A033549 -id:A033549 - cp's OEIS frontend

A033548 Honaker primes: primes P(k) such that sum of digits of P(k) equals sum of digits of k.

131, 263, 457, 1039, 1049, 1091, 1301, 1361, 1433, 1571, 1913, 1933, 2141, 2221, 2273, 2441, 2591, 2663, 2707, 2719, 2729, 2803, 3067, 3137, 3229, 3433, 3559, 3631, 4091, 4153, 4357, 4397, 4703, 4723, 4903, 5009, 5507, 5701, 5711, 5741, 5801, 5843

Offset: 1

Calculated by Jud McCranie

A090431(A049084(a(n))) = 0.

			131 is the 32nd prime and sum of digits of both is 5.

Proposed by G. L. Honaker, Jr.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A007605, A033549, A049084, A072439, A090431.

a033548 n = a033548_list !! (n-1)
a033548_list = filter ((== 0) . a090431 . a049084) a000040_list
-- Reinhard Zumkeller, Mar 16 2014

read("transforms") :
isA033548 := proc(n)
    if isprime(n) and digsum(n) = digsum(numtheory[pi](n)) then
        true ;
    else
        false;
    end if;
end proc:
A033548 := proc(n)
    local p, k;
    if n = 1 then
        131;
    else
        p := nextprime(procname(n-1)) ;
        while true  do
            if isA033548(p) then
                return p;
            end if;
            p := nextprime(p) ;
        end do:
    end if;
end proc:
seq(A033548(n),n=1..40) ; # R. J. Mathar, Jul 07 2021

Prime[ Select[ Range[ 2000 ], Apply[ Plus, IntegerDigits[ # ] ] == Apply[ Plus, IntegerDigits[ Prime[ # ] ] ] & ] ] (* Santi Spadaro, Oct 14 2001 *)
Select[ Prime@ Range@ 5927, Plus @@ IntegerDigits@ # == Plus @@ IntegerDigits@ PrimePi@ # &]  (* Robert G. Wilson v, Jun 07 2009 *)
nn=800;Transpose[Select[Thread[{Prime[Range[nn]],Range[nn]}],Total[IntegerDigits[First[#]]]== Total[ IntegerDigits[ Last[#]]]&]][[1]] (* Harvey P. Dale, Jun 13 2011 *)

is(n)=isprime(n) && sumdigits(n)==sumdigits(primepi(n)) \\ Charles R Greathouse IV, Jun 18 2015

from sympy.ntheory.factor_ import digits
from sympy import primepi, primerange
print([n for n in primerange(1, 5901) if (sum(digits(n)[1:])==sum(digits(primepi(n))[1:]))]) # Indranil Ghosh, Jun 27 2017, after Charles R Greathouse IV

a(n) = A000040(A033549(n)). - R. J. Mathar, Jul 07 2021

More terms from Robert G. Wilson v, Jun 07 2009

A071600 Numbers k such that k and prime(k) have the same number of 1's in their binary representation.

Original entry on oeis.org

1, 3, 13, 19, 21, 23, 25, 30, 44, 45, 47, 57, 60, 61, 71, 77, 98, 99, 101, 103, 107, 108, 110, 118, 121, 125, 158, 159, 178, 179, 184, 186, 187, 188, 209, 215, 218, 221, 237, 244, 246, 247, 248, 249, 251, 279, 287, 312, 334, 335, 346, 350, 359, 361, 362, 365

Offset: 1

Benoit Cloitre, Jun 01 2002

			221 = 11011101 in base 2, prime(221) = 1381 = 10101100101 in base 2, both have 6 "1's" in their binary representation, hence 221 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000120, A014499, A033549, A049084, A072439, A090455.

Select[Range[400],DigitCount[#,2,1]==DigitCount[Prime[#],2,1]&] (* Harvey P. Dale, Mar 09 2015 *)

for(n=1,1000,s=1; if(sum(i=1,length(binary(n)), component(binary(n),i))==sum(i=1,length(binary(prime(n))), component(binary(prime(n)),i)),print1(n,",")))

is(n)=hammingweight(n)==hammingweight(prime(n)) \\ Charles R Greathouse IV, Mar 07 2013

a(n) = A049084(A072439(n)); A000120(a(n)) = A000120(A072439(n)). - Reinhard Zumkeller, Jun 17 2002

A090455(a(n)) = 0, A000120(a(n)) = A014499(a(n)).

A090431 Difference between sums of digits of n and n-th prime.

Original entry on oeis.org

-1, -1, -2, -3, 3, 2, -1, -2, 4, -10, -2, -7, -1, -2, -5, -1, -6, 2, -3, -6, -7, -12, -6, -11, -9, 6, 5, 2, 1, -2, -6, 0, -5, -6, -6, 2, -3, 1, -2, -7, -12, -4, -4, -5, -8, -9, 7, 5, 2, -8, -2, -7, 1, 1, -4, 0, -5, 3, -2, -5, -6, -6, -1, 5, 4, 1, 6, 1, 1, -9, -3, -8, -6, -2

Offset: 1

Reinhard Zumkeller, Dec 01 2003

a(n) = A007953(n) - A007605(n);

a(A033549(n))=a(A049084(A033548(n)))=0; a(A049084(A090432(n)))<0; a(A049084(A090433(n)))>0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A239324 (partial sums).

a090431 n = a007953 n - a007605 n  -- Reinhard Zumkeller, Mar 16 2014

A117460 Primes prime(i) such that their sum-of-index-digits A007953(i) and their sum-of-digits A007605(i) are consecutive primes.

Original entry on oeis.org

2, 3, 5, 43, 113, 191, 373, 821, 1097, 1307, 1493, 1523, 1619, 1873, 1907, 2029, 2081, 2339, 3109, 3169, 3347, 3923, 4339, 4421, 4463, 4603, 5417, 5581, 6067, 6263, 6427, 6607, 6791, 6841, 6863, 7127, 7307, 7673, 7723, 7877, 8731, 9341, 10079, 10723

Offset: 1

Enoch Haga, Mar 18 2006

We select primes such that their sum-of-digits is some prime(j) and such that in addition the sum-of-digits of their index is prime(j-1).
Line 160 of the UBASIC program can be altered for <, >, or = relationships
Subset of A046704 - R. J. Mathar, Apr 17 2009

			"SOD" = "sum of digits": a(5) = 113, the prime whose index is 30. SOD(30) = 3 and SOD(113) = 5. Since 3 < 5 and 5 is nextprime to 3, adjoin 113 to the sequence.

Cf. A117461, A117458, A117459, A033548, A033549.

10 'use of str,mid,len,val 20 'in SOD prime index and SOD prime 30 Y=1 40 Y=nxtprm(Y) 50 C=C+1:print C;Y;"-"; 60 D=str(C):Z=str(Y) 70 E=len(D):F=len(Z) 80 for Q=2 to E 90 A=mid(D,Q,1):G=val(A) 100 I=I+G:print I; 110 next Q 120 for R=2 to F 130 B=mid(Z,R,1):H=val(B) 140 J=J+H:print J; 150 next R 160 if I=prmdiv(I) and J=prmdiv(J) and I>J and I=nxtprm(J) then stop 170 I=0:J=0 180 goto 40

{A000040(i): A007605(i) = A000040(j) and A007953(i) = A000040(j+1) for some j}. - R. J. Mathar, Apr 17 2009

Edited by R. J. Mathar, Apr 17 2009

A117461 Indices associated with primes in A117460. Both primes and their indices, after calculation of their respective digit sums, bear the relationship that both are prime and that sod(i) < sod(p) and sod(p) is the next prime after to sod(i), where sod is the sum of digits function.

Original entry on oeis.org

1, 2, 3, 14, 30, 43, 74, 142, 184, 214, 238, 241, 256, 287, 292, 308, 313, 346, 443, 449, 472, 544, 593, 601, 607, 623, 715, 737, 791, 814, 836, 854, 874, 881, 883, 913, 931, 973, 980, 995, 1088, 1156, 1237, 1307, 1316, 1343, 1381, 1396, 1462, 1565, 1622

Offset: 0

Enoch Haga, Mar 18 2006

A117458-A117459 is the opposite case where sod(i) > sod(p).
A117460-A117461 is sod(i) < sod(p).
A033548-A033549 is sod(i) = sod(p). - G. L. Honaker, Jr.

			a(4) = 30. Its associated prime is 113 with sod = 5; sod(a(4)) = 3. Since 3 < 5 and 5 is the next prime after 3, a(4) belongs in the sequence.

Cf. A007953 (sum of digits).

Cf. A117460, A117458, A117459, A033548, A033549.

10 'use of str,mid,len,val
20 'in SOD prime index and SOD prime
30 Y=1
40 Y=nxtprm(Y)
50 C=C+1:print C;Y;"-";
60 D=str(C):Z=str(Y)
70 E=len(D):F=len(Z)
80 for Q=2 to E
90 A=mid(D,Q,1):G=val(A)
100 I=I+G:print I;
110 next Q
120 for R=2 to F
130 B=mid(Z,R,1):H=val(B)
140 J=J+H:print J;
150 next R
160 if I=prmdiv(I) and J=prmdiv(J) and I

SOD's are calculated for these indices; if they and their associated prime SOD's are both prime and bear the relation in the Brief description above, they are added to the sequence.

A117477 Primes whose SOD and that of their indices are both prime and equal (indices may not be prime, but their SOD must be prime).

Original entry on oeis.org

131, 263, 1039, 1091, 1301, 1361, 1433, 2221, 2441, 2591, 2663, 2719, 2803, 3433, 3631, 4153, 4357, 4397, 5507, 5701, 5741, 5927, 6311, 6353, 6553, 6737, 6827, 6971, 7013, 7213, 7411, 7523, 7741, 8821, 9103, 11173, 11353, 11731, 11821, 12277, 12347

Offset: 1

Enoch Haga, Mar 19 2006

"SOD" = "sum of digits".

This sequence is a subset of A033548, the difference being that this sequence requires prime SODs.

			a(3) = 1039, the 175th prime. Both the SOD of the index and the prime are prime and equal: 13 = 13.

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

Cf. A117478, A033548-A033549, A117458-A117463.

sodQ[{n_,p_}]:=Module[{sodn=Total[IntegerDigits[n]],sodp=Total[IntegerDigits[p]]},AllTrue[ {sodn,sodp},PrimeQ] && sodn == sodp]; Select[With[{nn=1500},Table[{n,Prime[n]},{n,nn}]],sodQ][[;;,2]] (* Harvey P. Dale, Apr 20 2024 *)

20 'SOD prime index and SOD prime
30 Y=1
40 Y=nxtprm(Y)
50 C=C+1:print C;Y;"-";
60 D=str(C):Z=str(Y)
70 E=len(D):F=len(Z)
80 for Q=2 to E
90 A=mid(D,Q,1):G=val(A)
100 I=I+G:print I;
110 next Q
120 for R=2 to F
130 B=mid(Z,R,1):H=val(B)
140 J=J+H:print J;
150 next R
160 if I=prmdiv(I) and J=prmdiv(J) and I=J then stop
170 I=0:J=0
180 goto 40

Find primes whose indices, when SODs are computed, are both prime and SOD(i) = SOD(p)

A117478 Indices of associated primes in A117477.

Original entry on oeis.org

32, 56, 175, 182, 212, 218, 227, 331, 362, 377, 386, 397, 409, 481, 508, 571, 595, 599, 728, 751, 755, 779, 821, 827, 847, 869, 878, 896, 902, 922, 940, 953, 982, 1099, 1129, 1354, 1372, 1408, 1417, 1468, 1475, 1507, 1550, 1585, 1648, 1693, 1747, 1772, 1774

Offset: 0

Enoch Haga, Mar 19 2006

A subset of A033548-A033549 but here the SODs must be prime and equal

			a(3) = 182, with SOD 11. The associated prime is 1091, also SOD 11. SODs must be prime and equal.

Cf. A117477, A033548-A033549, A117458-A117463.

20 'SOD prime index and SOD prime
30 Y=1
40 Y=nxtprm(Y)
50 C=C+1:print C;Y;"-";
60 D=str(C):Z=str(Y)
70 E=len(D):F=len(Z)
80 for Q=2 to E
90 A=mid(D,Q,1):G=val(A)
100 I=I+G:print I;
110 next Q
120 for R=2 to F
130 B=mid(Z,R,1):H=val(B)
140 J=J+H:print J;
150 next R
160 if I=prmdiv(I) and J=prmdiv(J) and I=J then stop
170 I=0:J=0
180 goto 40

Find prime indices with associated primes where both SODs are the same and prime.

Typo in comment fixed by Franklin T. Adams-Watters, Dec 03 2009

A259067 Sum of digits of Honaker primes (A033548).

Original entry on oeis.org

5, 11, 16, 13, 14, 11, 5, 11, 11, 14, 14, 16, 8, 7, 14, 11, 17, 17, 16, 19, 20, 13, 16, 14, 16, 13, 22, 13, 14, 13, 19, 23, 14, 16, 16, 14, 17, 13, 14, 17, 14, 20, 23, 10, 11, 16, 17, 19, 20, 20, 23, 23, 23, 11, 13, 20, 13, 20, 17, 19, 10, 19, 13, 14, 16, 20, 10, 10, 13, 10, 13, 10, 13, 13, 19, 17, 13, 11, 14, 14, 14, 22, 16, 19, 20, 16, 20, 19, 20, 19, 20

Offset: 1

Zak Seidov, Jun 18 2015

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

Cf. A007605, A007953, A033548, A033549.

lista(nn) = forprime(p=2, nn, if ((sd=sumdigits(p)) == sumdigits(primepi(p)), print1(sd, ", "));); \\ Michel Marcus, Jun 18 2015

go(lim)=my(v=List(),n,s);forprime(p=2,lim, s=sumdigits(n++); if(sumdigits(p)==s, listput(v,s))); Vec(v) \\ Charles R Greathouse IV, Jun 18 2015

a(n) = A007953(A033548(n)) = A007953(A033549(n)).

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

A133608 Numbers n such that the sum of digits of n-th semiprime equals sum of digits of n.

Original entry on oeis.org

5, 6, 19, 40, 41, 42, 70, 71, 85, 89, 128, 148, 149, 166, 199, 246, 257, 271, 285, 327, 339, 346, 448, 449, 469, 484, 566, 592, 605, 617, 634, 643, 644, 676, 682, 694, 710, 713, 719, 740, 748, 751, 752, 753, 782, 793, 794, 797, 798, 815, 890, 901, 905, 961

Offset: 1

Jonathan Vos Post, Dec 27 2007

This is to A033549 as semiprimes A001358 are to primes A000040.

			a(1) = 5 because semiprime(5) = 14, whose sum of digits is 5, the same as its index as a semiprime.

Cf. A001358, A007953, A033549.

a = {}; c = 0; For[n = 4, n < 10000, n++, If[Sum[FactorInteger[n][[i, 2]], {i, 1, Length[FactorInteger[n]]}] == 2, c++; If[Plus @@ IntegerDigits[c] == Plus @@ IntegerDigits[n], AppendTo[a, c]]]]; a (* Stefan Steinerberger, Dec 29 2007 *)
SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]; SemiPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[SemiPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Select[Range@ 1000, fQ@# &] (* Robert G. Wilson v *)
nn=5000;With[{sp=Select[Range[nn],PrimeOmega[#]==2&]},Select[Range[ Length[sp]], Total[ IntegerDigits[sp[[#]]]] ==Total[ IntegerDigits[#]]&]] (* Harvey P. Dale, Oct 15 2012 *)

A007953(A001358(a(n))) = A007953(a(n)).

Corrected and extended by Stefan Steinerberger and Robert G. Wilson v, Dec 29 2007

cp's OEIS Frontend

A033548 Honaker primes: primes P(k) such that sum of digits of P(k) equals sum of digits of k.

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A071600 Numbers k such that k and prime(k) have the same number of 1's in their binary representation.

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PARI

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A090431 Difference between sums of digits of n and n-th prime.

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Haskell

A117460 Primes prime(i) such that their sum-of-index-digits A007953(i) and their sum-of-digits A007605(i) are consecutive primes.

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UBASIC

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A117461 Indices associated with primes in A117460. Both primes and their indices, after calculation of their respective digit sums, bear the relationship that both are prime and that sod(i) < sod(p) and sod(p) is the next prime after to sod(i), where sod is the sum of digits function.

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UBASIC

Formula

A117477 Primes whose SOD and that of their indices are both prime and equal (indices may not be prime, but their SOD must be prime).

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Mathematica

UBASIC

Formula

A117478 Indices of associated primes in A117477.

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