A354091 Fully multiplicative prime shift where the primes of the form 3k+2 are replaced by the next larger such prime, and primes of the form 3k and 3k+1 stay as they are.

1, 5, 3, 25, 11, 15, 7, 125, 9, 55, 17, 75, 13, 35, 33, 625, 23, 45, 19, 275, 21, 85, 29, 375, 121, 65, 27, 175, 41, 165, 31, 3125, 51, 115, 77, 225, 37, 95, 39, 1375, 47, 105, 43, 425, 99, 145, 53, 1875, 49, 605, 69, 325, 59, 135, 187, 875, 57, 205, 71, 825, 61, 155, 63, 15625, 143, 255, 67, 575, 87, 385, 83, 1125

Offset

1,2

Comments

Permutation of odd numbers. Preserves the prime signature.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16383

Index entries for sequences related to sigma(n)

Formula

Fully multiplicative with a(A003627(n)) = A003627(1+n), a(A007645(n)) = A007645(n).

For all n >= 1, A354092(a(n)) = n.

For all n >= 1, A046523(a(n)) = A046523(n) and A074941(a(n)) = A074941(n).

Example

The primes in A003627 are replaced by the next prime in that sequence, as: 2 -> 5 -> 11 -> 17 -> 23 -> 29 -> 41 -> ..., while other kinds of primes (A002476) stay intact, thus for 60 = 2^2 * 3^1 * 5^1, we have a(60) = 5^2 * 3^1 * 11^1 = 825.

Prog

(PARI) A354091(n) = { my(f=factor(n)); for(k=1, #f~, if(2==(f[k, 1]%3), for(i=1+primepi(f[k, 1]), oo, if(2==(prime(i)%3), f[k, 1]=prime(i); break)))); factorback(f); };

Crossrefs

Cf. A003627, A007645, A074941, A353815.

Cf. A354092 (left inverse), A354093 (inverse Möbius transform), A354094 (Möbius transform), A354095, A354096.

Cf. also A003961, A332818, A348746 for similar constructions.

Sequence in context: A357756 A181351 A266385 * A032532 A111108 A038245

Adjacent sequences: A354088 A354089 A354090 * A354092 A354093 A354094

Keyword

nonn,mult

Author

Antti Karttunen, May 17 2022

Status

approved