Carlos Rivera - cp's OEIS frontend

A335363 The smallest prime and start of a run of exactly n primes that remain prime after each digit is replaced by its square.

2, 19, 13, 131, 2111, 12815137, 103723971119

Offset: 1

Carlos Rivera, Jun 10 2020

a(7) > 4294967296.
Terms in A020449 are not terms of this sequence. - Chai Wah Wu, Jun 18 2020
a(8) > 10^13, if it exists. - Giovanni Resta, Jun 21 2020

			a(3)=13 because 13 -> 19 -> 181 are primes and 1641 is composite.

Cf. A020449, A068492.

a(7) from Giovanni Resta, Jun 20 2020

A335314 Full autoinsertable of reversed multidigit primes are such primes that remain prime after all the possible internal autoinsertions of the reversed prime, one at a time.

Original entry on oeis.org

127, 131, 149, 163, 191, 347, 383, 457, 463, 479, 521, 569, 571, 613, 643, 653, 659, 739, 757, 797, 941, 40471, 49991, 79627, 81869, 83407, 5916623

Offset: 1

Carlos Rivera, Jun 01 2020

If the prime has K digits all the possible internal autoinsertions are K-1.
Supposed to have only a finite quantity of terms.
If it exists, the next term a(28) is > 2^32.

			Example: 127 generates 2 primes 1'721'27 and 12'721'7
Example: 5916623 generates 6 primes: 5'3266195'916623, 59'3266195'16623, 591'3266195'6623, 5916'3266195'623, 59166'3266195'23, 591662'3266195'3

Cf. A335271.

forprime(p=11, 10^8, my(v=digits(p), d=#v, f=1); for(i=1, d-1, my(t=concat(concat(v[1..i], Vecrev(v)), v[i+1..d]), q=fromdigits(t)); if(!isprime(q), f=0; break)); if(f, print1(p, ", "))) \\ Hugo Pfoertner, Jun 01 2020

A335329 Primes p of the form 4k+1 such that the sum up to p of the primes of the same form is a square.

Original entry on oeis.org

29, 61, 197, 11789, 7379689, 161409881, 14881142931617, 34041319775377

Offset: 1

Carlos Rivera, Jun 01 2020

			5+13+17+29 = 64 = 8^2.
5+...+161409881 = 354203842652416 = 18820304^2.

Cf. A033998.

s=0; Select[Prime@ Range[10^9], Mod[#,4]==1 && IntegerQ@ Sqrt[s+=#] &] (* Robert Price, Sep 10 2020 *)
Module[{nn=74*10^5,k,a},k=Select[Prime[Range[nn]],Mod[#-1,4]==0&];a=Accumulate[ k];Select[ Thread[ {k,a}],IntegerQ[Sqrt[#[[2]]]]&]][[;;,1]] (* The program generates the first five terms of the sequence. *) (* Harvey P. Dale, Jul 19 2024 *)

s=0;forprime(p=5,10^9,if(p%4==1,s+=p;if(issquare(s),print1(p,", ")))) \\ Hugo Pfoertner, Jun 02 2020

10   'S1=sum of primes 4k+1, S1=sum of primes 4k+1
   20   'is S1 a square?
   30   S1=0:P=2:PM=2^32-10:K=1
   40   P=nxtprm(P):K=K+1:if P>PM then end
   50   if P@4=3 then goto 40
   60   S1=S1+P:SS1=isqrt(S1)
   70   if SS1*SS1=S1 then print K;P;S1;SS1;1
   80   goto 40

a(7) and a(8) from Martin Ehrenstein using Kim Walisch's primesieve, Jan 09 2021

A335271 Full autoinsertable primes are such primes that remain prime after all the possible internal autoinsertions, one at a time.

Original entry on oeis.org

131, 173, 179, 191, 197, 283, 293, 367, 383, 401, 547, 587, 641, 701, 709, 757, 797, 827, 12197, 12289, 53881, 54779, 68927, 37898818253

Offset: 1

Carlos Rivera, May 29 2020

Supposed to have only a finite quantity of terms. If the prime has K digits all the possible internal autoinsertions are K-1.

If it exists, the next term is > 2^32.

			The prime 131 can be inserted into itself in two positions: 1'131'31, 13'131'1. Both are primes.
The prime 68927 can be inserted into itself in four positions: 6'68927'8927, 68'68927'927, 689'68927'27, 6892'68927'7. All the four are primes.

Cf. A247098.

forprime(p=11,10^8,my(v=digits(p),d=#v,f=1);for(i=1,d-1,my(t=concat(concat(v[1..i],v),v[i+1..d]),q=fromdigits(t));if(!isprime(q),f=0;break));if(f,print1(p,", "))) \\ Hugo Pfoertner, May 30 2020

a(24) from Michael S. Branicky, Mar 27 2023

A334982 a(n) is the earliest start of a run of at least n consecutive primes that are also happy numbers.

Original entry on oeis.org

7, 19, 3067, 3313, 3313, 11550481, 11550481, 475389133, 1128863117, 1175267983, 19507818293, 176622893141, 176622893141, 495495126109, 4411512992681, 122634868247173, 122634868247173

Offset: 1

Carlos Rivera, May 18 2020

Shyam Sunder Gupta, Happy Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 7, 209-222.
Carlos Rivera, Puzzle 1002. Consecutive primes that are also happy numbers, The Prime Puzzles & Problems Connection.

Cf. A007770.

a(11)-a(17) from Giovanni Resta, May 23 2020

a(16) corrected by Giovanni Resta, May 30 2020

A241525 a(n) is the smallest start of a run of exactly n consecutive primes such that the sum of the digits of each prime is composite.

Original entry on oeis.org

19, 17, 13, 521, 509, 503, 499, 491, 14153, 25793, 25771, 37663, 37657, 98729, 98717, 98713, 98711, 98689, 98669, 98663, 98641, 98639, 98627, 98621, 98597, 98573, 69794393, 69794383, 268684679, 268684651, 268684627, 329788829, 545497787, 545497769, 545497759, 545497753, 545497747, 545497741, 545497727, 545497723, 545497691, 545497681, 545497679, 545497637, 545497633, 545497609

Offset: 1

Carlos Rivera, Apr 24 2014

No more terms below 2^32

			a(3)=13 because the run of the 3 consecutive primes {13, 17, 19} is such that the sum of digits for each prime is {4, 8, 10}.

Cf. A240598.

10   P=1:KM=0:K=0:'puzzle 1290, Meller
   20   P=nxtprm(P):if P>2^32-20 then end
   30   gosub *SODP:if S<>prmdiv(S) then K=K+1:Q=P:goto 20
   40   if K>KM then print K, Q:KM=K
   50   K=0:goto 20
  200   *SODP:S=0:L=alen(P)
  210   for I=1 to L:D=val(mid(str(P), I+1, 1))
  220   S=S+D:next I
  230   return

A240598 The smallest first term of a sequence of exactly n consecutive prime numbers each of which has the property that its digit sum is prime.

Original entry on oeis.org

11, 7, 5, 3, 2, 2063, 3253, 3251, 14293, 2442191, 2442179, 2442173, 2442151, 2442133, 2442113, 466343539, 793234063, 10158613657, 5200298339, 281201652541, 3140590111859, 1523243332991, 1631014452929, 1008266115029

Offset: 1

Carlos Rivera, Apr 08 2014

There is no requirement on the order of primes that arise as the digit sums.

a(25) > 2*10^13. - Giovanni Resta, Apr 09 2014

			a(15) = 2442113 because each of the following fifteen consecutive primes {2442113, 24422133, 2442151, 2442173, 2442179, 2442191, 2442197, 2442199, 2442227, 2442263, 2442287, 2442289, 2442311, 2442353, 2442359} has a sum of digits producing another prime number and the smallest is 2442113.
a(17) = 793234063  because each of the following seventeen consecutive primes {793234063 793234067 793234111 793234139 793234153 793234171 793234177 793234193 793234207 793234243 793234261 793234289 793234333 793234357 793234391 793234427 793234441} has a sum of digits producing another prime number and the smallest is 793234063.

Cf. A239790.

10   P=1:KM=0:K=0:'puzzle 1290, Meller
   20   P=nxtprm(P):if P>2^32-20 then end
   30   gosub *SODP:if S=prmdiv(S) then K=K+1:Q=P:goto 20
   40   if K>KM then print K, Q:KM=K
   50   K=0:goto 20
  200   *SODP:S=0:L=alen(P)
  210   for I=1 to L:D=val(mid(str(P), I+1, 1))
  220   S=S+D:next I
  230   return

a(18)-a(24) from Giovanni Resta, Apr 09 2014

A239790 The smallest multidigit prime of a sequence of n consecutive primes such that their digit sums are also a sequence of n consecutive primes.

Original entry on oeis.org

11, 41, 41, 191, 402131, 6340271501

Offset: 1

Carlos Rivera, Mar 26 2014

a(7), if it exists, is larger than 2*10^14. - Giovanni Resta, Apr 03 2014
a(7) <= 101100010001001200110001. - Jens Kruse Andersen, Aug 28 2016
a(7) <= 1212030150560200001. - Oscar Volpatti, Aug 25 2025

			a(4)=191 because 191, 193, 197, 199 generates 11, 13, 17, 19.
a(5)=402131 because 402131, 402133, 402137, 402139, 402197 generates 11,13,17,19,23.

Números y Algo Más, Primos cuya suma digital dan primos
Carlos Rivera, Puzzle 1229. Set of consecutive primes such that..., The Prime Puzzles & Problems Connection.
B. Sindelar, Two Sets of Consecutive Primes and their Sum of Digits Connection [Broken link]
Bill Sindelar, Jens Kruse Andersen, Marian Otremba, Two Sets of Consecutive Primes and their Sum of Digits Connection, digest of 12 messages in primenumbers Yahoo group, Aug 26 - Aug 30, 2016.

Cf. A007953, A046704, A240598.

isok(p, n) = if ((p > 10) && isprime(p), my(v=vector(n)); v[1] = p; for (i=2, n, v[i] = nextprime(v[i-1]+1);); my(vs=vector(n, i, sumdigits(v[i]))); if (!isprime(vs[1]), return(0)); for (i=2, n, if (vs[i] != nextprime(vs[i-1]+1), return(0));); return(1););
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Aug 28 2025

10   P=7:KM=0:'puzzle 1290, Meller
   20   P=nxtprm(P):if P>2^32-20 then end
   30   gosub *K:if K<=KM then goto 20
   40   print K,P,Q1:KM=K:goto 20
  100   *K
  110   Z=P:gosub *SODZ
  120   if SODZ<>prmdiv(SODZ) then return
  130   K=1:Q=SODZ:Q1=Q
  140   Z=nxtprm(Z):gosub *SODZ
  150   if SODZ<>nxtprm(Q) then return
  160   K=K+1:Q=nxtprm(Q):goto 140
  200   *SODZ:SODZ=0:L=alen(Z)
  210   for I=1 to L:D=val(mid(str(Z),I+1,1))
  220   SODZ=SODZ+D:next I
  230   return

a(6) from Giovanni Resta, Apr 03 2014

A235766 Smallest first term of a sequence of exactly n consecutive hoax numbers.

Original entry on oeis.org

22, 84, 12955, 291090, 9538589, 3541285143, 136063250955, 253282144742

Offset: 1

Carlos Rivera, Jan 15 2014

A hoax number is a composite number whose sum of digits is equal to the sum of the digits of its distinct prime factors.

			For n = 1, a(1) = 22 because 22 = 11*2 and the sum of the digits (SOD) in both sides is 4 but 23 is not composite.
For n = 2, a(2) = 84 because 84 = 2^2*3*7, SOD = 12; 85 = 5*17, SOD = 13 but 86 = 2*43 and SOD = 14 <> 9.

Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
Giovanni Resta, Hoax numbers.
Eric Weisstein's World of Mathematics, Hoax Number.

Cf. A019506.

a(6)-a(8) from Giovanni Resta, Jan 15 2014

A235680 The smallest first term of a sequence of n primes such that, after the first, each is equal to the previous prime plus the sum of all of its digits, plus the product of all of its nonzero digits.

Original entry on oeis.org

2, 191, 163, 151, 127, 1644997, 36778597, 935715673, 50682890749, 16390560362269, 63334172492839

Offset: 1

Carlos Rivera, Jan 13 2014

The following term produced in each sequence after the n-th is necessarily a composite integer.

			Example for n=8: a(8)=935715673 because after it the seven primes are 936311069, 936337351, 936490481, 936677149, 938391809, 938811763 and 939029537, with 936311069 = 935715673 + 9*3*5*7*1*5*6*7*3+(9+3+5+7+1+5+6+7+3) and so on...

C. Rivera, Puzzle 727. Extending the OEIS sequence A235680

Cf. A233692, A233783, A090009.

a(9) from Giovanni Resta, Jan 13 2014

a(10)-a(11) from Giovanni Resta, Feb 22 2014

cp's OEIS Frontend

User: Carlos Rivera

A335363 The smallest prime and start of a run of exactly n primes that remain prime after each digit is replaced by its square.

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A335314 Full autoinsertable of reversed multidigit primes are such primes that remain prime after all the possible internal autoinsertions of the reversed prime, one at a time.

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A335329 Primes p of the form 4k+1 such that the sum up to p of the primes of the same form is a square.

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A335271 Full autoinsertable primes are such primes that remain prime after all the possible internal autoinsertions, one at a time.

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A334982 a(n) is the earliest start of a run of at least n consecutive primes that are also happy numbers.

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A241525 a(n) is the smallest start of a run of exactly n consecutive primes such that the sum of the digits of each prime is composite.

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UBASIC

A240598 The smallest first term of a sequence of exactly n consecutive prime numbers each of which has the property that its digit sum is prime.

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UBASIC

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A239790 The smallest multidigit prime of a sequence of n consecutive primes such that their digit sums are also a sequence of n consecutive primes.

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A235766 Smallest first term of a sequence of exactly n consecutive hoax numbers.

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