A033548 -id:A033548 - cp's OEIS frontend

A174924 Semiprimes sp(k) = q * r such that sum of digits of sp(k) equals sum of digits of the semiprime index k.

14, 15, 55, 121, 122, 123, 214, 215, 265, 287, 407, 481, 482, 535, 667, 813, 851, 901, 951, 1119, 1149, 1174, 1537, 1538, 1639, 1681, 1961, 2059, 2117, 2165, 2209, 2245, 2246, 2386, 2419, 2458, 2501, 2513, 2537, 2603, 2629, 2641, 2642, 2643, 2807, 2845

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 02 2010

Numbers of the form q * r where q and r are primes, not necessarily distinct.
These numbers are also called semiprimes or 2-almost primes.
For primes with such a property see A033548

			sp(5) = 14 = 2 * 7 is the 5th semiprime, sum of digits sod(14) = 1+4 = 5, 1st term
sp(6) = 15 = 3 * 5 is the 6th semiprime, sum of digits sod(15) = 1+5 = 6. 2nd term
sp(40) = 121 = 11^2 is the 40th semiprime, sum of digits sod(121) = 1+2+1 = 4, 4th term
Additionally for the prime based (q=r=11) square 121: sod(q) + sod(r) = 2 * sod(11) = 4
The first 110 such semiprimes:
14, 15, 55, 121, 122, 123, 214, 215, 265, 287, 407, 481, 482, 535, 667, 813, 851, 901, 951, 1119,
1149, 1174, 1537, 1538, 1639, 1681, 1961, 2059, 2117, 2165, 2209, 2245, 2246, 2386, 2419,
2458, 2501, 2513, 2537, 2603, 2629, 2641, 2642, 2643, 2807, 2845, 2846, 2858, 2859, 2921,
3158, 3205, 3218, 3427, 3439, 4322, 4333, 4367, 4661, 4713, 4714, 4735, 4811, 5221, 5317,
5318, 5615, 5707, 5753, 6009, 6022, 6023, 6046, 6081, 6082, 6117, 6193, 6283, 6371, 6411,
6423, 6514, 6515, 6527, 6541, 6542, 6593, 6635, 6649, 6683, 6694, 6905, 7251, 7291, 7363,
7387, 8023, 8102, 8153, 8203, 8401, 8402, 8403, 8503, 8531, 9019, 9201, 9223, 9271, 9902

Cf. A033548, A046328

A276255 Sum of first n Honaker primes.

Original entry on oeis.org

131, 394, 851, 1890, 2939, 4030, 5331, 6692, 8125, 9696, 11609, 13542, 15683, 17904, 20177, 22618, 25209, 27872, 30579, 33298, 36027, 38830, 41897, 45034, 48263, 51696, 55255, 58886, 62977, 67130, 71487, 75884, 80587, 85310, 90213, 95222, 100729, 106430, 112141

Offset: 1

K. D. Bajpai, Aug 25 2016

A Honaker prime is a prime number prime(k) such that k and prime(k) have the same sum of digits.

			The first Honaker prime is 131, so a(1) = 131.
The second Honaker prime is 263, so a(2) = 131 + 263 = 394.

K. D. Bajpai, Table of n, a(n) for n = 1..7300

Cf. A033548.

Accumulate[Select[Prime@Range@3000, Plus @@ IntegerDigits@# == Plus @@ IntegerDigits@PrimePi@# &]] (* Bajpai *)
DeleteDuplicates[Accumulate[Table[Prime[n] * Boole[Plus@@IntegerDigits[n] == Plus@@IntegerDigits[Prime[n]]], {n, 1000}]]] (* Alonso del Arte, Aug 25 2016 *)

first(n)=my(v=vector(n),i,k,s); forprime(p=2,, if(sumdigits(k++)==sumdigits(p), v[i++] = s+=p); if(i==n, return(v))) \\ Charles R Greathouse IV, Aug 29 2016

use ntheory ":all"; my($s,$i)=(0,0); forprimes { say $s+=$ if sumdigits($) == sumdigits(++$i) } 1e7; # Dana Jacobsen, Aug 29 2016

A277111 Lesser of twin primes P(k) and P(k+1) such that Sd(P(k)) + Sd(P(k+1)) = Sd(k) + Sd(k+1), where Sd(x) is the sum of digits of x.

Original entry on oeis.org

1619, 2309, 2339, 12239, 28109, 35081, 37307, 37571, 50549, 51059, 51719, 62129, 64919, 65729, 87539, 89519, 91079, 113759, 121439, 121631, 160649, 170351, 174329, 182129, 191249, 205949, 215459, 223679, 231839, 254039, 270269, 285119, 301841, 317489, 319829

Offset: 1

Paolo P. Lava, Sep 30 2016

			P(256) = 1619, P(257) = 1621; Sd(256) + Sd(257) = 13 + 14 = 27 and Sd(1619) + Sd(1621) = 17 + 10 = 27.

Paolo P. Lava, First 60 couples [k, P(k)]

Cf. A001359, A033548.

T:=proc(w) local x, y, z; x:=w; y:=0; for z from 1 to ilog10(x)+1 do y:=y+(x mod 10); x:=trunc(x/10); od; y; end:
P:= proc(q) local a,b,k,n;
for n from 1 to q do if ithprime(n+1)-ithprime(n)=2 then if T(ithprime(n))+T(ithprime(n+1))=T(n)+T(n+1) then print(ithprime(n)); fi; fi; od; end: P(10^5);

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

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@# &]

A169645 Primes p = prime(k) of form 13//r, s//13 or t//13//u and sod(p) = sod(k).

Original entry on oeis.org

131, 1301, 1361, 1913, 3137, 7013, 7213, 11353, 12613, 13007, 13037, 13127, 13217, 13297, 13327, 13339, 13367, 13417, 13457, 13933, 15913, 18013, 22613, 29131, 31391, 41131, 41333, 51131, 54013, 57139, 57713, 63313, 64513, 65713, 68813, 70139, 71353, 74713

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 05 2010

Sum of digits of p = prime(k), p containing the string "13", equals sum of digits of the prime index k
A subsequence of A033548
Still no (published) proof if sequence is infinite

			13//1 = 131 = prime(32), r = 1, sod(k) = 5
19//13 = 1913 = prime(293), s = 19, sod(k) = 14
3//13//7 = 3137 = prime(446), t = 3, u = 7, sod(k) = 14

Cf. A033548, A166573, A175017

sodQ[{a_,b_}]:=SequenceCount[IntegerDigits[b],{1,3}]>0&&Total[ IntegerDigits[ a]] ==Total[IntegerDigits[b]]; Select[Table[ {n, Prime[n]},{n,7000}],sodQ][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 10 2018 *)

Corrected and extended by Harvey P. Dale, May 10 2018

A178613 The smaller member prime(i) of an emirp pair (prime(i),prime(j)), such that the digit sum of i equals the digit sum of j.

Original entry on oeis.org

37, 359, 769, 1409, 7687, 10711, 10853, 11243, 11593, 13441, 13751, 14423, 14551, 14879, 15307, 15661, 16879, 17959, 30853, 31193, 33863, 34589, 37307, 37489, 38449, 73369, 74959, 75239, 78259, 78839, 79669, 90089, 92779, 100267, 101531

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 30 2010

We consider base-10 emirp pairs (13,31) = (prime(6),prime(11)), (17,71) = (prime(7),prime(20)), (37,73) = (prime(12),prime(21)), ... (see A006567) and the digit sums of their prime indices (6,2=1+1), (7,2=2+0), (3=1+2,3=2=1),.. (see A156793).

If the digits sums of the two indices are the same, the smaller representative of the emirp pair is entered into the sequence.

			37 = prime(12) and 73 = prime(21) are an emirp pair with equal digit sums of the indices 1+2 = 3 = 2+1, which puts 37 into the sequence.
359 = prime(72) and 953 = prime(162) are an emirp pair with digit sums 7+2 = 9 = 1+6+2, which puts 359 into the sequence.
The 6th term is from the pair (10711 = prime(1306), 11701 = prime(1405)), see A033548
16th term: (17959 = prime(2059), 95971 = prime(9250)).
21st term: (34589 = prime(3694), 98543 = prime(9463)).

W. W. R. Ball, H. S. M. Coxeter: Mathematical Recreations and Essays, 13th edition, Dover Publications, 2010
C. Mauduit, J. Rivat: Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals of Mathematics, Vol. 171, No. 3, 1591-1646, 2010
H Schubart: Einfuehrung in die klassische und moderne Zahlentheorie Vieweg, Braunschweig, 1974

Cf. A006567, A156793.

f[n_] := Plus @@ IntegerDigits@ PrimePi@n; fQ[n_] := Block[{id = IntegerDigits@n}, rid = Reverse@ id; q = FromDigits@ rid; rid != id && PrimeQ@ FromDigits@ rid && n < q && f@n == f@q]; lst = {}; p = 13; While[p < 102148, If[ fQ@p, AppendTo[lst, p]]; p = NextPrime@p]; lst (* Robert G. Wilson v, Jul 31 2010 *)

More terms from Robert G. Wilson v, Jul 31 2010

A261142 Numbers n such that n, p=prime(n) and q=prime(p) have the same sum of digits.

Original entry on oeis.org

88, 248, 826, 1417, 1571, 1595, 3682, 3928, 4448, 5089, 5137, 6479, 7754, 8038, 8384, 8461, 9257, 9640, 10393, 10825, 10922, 11878, 13294, 14290, 14767, 14941, 15977, 16786, 17684, 17777, 17935, 18437, 18677, 19495, 20497, 20555, 21649, 22487, 23239, 23396

Offset: 1

Zak Seidov, Oct 22 2015

Hence both p and q are Honaker primes (A033548) and both n and p are terms in A033549.

			n=88, p=prime(n)=457 and q=prime(p)=3229 have the same sum of digits=16;
n=248, p=prime(n)=1571 and q=prime(p)=13217 have the same sum of digits=14;
n=660349, p=178115131 and q=178115131 have the same sum of digits=28.

Cf. A000040, A000720, A007953, A033548, A033549.

A281299 Primes p whose binary representation p_2 is the decimal representation of a prime q; and also the sum of the decimal digits of p equals the sum of the digits of p_2.

Original entry on oeis.org

5011, 7001, 11251, 22501, 32303, 32411, 90031, 101107, 104123, 108011, 111323, 121343, 122131, 124001, 125101, 141023, 224011, 233021, 235003, 241141, 321203, 324011, 421303, 432031, 442201, 510331, 511213, 520411, 801011, 1000183, 1000541, 1001191, 1005223, 1006231

Offset: 1

K. D. Bajpai, Jan 19 2017

Intersection of A037308 and A065720.

			a(1) = 5011 is a prime;
5011_2 = 1001110010011_10 is a prime;
5 + 0 + 1 + 1 = 7;
1 + 0 + 0 + 1 + 1 + 1 + 0 + 0 + 1 + 0 + 0 + 1 + 1 = 7; both the digit sums are equal.

Cf. A000040, A033548, A037308, A065720, A089971.

Select[Prime[Range[1000000]], PrimeQ[FromDigits[IntegerDigits[#, 2]]] && Plus @@ IntegerDigits[#] == Plus @@ IntegerDigits[FromDigits[IntegerDigits[#, 2]]] &]

eva(n) = subst(Pol(n), x, 10)
is(n) = ispseudoprime(n) && ispseudoprime(eva(binary(n))) && sumdigits(n)==sumdigits(eva(binary(n))) \\ Felix Fröhlich, Jan 19 2017

A344780 Semiprimes that are product of two distinct Honaker primes.

Original entry on oeis.org

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

cp's OEIS Frontend

A174924 Semiprimes sp(k) = q * r such that sum of digits of sp(k) equals sum of digits of the semiprime index k.

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A276255 Sum of first n Honaker primes.

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A277111 Lesser of twin primes P(k) and P(k+1) such that Sd(P(k)) + Sd(P(k+1)) = Sd(k) + Sd(k+1), where Sd(x) is the sum of digits of x.

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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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A343192 Happy Honaker primes.

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A169645 Primes p = prime(k) of form 13//r, s//13 or t//13//u and sod(p) = sod(k).

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A178613 The smaller member prime(i) of an emirp pair (prime(i),prime(j)), such that the digit sum of i equals the digit sum of j.

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A261142 Numbers n such that n, p=prime(n) and q=prime(p) have the same sum of digits.

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