Jeff Heleen - cp's OEIS frontend

A330519 Circulant determinant chain starting at 1007749.

1007749, 18791108, -186878195, -1050017609, -9619142907, -4098030816, -59235488169, -43035664380, -25214990766, -189627605493, 779029018560, 14532890711040, 364989611366055, 1929185790836472, -1121729573175075, 13492031465869425, 329600773788765624

Offset: 1

Jeff Heleen, Dec 17 2019

Form a circulant matrix from a given number. Find the determinant. Use that determinant to form the next circulant matrix. Repeat. If a determinant is negative, multiply each element of the next matrix by -1. 1007749 is the smallest number whose chain appears to increase without bound. (It would be nice to have a proof that this chain is unbounded!)
If we had started at 247, the next term in the chain, and all subsequent terms, would be 247, since (see A219324)
..........| 2 4 7 |
247 = det | 7 2 4 |. - N. J. A. Sloane, Dec 18 2019
..........| 4 7 2 |

Jeff Heleen, Circulant determinant chain for 1007749: gives table of n, a(n) for n from 1 to 36.
N. J. A. Sloane, Recent and Noteworthy Sequences in the OEIS, Dec. 18, 2012

Cf. A219324.

A257496 Doubly prime home primes (DPHP).

Original entry on oeis.org

9, 16, 26, 42

Offset: 1

Jeff Heleen, Apr 26 2015

Take all the factors generated during the search for a home prime, put them together in the order found (just like when looking for the home prime itself) and test for primality. If it is a prime you have a DPHP.
Further terms are 74, 75, 95.
The number of iterations must be more than 1, preventing numbers such as 12 (12->2*2*3->223) from being a DPHP. - Dana Jacobsen, May 06 2015
An additional restriction is that a number is not a DPHP if it appears as the result as a product in any home prime search.  So 25 is not a DPHP because a(10) => 2*5->25 => 5*11->511 => 7*73->773, and 25 appears in the process. - Dana Jacobsen, May 23 2015
49 and 77 do not currently have known home prime sequences, hence it is unknown if they are in the DPHP sequence. - Dana Jacobsen, May 06 2015

			Starting number 16 takes 4 iterations to get to the home prime of 31636373:
16 -> 2 * 2 * 2 * 2;
2222 -> 2 * 11 * 101;
211101 -> 3 * 11 * 6397;
3116397 -> 3 * 163 * 6373;
31636373 -> prime.
Now take all the factors found, in order: 2222211101311639731636373, a prime.
So 16 leads to a DPHP.

Jeffrey Heleen, Family Numbers: Mathemagical Black Holes, Recreational and Educational Computing, 5:5, pp. 6, 1990.
Jeffrey Heleen, Family numbers: Constructing Primes by Prime Factor Splicing, J. Recreational Math., Vol. 28 #2, 1996-97, pp. 116-119.

Patrick De Geest's World!Of Numbers, Wonplate 194
Patrick De Geest's World!Of Numbers, Home Primes

Cf. A037274.

A072712 Primes p such that p divides the (right) concatenation of all numbers from 1 to p written in base 10 (most significant digit on left).

Original entry on oeis.org

2, 3, 5, 8167, 371321

Offset: 1

Jeff Heleen, Aug 07 2002

Right concatenation, normal order.

Primes in A029455. - Derek Orr, Oct 04 2014

			p=17 is not a term since 1234567891011121314151617 is not divisible by 17.

Carlos Rivera, Puzzle 190. A follow up for the puzzle 188, The Prime Puzzles & Problems Connection.

Cf. A029447-A029542, A061931-A061978.

p=""; n=1; while(n<10^4, p=concat(p,Str(n)); if(eval(p)%n==0&&isprime(n), print1(n,", ")); n++) \\ Derek Orr, Oct 04 2014

Another term from Jeff Heleen, Oct 11 2009

A080718 1, together with numbers n that are the product of two primes p and q such that the multiset of the digits of n coincides with the multiset of the digits of p and q.

Original entry on oeis.org

1, 1255, 12955, 17482, 25105, 100255, 101299, 105295, 107329, 117067, 124483, 127417, 129595, 132565, 145273, 146137, 149782, 174082, 174298, 174793, 174982, 250105, 256315, 263155, 295105, 297463, 307183, 325615, 371893, 536539

Offset: 1

Jeff Heleen, Mar 06 2003

Except for 1, this sequence is a subsequence of A280928. More specifically, members of A280928 are also members of this sequence if and only if they are semiprime. - Ely Golden, Jan 11 2017
This sequence has no equivalent in odd bases. This is because any equivalent of A280928 in an odd base must have all terms having at least 3 prime factors. - Ely Golden, Jan 11 2017
All entries other than 1 are congruent to 4 mod 9, because p*q == p + q mod 9 (with p and q not both divisible by 3) implies p*q == 4 mod 9. - Robert Israel, May 05 2014

			1255 = 5*251, 12955 = 5*2591, 17482 = 2*8741, 100255 = 5*20051, 146137=317*461, etc.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..145 from Robert Israel)

The following sequences are all closely related: A020342, A014575, A080718, A280928, A048936, A144563.

filter:= proc(n) local F,p,q,Ln,Lpq;
  F:= ifactors(n)[2];
  if nops(F) > 2 or convert(F,`+`)[2]<>2 then return false fi;
  p:= F[1][1];
  if nops(F) = 2 then q:= F[2][1] else q:= F[1][1] fi;
  Ln:= sort(convert(n,base,10));
  Lpq:= sort([op(convert(p,base,10)),op(convert(q,base,10))]);
  evalb(Ln = Lpq);
end proc:
filter(1):= true:
A080718:= select(filter,[1, seq(4+9*i,i=1..10^6)]); # Robert Israel, May 04 2014

ptpQ[n_]:=Module[{sidn=Sort[IntegerDigits[n]],fi=Transpose[ FactorInteger[ n]]}, fi[[2]]=={1,1}&&Sort[Flatten[ IntegerDigits/@ fi[[1]]]]==sidn]; Join[{1}, Select[Range[4,550000,9],ptpQ]] (* Harvey P. Dale, Jun 22 2014 *)

from sympy import factorint
from itertools import count, islice
def agen(): # generator
    yield 1
    for k in count(4, 9):
        t = sorted(str(k))
        f = factorint(k)
        if sum(f.values()) == 2:
            p, q = min(f), max(f)
            if t == sorted(str(p)+str(q)):
                yield k
print(list(islice(agen(), 30))) # Michael S. Branicky, Apr 20 2025

Edited by N. J. A. Sloane, Jan 03 2009

Incorrect entry 163797 removed by Robert Israel, May 04 2014

A059758 Undulating palindromic primes: numbers that are prime, palindromic in base 10, and the digits alternate: ababab... with a != b.

Original entry on oeis.org

101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 18181, 32323, 35353, 72727, 74747, 78787, 94949, 95959, 1212121, 1616161, 323232323, 383838383, 727272727, 919191919, 929292929, 979797979, 989898989

Offset: 1

Jeff Heleen, Feb 11 2001

C. A. Pickover, "Wonders of Numbers", Oxford New York 2001, Chapter 52, pp. 123-124, 316-317.
C. W. Trigg, Palindromic Octagonal Numbers, Journal of Recreational Mathematics, 15:1, pp. 41-46, 1982-83.

Harvey P. Dale, Table of n, a(n) for n = 1..100
P. De Geest, More undulating primes
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Cf. A032758.

for l from 3 to 31 by 2 do for i from 0 to 9 do for j from 0 to 9 do it1 := sum(i*10^(2*k), k=0..(l-1)/2): it2 := sum(j*10^(2*k+1), k=0..(l-3)/2): if isprime(it1+it2) and i<>j then printf(`%d, `,it1+it2) fi: od: od: od: # James Sellers, Feb 13 2001

t = {}; t1 = {1, 3, 7, 9}; Do[p = 10 a + b; q = 10 b + a; t = Join[t, Select[Table[(p*10^(2 n + 1) - q)/99, {n, 4}], PrimeQ]], {a, t1}, {b, Range[0, 9]}]; Union[t] (* Jayanta Basu, Jun 23 2013 *)
uppQ[n_]:=Module[{idn=IntegerDigits[n]},OddQ[Length[idn]]&& PalindromeQ[ n] && Length[Union[Partition[idn,2]]]==1]; Select[Prime[Range[ 51*10^6]], uppQ] (* or *) Select[FromDigits/@Flatten[Table[Riffle[Table[n,i],k],{n,{1,3,7,9}},{i,5},{k,0,9}],2],#>9&&PrimeQ[#]&]//Sort (* The second program is significantly faster than the first. *) (* Harvey P. Dale, Feb 24 2018 *)

from sympy import isprime
A059758_list = []
for l in range(1,300):
    for a in '1379':
        for b in '0123456789':
            if a != b:
                p = int((a+b)*l+a)
                if isprime(p):
                    A059758_list.append(p) # Chai Wah Wu, Dec 21 2014

More terms from James Sellers, Feb 13 2001

A054970 Index numbers for palindromic hexagonal numbers.

Original entry on oeis.org

0, 1, 2, 6, 39, 55, 87, 182, 556, 644, 797, 917, 1593, 1685, 25141, 51425, 55556, 83527, 810311, 1620213, 1853942, 5555556, 17352586, 17835196, 25004441, 91071921, 170563673, 181737182, 184252876, 507354403, 1240058219, 1783816196, 2756800387

Offset: 1

Jeff Heleen, May 26 2000

Cf. A000384, A054969, A082721.

Select[Range[0, 10^6], PalindromeQ[PolygonalNumber[6, #]] &] (* Robert Price, Apr 27 2019 *)

Added a(1)=0 by Robert Price, Apr 27 2019

A054969 Palindromic hexagonal numbers.

Original entry on oeis.org

0, 1, 6, 66, 3003, 5995, 15051, 66066, 617716, 828828, 1269621, 1680861, 5073705, 5676765, 1264114621, 5289009825, 6172882716, 13953435931, 1313207023131, 5250178710525, 6874200024786, 61728399382716, 602224464422206, 636188414881636, 1250444114440521

Offset: 1

Jeff Heleen, May 26 2000

Robert Price, Table of n, a(n) for n = 1..33

Intersection of A000384 and A002113.

Cf. A054970, A082721.

Select[PolygonalNumber[6, Range[0, 10^6]], PalindromeQ] (* Robert Price, Apr 27 2019 *)

Added a(1)=0 by Robert Price, Apr 27 2019

A054910 Palindromic heptagonal numbers.

Original entry on oeis.org

0, 1, 7, 55, 616, 3553, 4774, 60606, 848848, 4615164, 5400045, 6050506, 7165445617, 62786368726, 65331413356, 73665056637, 91120102119, 345546645543, 365139931563, 947927729749, 3646334336463, 7111015101117, 717685292586717, 19480809790808491

Offset: 1

Jeff Heleen, May 26 2000

Intersection of A002113 (palindromic) and A000566 (heptagonal).

Robert Price, Table of n, a(n) for n = 1..29
P. De Geest, Palindromic Heptagonals

Cf. A054971, A000566.

Select[PolygonalNumber[7, Range[0, 10^6]], PalindromeQ] (* Robert Price, Apr 28 2019 *)

Added a(1)=0 by Robert Price, Apr 28 2019

A054971 n(5n-3)/2 is a palindromic heptagonal number.

Original entry on oeis.org

1, 2, 5, 16, 38, 44, 156, 583, 1359, 1470, 1556, 53537, 158476, 161656, 171657, 190914, 371778, 382173, 615769, 1207698, 1686537, 16943262, 88274141, 496329416, 765609041, 1820198063, 2497949426, 2850685772, 15911875336

Offset: 1

Jeff Heleen, May 26 2000

P. De Geest, Palindromic Heptagonals

Cf. A054910, A000566.

A054646 Smallest number to give 2^(2n) in a hailstone (or 3x + 1) sequence.

Original entry on oeis.org

1, 3, 21, 75, 151, 1365, 5461, 14563, 87381, 184111, 932067, 5592405, 13256071, 26512143, 357913941, 1431655765, 3817748707, 22906492245, 91625968981, 244335917283, 1466015503701, 5212499568715, 10424999137431

Offset: 1

Jeff Heleen, Apr 16 2000

In hailstone sequences, only even powers of 2 are obtained as a final peak before descending to 1. [I assume this should really say: "These are numbers whose 3x+1 trajectory has the property that the final peak before descending to 1 is an even power of 2." - N. J. A. Sloane, Jul 22 2020]

For n>1, this a bisection of A010120. For n=3,6,7,9,12,15,16,18,19,21, we have a(n)=(4^n-1)/3, the largest possible value because one 3x+1 step produces 2^(2n). - T. D. Noe, Feb 19 2010

			The "3x+1" sequence starting at 21 is 21, 64, 32, 16, 8, 4, 2, 1, ..., and is the smallest start which contains 64 = 2^(2*3). So a(3) = 21. - _N. J. A. Sloane_, Jul 22 2020

J. Heleen, Final Peak Sequences for Hailstone Numbers, 1993, preprint. [Apparently unpublished as of June 2017]

Index entries for sequences related to 3x+1 (or Collatz) problem

a054646 1 = 1
a054646 n = a070167 $ a000302 n  -- Reinhard Zumkeller, Jan 02 2013

For n > 1: a(n) = A070167(A000302(n)). - Reinhard Zumkeller, Jan 02 2013

cp's OEIS Frontend

User: Jeff Heleen

A330519 Circulant determinant chain starting at 1007749.

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A257496 Doubly prime home primes (DPHP).

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A072712 Primes p such that p divides the (right) concatenation of all numbers from 1 to p written in base 10 (most significant digit on left).

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A080718 1, together with numbers n that are the product of two primes p and q such that the multiset of the digits of n coincides with the multiset of the digits of p and q.

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A059758 Undulating palindromic primes: numbers that are prime, palindromic in base 10, and the digits alternate: ababab... with a != b.

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Maple

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A054970 Index numbers for palindromic hexagonal numbers.

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Mathematica

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A054969 Palindromic hexagonal numbers.

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Mathematica

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A054910 Palindromic heptagonal numbers.

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Mathematica