A057949 Numbers with more than one factorization into S-primes. See A054520 and A057948 for definition.

441, 693, 1089, 1197, 1449, 1617, 1881, 1953, 2205, 2277, 2541, 2709, 2793, 2961, 3069, 3249, 3381, 3465, 3717, 3933, 3969, 4221, 4257, 4389, 4473, 4557, 4653, 4761, 4977, 5229, 5301, 5313, 5445, 5733, 5841, 5929, 5985, 6237, 6321, 6417, 6489, 6633

Offset: 1

Author

Jud McCranie, Oct 14 2000

Keywords

• nonn

Comments

Numbers with k >= 4 prime factors (with multiplicity) that are congruent to 3 mod 4, no k-1 of which are equal. - Charlie Neder, Nov 03 2018

Examples

			2205 is in S = {1,5,9, ... 4i+1, ...}, 2205 = 5*9*49 = 5*21^2; 5, 9, 21 and 49 are S-primes (A057948).
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A054520, A057948, A057950.

Cf. A343826 (only 1 way), A343827 (exactly 2 ways), A343828 (exactly 3 ways).

Programs

PARI

ok(n)={if(n%4==1, my(f=factor(n)); my(s=[f[i,2] | i<-[1..#f~], f[i,1]%4==3]); vecsum(s)>=4 && vecmax(s)Andrew Howroyd, Nov 25 2018

Sage

def A057949_list(bound) :
    numterms = (bound-1)//4 + 1
    M = [1] * numterms
    for k in range(1, numterms) :
        if M[k] == 1 :
            kpower = k
            while kpower < numterms :
                step = 4*kpower+1
                for j in range(kpower, numterms, step) :
                    M[j] *= 4*k+1
                kpower = 4*kpower*k + kpower + k
    # Now M[k] contains the product of the terms p^e where p is an S-prime
    # and e is maximal such that p^e divides 4*k+1
    return [4*k+1 for k in range(numterms) if M[k] > 4*k+1]
# Eric M. Schmidt, Dec 11 2016

Extensions

Offset corrected by Eric M. Schmidt, Dec 11 2016