A033548 -id:A033548 - cp's OEIS frontend

A177061 Primes p formed from single-digit primes only, each used at most once.

2, 3, 5, 7, 23, 37, 53, 73, 257, 523, 2357, 2753, 3257, 3527, 5237, 5273, 7253, 7523

Offset: 1

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

List of (p,i): (2,1), (3,2), (5,3), (7,4), (23,9), (37,12), (53,16), (73,21), (257,55), (523,99), (2357,350), (2753,402), (3257,460), (3527,492), (5237,697), (5273, 699), (7253,928), (7523,953).
There are exactly eight primes whose digits are primes in strictly increasing order: 2, 3, 5, 7, 23, 37, 257, 2357. - James C. McMahon, Jul 04 2023
There are exactly six primes whose digits are primes in strictly decreasing order: 2, 3, 5, 7, 53, 73. - James C. McMahon, Aug 09 2023

			3//7 = 37 = prime(12) is the 6th term.
2//3//5//7 = 2357 = prime(350) is the 11th term
p = 7//5//2//3 = 7523 = prime(953) = A033548(59) is the last term.

E. I. Ignatjew, Mathematische Spielereien, Urania Verlag Leipzig/Jena/Berlin 1982

Cf. A000040, A168349, A168385, A347612

Select[FromDigits/@Flatten[Permutations/@Subsets[{2,3,5,7}],1],PrimeQ]// Union (* Harvey P. Dale, Sep 08 2021 *)

isok(p) = {my(d = digits(p)); if (#d == #Set(d) && vecmin(apply(isprime, d)) == 1, return (1)); return(0);}
lista() = {forprime(p=1, 100000, if (isok(p), print1(p, ", ")););} \\ Michel Marcus, Aug 07 2020

Edited by Assoc. Eds. OEIS, May 09 2010
Missing term 5273 added by Eren Donmez, Aug 07 2020
Cross reference added by Harvey P. Dale, Sep 09 2021

A090432 Primes prime(k) having a greater sum of digits than does k.

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 29, 31, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, 89, 97, 113, 127, 137, 139, 149, 157, 167, 173, 179, 181, 191, 193, 197, 199, 229, 233, 239, 257, 269, 277, 281, 283, 293, 307, 349, 353, 359, 367, 373, 379, 383, 389, 397, 409, 419, 439

Offset: 1

Reinhard Zumkeller, Dec 01 2003

A090431(a(n)) < 0.

Cf. A007605, A007953, A033548, A090431, A090433, A090456.

Prime[#]&/@Select[Range[100],Total[IntegerDigits[Prime[#]]]-Total[IntegerDigits[#]]>0&] (* Michel Lagneau, Nov 07 2015 *)

isok(n) = sumdigits(prime(n)) > sumdigits(n); \\ Michel Marcus, Nov 07 2015

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.

A176012 Chen primes A109611(k) which have the same sum-of-digits as their index k.

Original entry on oeis.org

227, 827, 1201, 1621, 2179, 2333, 2441, 2711, 3041, 3251, 3329, 3541, 5147, 5167, 5701, 5711, 6131, 6661, 6833, 7321, 7331, 8501, 9209, 9239, 10271, 13807, 14251, 14449, 14629, 15731, 15761, 16007, 16139, 16619, 16741, 17291, 19421, 20231, 20441, 20507, 22441

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 06 2010

The associated indices k are:
38, 98, 130, 163, 199, 209, 218, 236, 260, 272, 278, 292, 386, 388, 418, 419, 443, 469, 479, 508,...
The indices of A109611(k) in the primes A000040 are A000720(A109611(k)) =
49, 144, 197, 257, 327, 345, 362, 395, 436, 458, 469, 496, 686, 688, 751, 752, 799, 859, 880, 933, 934, 1060, ..
Some entries are also Honaker primes (A033548): 2441, 5701, 5711, 15761, 26119, 31517, 34471, 37019, 44221,...

			a(1) = 227 = A109611(38) where 2+2+7 = 11 = 3+8.
a(2) = 827 = A109611(98), where 8+2+7 =17= 9+8.

M. du Sautoy: Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Beck, 4. Auflage, 2005

B. Green and T. Tao, Restriction Theory of the Selberg Sieve, with Applications, J. Théor. Nombres Bordeaux 18, 2006.

Cf. A033548, A109611, A174924.

{A109611(k): A007953(A109611(k)) = A007953(k) }.

9241 replaced by 9239, and lists of examples reduced by R. J. Mathar, Jun 07 2010

A176465 Palindromic primes p(k) = palprime(k) such that their sum of digits ("sod") equals sum of digits of their palprime index k.

Original entry on oeis.org

13331, 1022201, 1311131, 3001003, 3002003, 100707001, 102272201, 103212301, 103323301, 103333301, 104111401, 105202501, 105313501, 105323501, 106060601, 111181111, 111191111, 112494211, 121080121, 140505041, 160020061, 160161061

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 18 2010

p(k) = palprime(k) (see A002385) with sod(p(k)) = sod(k)
List of (p(k),k):
(13331,29) (1022201,116) (1311131,173) (3001003,304) (3002003,305)
(100707001,790) (102272201,818) (103212301,832) (103323301,835) (103333301,836)
(104111401,850) (105202501,862) (105313501,865) (105323501,866) (106060601,875)
(111181111,961) (111191111,962) (112494211,979) (121080121,1096) (140505041,1379)
(160020061,1672) (160161061,1678) (160171061,1679) (181111181,1958) (300151003,2209)
(310131013,2344) (313222313,2387) (320444023,2488) (321242123,2495) (341040143,2765)
(341222143,2767) (342020243,2774) (342202243,2776) (342212243,2777) (342313243,2779)
(343050343,2788) (700090007,3488) (730111037,3884) (910212019,4858)

			p(1) = 13331 = palprime(29), sod(p(1)) = 1+3+3+3+1 = 11 = sod(29), first term
p(8) = 103212301 = palprime(832), sod(p(8)) = 1+0+3+2+1+2+3+1 = 13 = 8+3+2 = sod(832), 8th term
p(?) = 156300010003651 = palprime(99643), sod(p(?)) = 31 = sod(99733)
Note successive p(i) and p(i+1) which are also consecutive palindromic primes (i = 4, 9, 13, 16, 22, 33)

A. H. Beiler: Recreations in the Theory of Numbers: The Queen of Mathematical Entertains. Dover Publications, New York, 1964
M. Gardner: Mathematischer Zirkus , Ullstein Berlin-Frankfurt/Main-Wien, 1988
K. G. Kroeber: Ein Esel lese nie. Mathematik der Palindrome, Rowohlt Tb., Hamburg, 2003

A000213, A002385, A033548, A155214, A176111

A176790 Honaker primes of the form k^2 + 1.

Original entry on oeis.org

3137, 4357, 13457, 80657, 115601, 184901, 309137, 341057, 1008017, 1073297, 4227137, 5541317, 11806097, 16974401, 18576101, 22848401, 24443137, 24542117, 27625537, 28132417, 30913601, 39112517, 42432197, 46049797, 46321637, 52417601, 71132357, 84713617, 92736901

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 26 2010

The intersection of A033548 with A002522 or with A002496.
The list of associated n is: 56, 66, 116, 284, 340, 430, 556, 584, 1004, 1036, 2056, ...
The associated indices in A002496 are: 14, 15, 21, 48, 53, 61, 73, 76, 113, 115, 215, 243, 341, 395, 414, ...

			a(1) = 3137 = 56^2 + 1 = A033548(24).
a(2) = 4357 = 66^2 + 1 = A033548(31).

M. Aigner, Diskrete Mathematik, Vieweg u. Teubner, 6. Aufl., 2006.
E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, Berlin, 1985.
H. Scheid, Zahlentheorie, Spektrum Akademischer Verlag, 4. Aufl., 2006.

K. D. Bajpai, Table of n, a(n) for n = 1..1570

Cf. A000040, A002496, A033548, A176012, A176111, A176465.

fHQ[n_]:=Plus@@IntegerDigits@n==Plus@@IntegerDigits@PrimePi@n;Select[Range[10000]^2+1, PrimeQ[#] && fHQ[#] &]  (* K. D. Bajpai, Apr 06 2021 *)

for(n =1, 50000, my(k=n^2+1); if(isprime(k) && vecsum(digits(k))==vecsum(digits(primepi(k))), print1(k, ", "))); \\ K. D. Bajpai, Apr 06 2021

Comments tightened by R. J. Mathar, Jun 07 2010

a(21)-a(29) from K. D. Bajpai, Apr 06 2021

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)

A066521 Primes p(k) such that the product of digits of p(k) equals the product of digits of k.

Original entry on oeis.org

17, 181, 443, 491, 1163, 2131, 2143, 2153, 2621, 4253, 4621, 7691, 11483, 11593, 11813, 23819, 26951, 27179, 32261, 32443, 33811, 35171, 35227, 37643, 41543, 42433, 42443, 44623, 44917, 45971, 51473, 54517, 58193, 61223, 63131, 64591

Offset: 1

Jason Earls, Jan 05 2002

Numbers with product of digits=0, like prime(80) = 409, are not included. - Harry J. Smith, Feb 20 2010

			181 is the 42nd prime and the product of digits of both is 8.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A007954, A033548.

Select[Table[{n,Prime[n]},{n,6500}],DigitCount[#[[1]],10,0]==0 && Times@@ IntegerDigits[#[[1]]]==Times@@IntegerDigits[#[[2]]]&][[All,2]] (* Harvey P. Dale, Oct 17 2020 *)

ProdD(x)= { local(p=1); while (x>9 && p>0, p*=x%10; x\=10); return(p*x) } { n=0; for (m=1, 10^10, p=prime(m); d=ProdD(m); if (ProdD(p) == d && d, write("b066521.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 20 2010

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

cp's OEIS Frontend

A177061 Primes p formed from single-digit primes only, each used at most once.

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A090432 Primes prime(k) having a greater sum of digits than does k.

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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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A176012 Chen primes A109611(k) which have the same sum-of-digits as their index k.

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A176465 Palindromic primes p(k) = palprime(k) such that their sum of digits ("sod") equals sum of digits of their palprime index k.

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A176790 Honaker primes of the form k^2 + 1.

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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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