K. D. Bajpai - cp's OEIS frontend

A344780 Semiprimes that are product of two distinct Honaker primes.

34453, 59867, 120191, 136109, 137419, 142921, 170431, 178291, 187723, 205801, 250603, 253223, 273257, 275887, 280471, 286933, 290951, 297763, 319771, 339421, 342163, 348853, 354617, 356189, 357499, 357943, 367193, 376879, 401777, 410947, 413173, 422999, 449723

Offset: 1

K. D. Bajpai, May 28 2021

Subsequence of A006881.

a(1) = 34453 is the only number <= 5*10^6 that is a triangular number.

			34453 = 131*263 which are distinct Honaker primes.
120191 = 263*457 which are distinct Honaker primes.

Cf. A006881, A033548, A144482, A144856, A333788.

isA006881 := proc(n)
    if numtheory[bigomega](n) =2 and A001221(n) = 2 then
        true ;
    else
        false ;
    end if;
end proc:
isA344780 := proc(n)
    if isA006881(n) then
        for p in ifactors(n)[2] do
            if not isA033548(op(1,p)) then
                return false;
            end if;
        end do:
        true ;
    else
        false;
    end if;
end proc:
for n from 1  do
    if isA344780(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Jul 07 2021

fHQ[n_] := Plus @@ IntegerDigits@n == Plus @@ IntegerDigits@PrimePi@n;
lst = {}; Do[If[Plus @@ Last /@ FactorInteger[n] == 2, a = Length[First /@ FactorInteger[n]]; If[a == 2, b = First /@ FactorInteger[n]; c = b[[1]]; d = b[[2]]; If[fHQ[c] && fHQ[d], AppendTo[lst, {n,c,d}]]]], {n, 2000000}]; lst

A343192 Happy Honaker primes.

Original entry on oeis.org

263, 1039, 1933, 2221, 3067, 3137, 5741, 6343, 6353, 6971, 7481, 8821, 9103, 10247, 11251, 12347, 13037, 13339, 13457, 13933, 14437, 16451, 17317, 18041, 21617, 26309, 26339, 30091, 30293, 31177, 32009, 34471, 35227, 36307, 36433, 37117, 41131, 41333, 41801, 43781

Offset: 1

K. D. Bajpai, Apr 07 2021

Intersection of A033548 and A035497 or A007770.

			263 is a Honaker prime: the number of primes up to 263 is 56 and 2 + 6 + 3 = 11 = 5 + 6. 263 is also a Happy number: iterating the sum of squares of digits terminates in 1, i.e., 263 -> 4 + 36 + 9 = 49 -> 16 + 81 = 97 -> 81 + 49 = 130 -> 1 + 9 + 0 = 10 -> 1 + 0 = 1. Thus 263 is a Happy Honaker prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A007770, A033548, A035497, A046519.

Select[Prime[Range[20000]], FixedPoint[Total[IntegerDigits[#]^2] &, #, 10] == 1 && Plus @@ IntegerDigits@# == Plus @@ IntegerDigits@PrimePi@# &]

A343139 Numbers k that satisfy the condition digitsum(k) = digitsum(pi(k)) where pi is the prime counting function.

Original entry on oeis.org

15, 27, 51, 63, 120, 130, 131, 142, 153, 164, 208, 218, 230, 242, 252, 262, 263, 274, 305, 318, 327, 338, 348, 360, 370, 381, 392, 413, 424, 435, 446, 456, 457, 702, 712, 722, 732, 805, 860, 901, 912, 922, 932, 1016, 1027, 1038, 1039, 1048, 1049, 1059, 1071, 1080

Offset: 1

K. D. Bajpai, Apr 06 2021

a(7) = 131 is the first prime in this sequence.

A033548 (Honaker primes) is a subsequence of this sequence.

			153 is a term because the number of primes up to 153 is 36 and 1 + 5 + 3 = 9 = 3 + 6.
435 is a term because number of primes up to 435 is 84 and 4 + 3 + 5 = 12 = 8 + 4.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000720, A007953, A010846, A033548, A033549.

fHQ[n_] := Plus @@ IntegerDigits@n == Plus @@ IntegerDigits@PrimePi@n; Select[Range[3000], fHQ[#] &]

for(n=1, 5000, if(sumdigits(n)==vecsum(digits(primepi(n))), print1(n, ", " )));

upto(n) = { my(q = 2, ulim = nextprime(n), pi = 0, res = List()); forprime(p = 3, ulim, pi++; for(i = q, p-1, if(sumdigits(i) == sumdigits(pi), listput(res, i) ) ); q = p ); res } \\ David A. Corneth, May 26 2021

from sympy import primepi
def sd(n): return sum(map(int, str(n)))
def ok(n): return sd(n) == sd(primepi(n))
print(list(filter(ok, range(1, 1081)))) # Michael S. Branicky, May 28 2021

A337491 Numbers k such that exactly one of 2k + 3 and 4k + 3 is prime.

Original entry on oeis.org

8, 11, 13, 16, 22, 26, 28, 29, 31, 35, 37, 38, 41, 43, 44, 50, 53, 56, 59, 64, 65, 68, 70, 73, 74, 76, 80, 85, 86, 88, 91, 97, 98, 107, 109, 112, 113, 116, 118, 121, 122, 125, 127, 133, 134, 136, 137, 139, 142, 145, 146, 149, 151, 152, 155, 160, 161, 167, 170

Offset: 1

K. D. Bajpai, Aug 29 2020

nonn

Integers that are in A067076 or in A095278, but not in both. - Michel Marcus, Aug 29 2020

			a(1) = 8 is a term because 2*8 + 3 = 19 is a prime; but 4*8 + 3 = 35 = (5*7) is a composite number.
a(4) = 16 is a term because 2*16 + 3 = 35 = (5*7) is a composite number; but 4*16 + 3 = 67  is a prime.
a(6) = 26 is a term because 2*26 + 3 = 55 = (5*11) is a composite number; but 4*26 + 3 = 107  is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5000

Cf. A063908, A067076, A095278, A337480.

A337491:=n->`if`((isprime(2*n+3) xor isprime(4*n+3)), n, NULL): seq(A337491(n), n=1..500);

Select[Range[0, 250], Xor[PrimeQ[2 # + 3], PrimeQ[4 # + 3]] &]
Select[Range[200],Total[Boole[PrimeQ[{2,4}#+3]]]==1&] (* Harvey P. Dale, Jan 26 2023 *)

isok(k) = bitxor(isprime(2*k+3), isprime(4*k+3)); \\ Michel Marcus, Aug 29 2020

A337480 Numbers k such that exactly one of 6k + 5 and 12k + 5 is prime.

Original entry on oeis.org

6, 12, 13, 17, 18, 19, 23, 26, 27, 28, 31, 33, 39, 41, 44, 47, 48, 49, 52, 53, 54, 56, 57, 59, 67, 68, 69, 74, 76, 78, 83, 86, 87, 88, 91, 93, 94, 97, 101, 109, 112, 114, 116, 117, 124, 126, 128, 129, 132, 133, 137, 139, 141, 144, 146, 147, 151, 154, 159, 161

Offset: 1

K. D. Bajpai, Aug 28 2020

nonn

			a(5) = 18 is a term because 6*18 + 5 = 113 is prime; but 12*18 + 5 = 221 = (13*17) is a composite number.
a(8) = 26 is a term because 6*26 + 5 = 161 = (7*23) is a composite number; but 12*26 + 5 = 317 is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A063913, A145487, A145490, A172287.

A337480:=k->`if`(isprime(6*k+5) xor isprime(12*k+5),k, NULL): seq(A337480(k), k=1..1000);

Select[Range[0, 250], Xor[PrimeQ[6 # + 5], PrimeQ[12 # + 5]] &]

for(k=1, 1000, if (bitxor(isprime(6*k+5), isprime(12*k+5)), print1(k, ", ")));

A331203 Numbers k such that k/(digsum(k)) is an integer cube.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 72, 243, 320, 486, 512, 640, 704, 832, 960, 1000, 1088, 1125, 2000, 2401, 3000, 3430, 4000, 4116, 4802, 5000, 5145, 5831, 6000, 6174, 6517, 6860, 7000, 7546, 8000, 8575, 8918, 9000, 9216, 9947, 19683, 35152, 35937, 41743, 43940, 46137

Offset: 1

K. D. Bajpai, Jan 12 2020

If m belongs to the sequence, then 1000*m also belongs to the sequence. - Rémy Sigrist, Jan 12 2020

			a(11) = 243: 243/(2 + 4 + 3) = 27 = 3^3.
a(12) = 320: 320/(3 + 2 + 0) = 64 = 4^3.

Cf. A001102, A005349, A007953, A028839, A059094.

[n : n in[1 .. 1000] | IsIntegral((n/(&+Intseq(n)))^(1/3))];

Select[Range[100000], IntegerQ[CubeRoot[#/Total[IntegerDigits[#]]]] &]

is(n) = my (k=n/sumdigits(n)); type(k)==type(42) && ispower(k,3) \\ Rémy Sigrist, Jan 12 2020

A331346 Primes using all the square digits {0, 1, 4, 9} and no others.

Original entry on oeis.org

1049, 1409, 4019, 4091, 9041, 10499, 10949, 14009, 49019, 49109, 90149, 90401, 94109, 99041, 99401, 100049, 101149, 101419, 101449, 104009, 104119, 104149, 104491, 104911, 104999, 109049, 109141, 109441, 110419, 110491, 111049, 111409, 114901, 140009, 140191, 140419

Offset: 1

K. D. Bajpai, Jan 14 2020

Subsequence of A061246.

			a(1) = 1049 is prime containing all the square digits (0, 1, 4, 9) and no others.
a(2) = 1409 is prime containing all the square digits (0, 1, 4, 9) and no others.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A061246, A066591, A323387.

[p:p in PrimesUpTo(150000)|Set(Intseq(p)) eq {0,1,4,9}]; // Marius A. Burtea, Jan 14 2020

f:= proc(n) local L,x;
  L:= convert(n,base,4);
  if convert(L,set) <> {0,1,2,3} then return NULL fi;
  L:= subs(2=4,3=9,L);
  x:= add(L[i]*10^(i-1),i=1..nops(L));
  if isprime(x) then x else NULL fi
end proc:
map(f, [$4^3..4^6]); # Robert Israel, Jan 16 2020

Select[FromDigits /@ Tuples[{0, 1, 4, 9}, 6], PrimeQ[#] && Union[IntegerDigits[#]] == {0, 1, 4, 9} &]

A330438 Numbers k such that k^2-2 and k^3-2 are prime.

Original entry on oeis.org

9, 15, 19, 27, 37, 121, 135, 145, 211, 217, 259, 265, 267, 279, 355, 357, 387, 391, 435, 489, 525, 561, 615, 621, 727, 951, 987, 1029, 1119, 1141, 1177, 1251, 1287, 1357, 1435, 1491, 1561, 1617, 1717, 1785, 1819, 1839, 1875, 1909, 1989, 2001, 2077, 2107, 2211

Offset: 1

K. D. Bajpai, Dec 14 2019

nonn

Intersection of A028870 and A038599.

			a(1) = 9: 9^2 - 2 = 79; 9^3 - 2 = 727; both results are prime.
a(2) = 15: 15^2 - 2 = 223; 15^3 - 2 = 3373; both results are prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A028870, A028873, A038599, A038600, A108701.

[n : n in [1 .. 100] | IsPrime (n^2 - 2) and IsPrime (n^3 - 2)];

filter:= k -> isprime(k^2-2) and isprime(k^3-2):
select(filter, [$2..10000]); # Robert Israel, Dec 24 2019

Select[Range[10000], PrimeQ[#^3 - 2] && PrimeQ[#^2 - 2] &]

A330508 Numbers k such that k + 6^t is semiprime for t = 0 to 9.

Original entry on oeis.org

61273, 109441, 160213, 274501, 275473, 311593, 360673, 394201, 477181, 486061, 514993, 522085, 617137, 620053, 715477, 725485, 803833, 812677, 847117, 1063585, 1146913, 1182577, 1215865, 1232917, 1409425, 1508113, 1587241, 1768993, 1863073, 1895413, 2085517, 2095177

Offset: 1

K. D. Bajpai, Dec 16 2019

nonn

a(2620) = 530079693 is the first multiple of 3 in this sequence; there are no multiples of 2. - Charles R Greathouse IV, Dec 20 2019

			a(1) = 61273:
  61273 + 6^0  =    61274 =   2 *  30637;
  61273 + 6^1  =    61279 = 233 *    263;
  61273 + 6^2  =    61309 =  37 *   1657;
  61273 + 6^3  =    61489 =  17 *   3617;
  61273 + 6^4  =    62569 =  13 *   4813;
  61273 + 6^5  =    69049 =  29 *   2381;
  61273 + 6^6  =   107929 =  37 *   2917;
  61273 + 6^7  =   341209 =  11 *  31019;
  61273 + 6^8  =  1740889 = 197 *   8837;
  61273 + 6^9  = 10138969 =  89 * 113921;
all ten results are semiprime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A076274.

Cf. A001358, A082919, A096173, A104238, A105041.

f:=func; [k:k in [1..2100000]|forall{m:m in [0..9]|f(k+6^m)}]; // Marius A. Burtea, Dec 20 2019

fX[n_] = PrimeOmega[n] == 2; Select[Range[2000000], AllTrue[# + 6^{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, fX] &]

issemi(n)=bigomega(n)==2
is(n)=for(t=0,9, if(!issemi(n+6^t), return(0))); 1 \\ Charles R Greathouse IV, Dec 20 2019

A329727 Numbers k such that k^3 +- 2 and k +- 2 are prime.

Original entry on oeis.org

129, 1491, 1875, 2709, 5655, 6969, 10335, 14325, 14421, 17319, 26559, 35109, 37509, 43719, 50229, 52629, 101871, 102795, 104325, 105501, 120429, 127599, 132699, 136395, 137829, 157521, 172425, 173685, 179481, 186189, 191829, 211371, 219681, 221199, 229215, 234195

Offset: 1

K. D. Bajpai, Nov 19 2019

nonn

All terms in this sequence are divisible by 3.

			a(1) = 129:
  129^3 + 2 = 2146691;
  129^3 - 2 = 2146687;
  129   + 2 =     131;
  129   - 2 =     127; all four results are prime.
a(2) = 1491:
  1491^3 + 2 = 3314613773;
  1491^3 - 2 = 3314613769;
  1491   + 2 =       1493;
  1491   - 2 =       1489; all four results are prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Intersection of A038599, A067200, and A087679.

Cf. A040976, A052147, A090121, A268043, A268186.

[k:k in [1..250000]|forall{m:m in [-2,2]|IsPrime(k+m) and IsPrime(k^3+m)}]; // Marius A. Burtea, Nov 20 2019

Select[Range[500000], PrimeQ[#^3 + 2] && PrimeQ[#^3 - 2] && PrimeQ[# + 2] && PrimeQ[# - 2] &]

isok(k) = isprime(k-2) && isprime(k+2) && isprime(k^3-2) && isprime(k^3+2); \\ Michel Marcus, Nov 24 2019

list(lim)=my(v=List(),p=127,k); forprime(q=131,lim+2,if(q-p==4 && isprime((k=p+2)^3-2) && isprime(k^3+2), listput(v,k)); p=q); Vec(v) \\ Charles R Greathouse IV, May 06 2020

cp's OEIS Frontend

User: K. D. Bajpai

A344780 Semiprimes that are product of two distinct Honaker primes.

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Maple

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A343192 Happy Honaker primes.

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A343139 Numbers k that satisfy the condition digitsum(k) = digitsum(pi(k)) where pi is the prime counting function.

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A337491 Numbers k such that exactly one of 2*k + 3 and 4*k + 3 is prime.

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Maple

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PARI

A337480 Numbers k such that exactly one of 6*k + 5 and 12*k + 5 is prime.

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Maple

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A331203 Numbers k such that k/(digsum(k)) is an integer cube.

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Magma

Mathematica

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A331346 Primes using all the square digits {0, 1, 4, 9} and no others.

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Magma

Maple

A337491 Numbers k such that exactly one of 2k + 3 and 4k + 3 is prime.

A337480 Numbers k such that exactly one of 6k + 5 and 12k + 5 is prime.