 4, 7, 2, 5, 19, 8, 8, 10, 6, 11, 0, 13, 20, 16, 2, 41, 14, 14, 2, 17, 65, 11, 0, 19, 36, 19, 2, 17, 30, 20, 10, 22, 12, 39, 4, 25, 76, 48, 12, 17, 20, 28, 64, 21, 58, 23, 0, 29, 28, 32, 4, 73, 211, 31, 2, 34, 18, 31, 14, 35, 132, 38, 2, 49, 26, 43, 10, 40, 114, 223, 0, 43, 60, 33, 2

Offset: 1

Antti Karttunen, Dec 09 2024

nonn

No negative terms.

Cf. A000010, A003961, A003972, A008683, A048673, A051953, A378521, A378747.

Positions of 0's is given by {2} U A108605.

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A378747(n) = (A048673(n)-(sigma(n)-n));
A378748(n) = sumdiv(n,d,moebius(d)*A378747(n/d));

a(n) = Sum_{d|n} A008683(d)*A378747(n/d).
a(n) = A378521(n) - A051953(n).
For n > 1, a(n) = A000010(n) + A000010(A003961(n))/2 - n.

A108610 Semiprimes with prime sum of decimal digits and prime sum of prime factors.

Original entry on oeis.org

34, 58, 142, 214, 274, 298, 382, 454, 478, 562, 694, 838, 922, 1042, 1138, 1198, 1282, 1318, 1642, 1714, 2038, 2098, 2122, 2182, 2302, 2458, 2638, 2854, 2902, 2962, 3334, 3394, 3442, 3574, 3754, 3862, 4054, 4162, 4258, 4474, 4618, 4762, 5314, 5374, 5422

Offset: 1

Zak Seidov, Jun 12 2005

Intersection of A108605 and A108606. All terms are even. Cf. A001358 semiprimes, A101605 3-almost primes, A108605 semiprimes with prime sum of factors, A108606 semiprimes with prime sum of digits.

			34=2*17 (semiprime), with 3+4=7 and 2+17=19 both prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A101605, A108605, A108606.

psddQ[n_]:=!IntegerQ[Sqrt[n]]&&PrimeOmega[n]==2&&PrimeQ[Total[ IntegerDigits[n]]] && PrimeQ[Total[Transpose[FactorInteger[n]][[1]]]]; Select[Range[5500],psddQ] (* Harvey P. Dale, Oct 03 2012 *)

A176811 Number of primes between 2(lesser of n-th twin prime pair) and 2(greater of n-th twin prime pair).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1

Offset: 1

Juri-Stepan Gerasimov, Apr 26 2010

nonn

Number of primes between 2*A001359(n) and 2*A006512(n).
Number of primes between A108605(n) and A176810(n).
Number of primes between 2*A077800(2n-1) and 2*A077800(2n).

			a(1)=1 because 2*3 < 7 (prime) < 2*5;
a(2)=2 because 2*5 < 11 (prime) < 13(prime) < 2*7;
a(3)=1 because 2*11 < 23 (prime) < 2*13.

Cf. A001359, A006512, A077800, A108605, A176810.

A001359 := proc(n) option remember; if n = 1 then 3; else for a from procname(n-1)+2 by 2 do if isprime(a) and isprime(a+2) then return a; end if; end do: end if; end proc:
A006512 := proc(n) A001359(n)+2 ; end proc:
A176811 := proc(n) numtheory[pi](2*A006512(n)) - numtheory[pi](2*A001359(n)) ; end proc:
seq(A176811(n),n=1..120) ; # R. J. Mathar, Apr 27 2010

PrimePi[2*#[[2]]]-PrimePi[2*#[[1]]]&/@Select[Partition[Prime[Range[1000]],2,1],#[[2]]- #[[1]] == 2&] (* Harvey P. Dale, Jul 21 2023 *)

Terms corrected starting at a(34) by R. J. Mathar, Apr 27 2010

A134622 Products pq ("semiprimes") of two primes p and q >= p such that q+2 is a prime.

Original entry on oeis.org

6, 9, 10, 15, 22, 25, 33, 34, 51, 55, 58, 77, 82, 85, 87, 118, 119, 121, 123, 142, 145, 177, 187, 202, 203, 205, 213, 214, 221, 274, 287, 289, 295, 298, 303, 319, 321, 355, 358, 377, 382, 394, 411, 413, 447, 451, 454, 478, 493, 497, 505, 533, 535, 537, 538, 551, 562, 573

Offset: 1

N. J. A. Sloane, Oct 17 2009

nonn

Cf. A108605.

p:=ithprime;
t1:=[];
for i from 1 to 100 do
p1:=p(i);
for j from i to 100 do
p2:=p(j);
if isprime(p2+2) then t1:=[op(t1),p1*p2]; fi;
od: od:
t2:=sort(t1);

Select[Range[573],PrimeOmega[#]==2&&PrimeQ[Divisors[#][[-2]]+2]&] (* James C. McMahon, Apr 08 2025 *)

A152126 If f(x) = x^3+x^5+x^11+x^17+x^29+x^41+..., where the exponents are the smaller twin of twin prime pairs, consider {f(x)}^2 and write the exponents of that expansion down : x^6+2x^8+x^10+2x^12+.... The proposed sequence is that sequence of exponents.

Original entry on oeis.org

6, 8, 10, 14, 16, 20, 22, 28, 32, 34, 40, 44, 46, 52, 58, 62, 64, 70, 74, 76, 82, 88, 100, 104, 106, 110, 112, 118, 124, 130, 136, 140, 142, 148, 152, 154, 160, 166, 172, 178, 182, 184, 190, 194, 196, 200, 202, 208, 214, 220, 226, 230, 232, 238, 242, 244, 250, 256, 262, 268, 272, 274, 280, 284, 286, 292

Offset: 1

Paul Bruckman (pbruckman(AT)hotmail.com), Nov 25 2008

nonn

I would also like to tabulate the corresponding sequences for 3rd and higher powers of f(x) in separate sequences, maybe as far as 12th powers of f, assigning new numbers to each such sequence. For example, for the 3rd power, the sequence would begin {9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 31, etc. (the first "gap" appearing at 29). Each sequence would show no more than perhaps 500 (?) terms, or whatever number is needed to display a first gap.

Numbers of the form A001359(i)+A001359(j), including those of A108605 related to i=j. - R. J. Mathar, Nov 28 2008

			I would like to show that some power of f(x) (as low a power as possible) contains no gaps. By this, I mean that the sequence of numbers in the m-th power of f should have the same parity as m and should start with 3m and that the sequence of odd (or even) numbers should have no gaps.

Corrected coefficient [x^10](f^2) in definition, inserted 34, extended. - R. J. Mathar, Nov 28 2008

A166305 Even semiprimes k such that the largest prime factor + 8 is a prime. Also semiprimes k such that k+16 is semiprime.

Original entry on oeis.org

6, 10, 22, 46, 58, 106, 118, 142, 178, 202, 262, 298, 346, 382, 466, 526, 538, 718, 778, 802, 862, 898, 958, 982, 1126, 1138, 1186, 1198, 1306, 1366, 1402, 1438, 1486, 1522, 1642, 1822, 1858, 1966, 2026, 2062, 2122, 2218, 2326, 2386, 2446, 2458, 2566, 2578

Offset: 1

Giovanni Teofilatto, Oct 11 2009

nonn

Cf. A108605, A165986.

A006530 := proc(n) numtheory[factorset](n) ; max(op(%)) ; end proc: isA166305 := proc(n) if type(n,'even') then if numtheory[bigomega](n) = 2 then isprime(A006530(n)+8) ; else false; end if; else false; end if; end proc: for n from 4 to 3000 by 2 do if isA166305(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Jan 30 2010

Select[2*Range[1500],PrimeOmega[#]==PrimeOmega[#+16]==2&] (* Harvey P. Dale, Dec 28 2013 *)

Extended by R. J. Mathar, Jan 30 2010

A258400 Perfect powers m^k such that m, k and m+k are primes.

Original entry on oeis.org

8, 9, 25, 32, 121, 289, 841, 1681, 2048, 3481, 5041, 10201, 11449, 18769, 22201, 32041, 36481, 38809, 51529, 57121, 72361, 78961, 96721, 120409, 131072, 175561, 185761, 212521, 271441, 323761, 358801, 380689, 410881, 434281, 654481, 674041, 683929, 734449

Offset: 1

Carlos Eduardo Olivieri, May 28 2015

nonn

Necessarily either m or k = 2, thus if a(n) is even, it is a power of 2 with odd prime exponent, otherwise (if a(n) is odd), it is a square of odd prime.
For each term m^k, there will be another k^m.
a(3), a(5), a(11) are of the form n! + 1.
Let F(m,k) = m*k, such that m^k = a(n), so A108605 is a subsequence of F. For example a(1) = 2^3 and F(2,3) = A108605(1).

			a(1) = 8, because 8 = 2^3 and 2+3 = 5.
a(4) = 32, because 32 = 2^5 and 2+5 = 7.
a(5) = 121, because 121 = 11^2 and 11+2 = 13.
a(25) = 131072, because 131072 = 2^17 and 2+17 = 19.

Subsequence of A001597, A000961.

Cf. A000079, A001248, A108605, A109611.

SmallestDivisor[n_] := If[n == 1, 1, Divisors[n][[2]]]; perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; ppl = Select[Range[200000], perfectPowerQ]; base[n_] := ppl[[n]]^(1/exp[n]); exp[n_] := SmallestDivisor[GCD @@ FactorInteger[ppl[[n]]][[All, 2]] ]; pp2l = Table[ {base[n], exp[n]}, {n, Length[ppl]}]; p[n_] := pp2l[[n]][[1]]; q[n_] := pp2l[[n]][[2]]; lt = Select[Range[Length[pp2l]], PrimeQ[p[#]] && PrimeQ[q[#]] && PrimeQ[p[#] + q[#]] &]; ppl[[lt]]
Select[Range[10^6], Length[f = FactorInteger@ #] == 1 && PrimeQ@ f[[1, 2]] && PrimeQ@ Total@ f[[1]] &] (* Giovanni Resta, Jun 23 2015 *)

a(28)-a(38) from Giovanni Resta, Jun 23 2015

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A378748 Möbius transform of A378747.

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A108610 Semiprimes with prime sum of decimal digits and prime sum of prime factors.

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A176811 Number of primes between 2*(lesser of n-th twin prime pair) and 2*(greater of n-th twin prime pair).

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A134622 Products pq ("semiprimes") of two primes p and q >= p such that q+2 is a prime.

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A152126 If f(x) = x^3+x^5+x^11+x^17+x^29+x^41+..., where the exponents are the smaller twin of twin prime pairs, consider {f(x)}^2 and write the exponents of that expansion down : x^6+2x^8+x^10+2x^12+.... The proposed sequence is that sequence of exponents.

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A166305 Even semiprimes k such that the largest prime factor + 8 is a prime. Also semiprimes k such that k+16 is semiprime.

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Maple

Mathematica

Extensions

A258400 Perfect powers m^k such that m, k and m+k are primes.

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Mathematica

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A176811 Number of primes between 2(lesser of n-th twin prime pair) and 2(greater of n-th twin prime pair).