A373467 Palindromes with exactly 7 (distinct) prime divisors.

20522502, 21033012, 22444422, 23555532, 24266242, 25777752, 26588562, 35888853, 36399363, 41555514, 41855814, 42066024, 43477434, 43777734, 44888844, 45999954, 47199174, 51066015, 51666615, 52777725, 53588535, 53888835, 55233255, 59911995, 60066006, 60366306, 61777716, 62588526, 62700726

Offset

1,1

Links

Harvey P. Dale, Table of n, a(n) for n = 1..7000

Formula

Intersection of A002113 and A176655.

Example

Obviously all terms must be palindromic; let us consider the prime factorization:
a(1) = 20522502 = 2 * 3^2 * 7 * 11 * 13 * 17 * 67 has exactly 7 distinct prime divisors, although the factor 3 appears twice in the factorization. (Without the second factor 3 the number would not be palindromic.)
a(2) = 21033012 = 2^2 * 3 * 7 * 11 * 13 * 17 * 103 has exactly 7 distinct prime divisors, although the factor 2 appears twice in the factorization. (Without the second factor 2 the number would not be palindromic.)
a(3) = 22444422 = 2 * 3 * 7 * 11 * 13 * 37 * 101 is the product of 7 distinct primes (cf. A123321), hence the first squarefree term of this sequence.

Mathematica

nPal[n_]:=Module[{id=IntegerDigits[n], lid, flid2, revleft1, oklpl, lfpt1, new1, lfpt2, revleft2, new2}, lid=Length[id]; flid2=Floor[lid/2]; revleft1=Reverse[Take[id, flid2]]; oklpl=If[OddQ[lid], flid2+1, flid2]; lfpt1=Take[id, oklpl]; new1=FromDigits[Join[lfpt1, revleft1]]; lfpt2=IntegerDigits[FromDigits[lfpt1]+1]; revleft2=If[EvenQ[lid], Reverse[lfpt2], Reverse[Drop[lfpt2, -1]]]; new2=FromDigits[Join[lfpt2, revleft2]]; Which[Union[id]=={9}, n+2, new1>n, new1, True, new2]]; Select[NestList[nPal[#]&, 20522502, 10000], PrimeNu[#]==7&] (* Harvey P. Dale, Nov 29 2025 *)

Prog

(PARI) A373467_upto(N, start=vecprod(primes(7)), num_fact=7)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && listput(L, start)); L}

Crossrefs

Cf. A046333 (same with bigomega = 7: counting prime factors with multiplicity), A046397 (same but only squarefree terms), A373465 (same with omega = 5), A046396 (same with omega = 6).

Cf. A002113 (palindromes), A176655 (omega(.) = 7), A123321 (products of 7 distinct primes).

Sequence in context: A116497 A346685 A133543 * A294031 A321670 A356071

Adjacent sequences: A373464 A373465 A373466 * A373468 A373469 A373470

Keyword

nonn,base

Author

M. F. Hasler, Jun 06 2024

Status

approved